Mth 173 Assignments Table

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Module 1: Review and Exploration of Rates of Change The Major Quiz should be taken within a week of completing Module 1
Module 1 Objectives Given a set of data construct a graph, an approximate trendline, a selected set of points and an appropriate mathematical model of the data. Apply the rate of change definition to determine the average rate of change of a quantity on an interval, or given the average rate the change in the quantity on the interval. Apply the difference quotient and limits to get the rate-of-change function for a given quadratic function. Solve problems requiring the application of properties of basic functions. (Basic functions include quadratic, exponential, power, trigonometric, polynomial and rational functions). Graph generalized basic functions using transformations and symmetry. Using one or more trapezoids to approximate a function determine the slopes and areas associated with each, and the interpretation of each of these quantities. Apply proportionality and variation to solve problems. Construct combinations of basic functions, including sums, products, quotients and composites. Test a given basic function on a given interval for invertibility, and if possible find its inverse. Solve algebraic, trigonometric, exponential and logarithmic equations. Construct a numerical solution of a given rate-of-change equation. Find the given limit of a given expression. Determine intervals of continuity for a given function. By the end of Module 1 you will have solidified and perhaps expanded your understanding, obtained in prerequisite courses, of linear, quadratic, power, exponential, logarithmic and trigonometric functions and the analytical tools required to work with these functions explored the idea of the rate of change of one quantity with respect to the other developed the idea of rate of change into the idea of the derivative and integral and applied this insight in the case of linear and quadratic functions. At that time you should fully understand the precalculus required for success in this course, as well as the content of the following summaries. You should scan the summary at least once a week and should find that parts of it become increasingly clear. This should help you to organize your knowledge as you progress through the material. Trapezoidal Approximation Graphs We construct an approximation to the graph of a given function by partitioning its domain into trapezoids and labeling information relevant to the interpretation of the graph. If the function represents a quantity Q(t) which depends on clock time t, then the width Dt of the trapezoid represents the duration of the time interval associated with trapezoid, and the rise associated with the trapezoid represents the change DQ in the quantity Q over that time interval, so that the slope of the trapezoid is slope = rise / run = DQ / Dt, which represents the average rate, with respect to clock time, at which quantity Q is changing for that time interval. If the function represents the rate dQ / dt at which some quantity Q changes with respect to clock time, then the average altitude of a trapezoid represents the approximate average rate rateAve = (dQ / dt)Ave and the width of the trapezoid represents the duration Dt of the time interval. The area of the trapezoid therefore represents the approximate change DQ in the quantity. The run Dt associated with the trapezoid represents the duration of the time interval while the rise D (dQ / dt) represents the change in the rate dQ / dt. The slope associated with the trapezoid is therefore slope = rise / run = D (dQ / dt) / Dt and represents the average rate at which the rate dQ / dt changes for the time interval. The Idea of the Derivative We know that to find the average rate of change of a quantity over a time interval we can divide the change in the quantity by the duration of the time interval. When the quantity is given by a function y(t), the change in the value of the function between clock times t and t + Dt is y(t + Dt) - y(t) and the duration of the time interval is Dt, so that the average rate of change between these clock times is [ y(t + Dt) - y(t) ] / Dt. The limiting value of this average rate of change, as Dt -> 0, is called the derivative of the function y with respect to t, denoted dy / dt or y ' ( t ) . If y(t) is a linear function y(t) = m t + b of clock time, then dy / dt = lim {t -> 0} [ y(t + Dt) - y(t) ] / Dt is easily simplified to give us dy / dt = m. That is, the derivative of a linear function is constant, and is equal to the slope of the graph of that function. If y(t) is a quadratic function y(t) = a t^2 + b t + c, then dy / dt = lim {t -> 0} [ y(t + Dt) - y(t) ] / Dt can be simplified to give dy / dt = 2 a t + b. Thus the derivative of a quadratic function is a linear function. Since a linear function changes at a constant rate, and a linear function is the rate at which a quadratic function changes, a quadratic function has the distinction that its rate of change changes at a constant rate. The value dy / dt of the derivative of this quadratic function at a given clock time is equal to the slope of the parabolic graph of the function at that clock time. Formulas for the derivatives of power functions, exponential and logarithmic functions, and trigonometric functions, as well as rules for calculating derivatives of combined functions, will be developed during the first semester of this course. If we interpret y(t) as the depth of water in a uniform cylinder as water flows from a hole at the bottom of the cylinder, then as we have observed through a modeling exercise y(t) is very closely modeled by a quadratic function of clock time. The rate at which water depth changes, as calculated from a table of observed y vs. t values, appears to be a linear function of clock time. The rate at which water depth changes is approximated by the derivative dy / dt of the quadratic model y(t). As we have seen this derivative is a linear function of clock time, which agrees with our observation that the rate is linear in time. For more detailed information on the quadratic model of the flow, you may if you wish consult the Precalculus I homepage. If you are interested in the physics of the situation you may consult the Physics I homepage. The Idea of the Integral We know that to find the change in a quantity over a time interval we can multiply the rate of change in the quantity by the duration of the time interval. Given that the rate of change of some quantity is given by a function r(t), then the approximate average value of this rate of change between clock times t1 and t2 is [ r(t1) + r(t2) ] / 2. It follows that the change in the quantity is approximately equal to the product [ r(t1) + r(t2) ] / 2 * (t2 - t1) of the approximate average rate of change and the duration of the time interval. This process is equivalent to finding the area of the trapezoid beneath the line segment connecting the points (t1, r(t1)) and (t2, r(t2)). The average altitude of this trapezoid represents the approximate average rate at which the function changes and the width of the trapezoid represents the duration of the time interval. If we directly observe the average rate of change of a quantity at a series of clock times, we can find the approximate change in the quantity between each successive pair of clock times. We can then add the approximate changes to get the total approximate change. The smaller the time interval Dt = t2 - t1, then the closer the graph of the trapezoid will tend to be to the graph of the actual function, and the closer the approximate change thus calculated will be to the actual change. For any reasonably well-behaved rate function, the accuracy of the approximation is approximately proportional to the square of Dt, which means that the approximation improves very rapidly as Dt decreases. The process of finding the change in a quantity from its rate function is called integration. The specific process outlined above is a form of approximate integration. Using approximate integration we can find for a given rate function r(t) the approximate change in a quantity from some clock time t = a to another clock time t = b. The precise change in the quantity would be called the definite integral of the rate function r(t) between clock times t = a and t = b.
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00	Early_Content_Assignments Design of the Course
Asst 01	qa_01			Modeling Project #1 simulated flow data			Class Notes #01-03		query_01
Class Notes Summarized: #01: Quadratic Model of Depth vs. Time for water flowing from a uniform cylinder. Analyzing Depth vs. Clock Time Data Given a set of depth vs. clock time data we can calculate average rates of depth change between every pair of successive clock times. If we hypothesize a quadratic model we can choose three points on our approximate 'best-fit' curve to use to create a model function y = a t^2 + b t + c. Substituting the y and t coordinates of our three points we obtain linear three equations in a, b and c, which can be solved simultaneously for these parameters. This will give us our model. Solving the Equations We solve the simultaneous equations by the process of elimination. Using the Model We can use the model to predict depth at a given clock time or to find clock time at which a given depth occurs. #02: Rates of change for the depth vs. time model. Average Rates of Change for the Depth vs. Clock Time Model We can find the average rate of depth change between any two clock times, given a depth vs. clock time function. We evaluate the function at the two clock times to determine the depths corresponding to these clock times, then we calculate the change in depth and the difference in the clock times `dy / `dt. We use these differences to calculate the average rate. Precise Rate of Depth Change for the Model For a specific quadratic function we can symbolically calculate the average rate between clock times t and t + `dt; imagining that `dt approaches zero we obtain the actual rate-above-change function dy / dt, or y'(t). #03: The Rate of Change of a Quadratic Depth Function: Differentiation and Integration Average Rates of Change for Depth Functions We can find the average rate of depth change between any two clock times, given a depth vs. clock time function. We evaluate the function at the two clock times to determine the depths corresponding to these clock times, then we calculate the change in depth and the difference in the clock times `dy / `dt. We use these differences to calculate the average rate. For a specific quadratic function we can symbolically calculate the average rate between clock times t and t + `dt; imagining that `dt approaches zero we obtain the actual rate-above-change function dy / dt, or y'(t). Generalizing to y = a t^2 + b t + c We can generalize the above process to the general quadratic function y = a t^2 + b t + c, obtaining the general rate-of-change function y'(t) = dy / dt = ` a t + b. What the Rate Function tells us about the Depth Function From the rate function y'(t) of an unknown quadratic function we can determine the constants a and b for the function y(t) = a t^2 + b t + c. Using this knowledge we easily find the difference in the depths between two given clock times. The only thing we cannot find from the rate function is the constant c, which we need to determine the actual depth at a given time. However, without c we can still find the depth change between two given clock times. Video Links Click on the specific video links for video explanations of these topics.
Objectives: Relate an ordered sequence of points in the y vs. t plane to slopes, average rates of change and other graph characteristics, and interpret. Be able to solve linear and quadratic equations and inequalities, and while you’re at it be able to deal with piecewise definitions. These are standard prerequisite procedures and ideas. However quadratic inequalities are often challenging at this level. 1. Relate an ordered sequence of points of the y vs. t plane, the corresponding partition of an interval of the t axis, the slopes of the line segments between the points, the slope corresponding to a subinterval of the partition, the average rate of change of y with respect to t on each subinterval of the partition, the change in t and the change in y on each subinterval of the partition, and the interpretations when t is clock time and y is depth or price. 01.01: Relate{(t_i,y_i) \| 0 <= i <= n} U {trendline} U {a = t_0, b = t_n, slope_i, aveRate_i, rise_i, run_i, `dt_i, `dy_i} U {a = t_0, b = t_n, a = t_0 < t_1 < … < t_n = b, partition of the interval [a, b] of the t axis} where: · (t_i, y_i) is a point in the y vs. t plane, t_i < t_(i+1) · slope_i is the slope of the line segment from (t_(i-1), y_(i-1) ) to (t_i, y_i) · aveRate is the average rate of change of y with respect to t corresponding to the t subinterval [t_(i-1), t_i ) Interpret for y = depth of water in a container, t = clock time. Interpret for y = price of a stock, t = clock time. Motivation: Partitions are fundamental, graphical representation is important, rate is the most fundamental quantity in calculus, which is useless without the ability to interpret. Feasibility: Partitions are easy to understand. Rates, rise, run and slope are familiar prerequisite concepts. Limited vernacular example: 2: Relate the following: a set of more than three data points in a coordinate plane: hand-sketched graph and a smooth curve representing the data, three selected representative points on the curve algebraically-determined quadratic function fitting the three selected points deviations of data points from curve, and residuals observed patterns in the residuals evaluated the quality of the model predicted value of y given the value of t based on model value(s) of t given the value of y based on model the vertex of the parabolic graph of the function graph of model constructed using transformations, starting with the y = x^2 function transformed graph expressed in the notation y = A f(x - h) + k, where f(x) = x^2 interpretation for y = water depth vs. t = clock time for water flowing from a hole in the side of a uniform cylinder interpretation for y = stock price vs. t = clock time Technical definition: Relate {data points (t_i,y_i) \| 0 <= i <= n} U {hand-sketched y vs. t graph of points, hand-sketched smooth trendline, selection of three points on trendline, three simultaneous equations for parameters of quadratic function through three selected points} U {solution of equations, quadratic model, t value(s) corresponding to given y value, y value(s) corresponding to given t value} U {deviation of model from each (t_i, y_i), average deviation of model from data, trend of deviations} U {vertex of quadratic model, construction of graph of quadratic function from basic points, construction of graph of quadratic function by slope characteristics} U {depth vs. clock time interpretation, stock price vs. clock time interpretation}
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Asst 02	qa_02			View Calculus I clips on disk 1, listed in the HTML file as Disk 1 (Gen 1). think about questions posed in documentation PHeT Exercise 1
Class Notes Summarized: #04: The Concepts of Differentiation and Integration in the Context of Rate Functions Depth and Rate-of-Depth-Change Functions The quadratic depth function y = a t^2 + b t + c implies a linear rate-of -depth-change function y ' = 2 a t + b. A linear rate-of-depth-change function y ' = m t + d implies a quadratic depth function y = 1/2 m t^2 + d t + c, where c is an arbitrary constant number while m and d are known if y ' is known. Thus the rate-of-depth-change function allows us to determine the change in depth between any two clock times; however to find the absolute depth at a clock time we must evaluate arbitrary constant c, which we can do if we know the depth at a given clock time. The process of obtaining a rate function from a quantity function is called differentiation, and the rate function is called the derivative of the quantity function. The process of obtain the change-of-quantity function is called integration, and the quantity function is called the antiderivative or integral of the rate function. Solution of Homework Problem from Modeling Project #2: Number of Decays obtained from Rate of Decay Function From the function giving the rate at which a radioactive substance decays we estimate the number of decays over a substantial time interval. The process is depicted using a trapezoidal approximation graph.
Objectives: Continue to master the objectives of Assignment 1.
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Asst 03	qa_03		Week 2 Quiz #1	Brief Synopsis: The_Idea_of_the_Derivative			Class Notes #04		query_03
Objectives: 1. Where the sequence of average slopes of a y vs. t graph, over a series of intervals, has an identifiable pattern, identify and continue the pattern and use to project new graph points. 2. Relate the following: a quadratic function y(t) = a t^2 + b t + c on an interval the value of the difference quotient (y(t+`dt) – y(t)) / ((t + `dt) – t) for the interval the limit as `dt -> 0 of (y(t+`dt) – y(t)) / ((t + `dt) – t) for arbitrary t the function y ‘ = m t + b equal to the limit of the preceding the average rate of change of y with respect to t on interval average value of y ‘ (t) on interval change in y on interval derivative of y(t) derivative of y ‘ (t) antiderivative of y(t) antiderivative of y ‘ (t) definite integral of y ‘ (t) on interval Technically: Relate: { y(t) = a t^2 + b t + c, interval t_0 <= t <= t_f } U { (y(t+`dt) – y(t)) / ((t + `dt) – t), limit as `dt -> 0 of (y(t+`dt) – y(t)) / ((t + `dt) – t), y ‘ = m t + b } U { average rate of change of y with respect to t on interval, average value of y ‘ (t) on interval, change in y on interval } U {derivative of y(t), derivative of y ‘ (t), antiderivative of y(t), antiderivative of y ‘ (t), definite integral of y ‘ (t) on interval } Of the last four listed subsets, all the elements of any one can be related to the elements of the untion of the other three with the first listed subset. Be able to do so.
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Asst 04	qa_04		Week 2 Quiz #2	Modeling Project #2 (Not Currently Assigned): Collaborative Investigation 1					query_04
Objectives: 1. Relate the following: initial principle interest rate growth rate number of annual compounding (including infinite) duration of investment doubling time principle function exponential function y = A b^t form of exponential function definition of e 2. Relate the following: initial quantity growth rate growth factor exponential function (t_1, y_1) (t_2, y_2) y = A b^t form of exponential function y = A * 2^(k t) form of exponential function y = A * e^(k t) form of exponential function value of y for given t value of t for given y vertically shifted exponential functions horizontal asymptote doubling time halflife construction of graph from basic points slope characteristics of graph construction of graph from point and halflife or doubling time
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Asst 05	qa_05	Ch 1.1		Brief Synopsis: Trapezoidal_Approximation_Graphs PHeT Exercise 2	text_05		Class Notes #05		query_05
Class Notes Summarized: #05: Growth of an Exponential Function; Trapezoidal Representation of Approximate Derivatives and Integrals Growth Rate, Growth Factor and the Quantity Function; Doubling Time An exponential function is characterized by a growth rate r, a growth factor (1+r) and a quantity function Q(t) = Q0 (1+r)^t. Alternative forms of the exponential function include Q(t) = Q0 b^t and Q(t) = Q0 e^(kt). Any function of this form has a doubling time tD such that for any t, Q(t+tD) = 2 Q(t), which we demonstrate algebraically and depict graphically. Representation by and Interpretation of Trapezoids Given a velocity vs. clock time function we can construct a trapezoid between any two graph points, with vertical altitudes running from the horizontal axis to the respective graph points. These altitudes represent the initial and final velocities over the corresponding time interval. The area of this trapezoid represents the product of the average of the initial and final velocities and the duration of the time interval, and therefore the distance that an object would move during the time interval at this average velocity. If the velocity function is not linear during time interval, it is very unlikely that the actual average velocity will equal the average of the initial and final velocities, and the distance so calculated will be an approximation rather than a precise value. The slope of the line segment between the graph points will represent the average rate at which the velocity changes (change in velocity divided by change in clock time). In general if the graph represents the rate at which some quantity changes vs. clock time, a trapezoid can be constructed to approximate the change in the quantity between to given clock times, with the change in the quantity represented by the area of the trapezoid. More accurate approximations can be obtained by subdividing the trapezoid into a series of 'thinner' trapezoids, on which the line segments between graph points more nearly approximate the actual function. The area under the graph between two clock time therefore represents the integral of the rate function between the two clock times. This integral represents the change in the quantity between these two clock time. If the graph represents some quantity vs. clock time, then a similar trapezoid or series of trapezoids will have line segments between graph points which represent average slope between graph points, and which therefore represent average rates of change between the corresponding clock times. These average rates of change represent the approximate derivatives of the function depicted by the graph. If the clock times on a series of trapezoids are uniformly spaced, then if the slopes represent rates of change, then at any graph point the change in slope at that point divided by the uniform time interval between graph points will represent the approximate rate at which the slope changes at the graph point. Since the slope represents the rate at which the function changes, this rate of slope change will represent the rate at which the rate changes. This quantity is and approximate second derivative of the function. By interpreting the altitude and width of a trapezoid, we can interpret what the product of average altitude and width represents, and we can interpret what is represented by the change in altitude divided by the width.
Objectives: 1. As in Objective 1 of Assignment 1: Relate an ordered sequence of points of the y vs. t plane, the corresponding partition of an interval of the t axis, the slopes of the line segments between the points, the slope corresponding to a subinterval of the partition, the average rate of change of y with respect to t on each subinterval of the partition, the change in t and the change in y on each subinterval of the partition, and the interpretations when t is clock time and y is depth or price. In addition divide the region beneath the graph into trapezoids, one trapezoid for each interval of the partition, and Relate the following: Interpretation of average ‘graph altitude’ of a trapezoid. Interpretation of area of each trapezoid. Interpretation of accumulated areas. Use of accumulated areas to find approximate area between two t values average ‘graph altitude’ of each trapezoid trapezoid areas labeling of trapezoidal graph table of labels Technically: Relate: {(t_i,y_i) \| 0 <= i <= n} U {slope_i, aveRate_i, rise_i, run_i, `dt_i, `dy_i \| 1 <= i <= n} U {area_i, aveAlt_i, accum_area_i \| 1 <= i <= n} U {t_i, t_j, area beneath graph from t_i to t_j \| 1 <= i <= n, 1 <= j <= n}
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Asst 06	qa_06	Ch 1.2	Week 3 Quiz #1 Week 3 Quiz #2	Modeling Project #3	text_06	Outline of Introductory Topics through Major Quiz	Class Notes #06		query_06
Class Notes Summarized: #06: Project #3; Derivative of y = a x^3; The Differential Depth Function and Rate of Depth Change Function We work a review problem involving a quadratic depth function. Analyzing volume vs. diameter data for sandpiles Testing the proportionality y = a x^3 for sandpile volume y vs. diameter x, we obtain questionable results. Looking at the proportionality for different sandpiles, and comparing with a DERIVE best fit for y vs. x data, we conclude that for the data considered y = .002 x^3 is a good, if not perfect, model. The derivative of y = a x^3 Using the definition of the derivative we do a little algebra and determine that the derivative of y = a x^3 is y ' = 3 a x^2. Applying the derivative: the differential For a given diameter x we easily determine the rate y' = dy / dx= 3 a x^2 at which the volume y of a sandpile is changing, with respect to changes in x, at that diameter. Using this rate we can estimate the volume change for a given small change `dx in x. The volume change will simply be the product of the rate y' and change `dx in x: `dy = y' * `dx, or `dy = dy / dx * `dx. The Differential The essence of the concept of the differential is that the change `dy in y corresponding to a change `dx in x is `dy = y'(x) `dx.* Linked Outline of Introductory Topics through 9/04/98: Study these for the upcoming 9/13/98 major quiz
Objectives: Identify quantities which are proportional to various powers of the linear dimensions of a three-dimensional geometric object, specifically to the first, second and third powers, as well as to the -1 and -2 powers. Given simultaneous values of y and x, and the proportionality y = k x^n, determine the value of k, use this value to model y vs. x as a power function, construct the graph of the function, find values of y given values of x, find values of x given values of y. Given the nature of the proportionality between x and y, determine the ratio y_2 / y_1 of two y values as the appropriate power of the ratio x_2 / x_1 of the corresponding x values. 1. Apply midpoint and distance formulas and relate to the Pythagorean Theorem and similarity of triangles. Technically: Relate {(x_1, y_1), (x_2, y_2), (x_mid, y_mid), d( (x_1, y_1), (x_2, y_2) ), Pythagorean Theorem}, where (x_mid, y_mid) is midpoint between (x_1, y_1) and (x_2, y_2) d( (x_1, y_1), (x_2, y_2) ) is distance between points 2. Relate the following: the function r(t) such that r(t) is rate of change of y(t) with respect to t (i.e., r = y ‘) the value of y when t = t_0, where t_0 can be symbolic or numerical an increment `dt, symbolic or numerical a uniform partition of the interval [a, b] of the t axis: a = t_0, b = t_n, a = t_0 < t_1 < … < t_n = b, where for each <= i <= n we have t_i – t_(i-1) = `dt the approximate change in y for each interval based on the value of r at the beginning of the interval, and on `dt the approximate total change in depth for interval a <= t <= b, in the application where y is depth function and r is rate-of-depth-change function Technically: Relate {function r(t) \| r(t) is rate of change of y(t) with respect to t (i.e., r = y ‘)} U {value of y when t = t_0, increment `dt } U { uniform partition of the interval [a, b] of the t axis: a = t_0, b = t_n, a = t_0 < t_1 < … < t_n = b \| t_i – t_(i-1) = `dt, 1 <= i <= n } U {approximate change in y for ith interval based on r(t_(i-1)) and `dt \| 1 <= i <= n } U { approximate total change in depth for interval a <= t <= b } U { application when y is depth function and r is rate-of-depth-change function } 3. Relate for some linear dimension x of a set of geometrically similar objects and a quantity y proportional or inversely proportional to x: the linear dimensions x_1 and x_2 of two objects and the value y_1 for that object the value y_2 corresponding to the second object the ratio of the linear dimensions the ratio of y values the ratio of x values the equation governing the proportionality the value of the proportionality constant a graph of y vs. x 4. Relate for some linear dimension x of a set of geometrically similar objects in at least two dimensions, and a quantity y proportional or inversely proportional to the area of an object: the linear dimensions x_1 and x_2 of two objects and the value y_1 for that object the value y_2 corresponding to the second object the ratio of the linear dimensions the ratio of y values the ratio of x values the equation governing the proportionality the value of the proportionality constant a graph of y vs. x 5. Relate for some linear dimension x of a set of geometrically similar objects in three dimensions, and a quantity y proportional or inversely proportional to the volume of an object: the linear dimensions x_1 and x_2 of two objects and the value y_1 for that object the value y_2 corresponding to the second object the ratio of the linear dimensions the ratio of y values the ratio of x values the equation governing the proportionality the value of the proportionality constant a graph of y vs. x 6. Relate for some power p: the proportionality y = x^p x values x_1 and x_2 value y_1 corresponding to x_1 the value y_2 corresponding to the second object the ratio of the x values the ratio of the y values the value of the proportionality constant a graph of y vs. x 6. Relate for powers p and p ': the proportionality y = k x^p x values x_1 and x_2 value y_1 corresponding to x_1 the value y_2 corresponding to the second object the ratio of the x values the ratio of the y values the value of the proportionality constant a graph of y vs. x the proportionality z = k ' y^p y values y_1 and y_2 value z_1 corresponding to y_1 the value z_2 corresponding to the second object the ratio of the y values the ratio of the z values the value of the proportionality constant a graph of z vs. y a graph of z vs. x the proportionality equation relating z and x the proportionality constant for the equation relating z and x 7. Construct the graph of the y = k x^p power function using the basic points corresponding to x = -1, 0, 1/2, 1 and 2, and using transformations construct the graph of y = A ( x - h) ^ p + c. Technically: Relate {p, y = x^p} U {basic points (x, y) \| x = -1, 0, 1/2, 1, 2} U {graph of points} U { y = f(x) = A (x - h)^p + c} U values of A, h, c} U {graph of y = x^p} U {constructed graph of basic points transformed by f(x)}
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Asst 07	qa_07	Ch 1.3	Week 3 Quiz #3 Week 4 Quiz #1	Inverse Functions and Logarithms	text_07		Class Notes #07, 08		query_07
Class Notes Summarized: #07: The Differential; Tangent Line Approximation to Differential Sandpile Interpretation of the Differential Using the idea of the differential we approximate the volume of a sandpile with volume function y = .0031 x^3 at diameter x = 30.1, given its volume at x = 30. We interpret this result in terms of the tangent line to the graph of the y vs. x function at the x = 30 point. The Tangent Line We use the same function as before to obtain an equation for the tangent line at the x = 30 point. We use this equation to approximate the original function in the vicinity of the x = 30 point. Testing a Proportionality Testing a table of velocity of a falling object vs. distance fallen to see if velocity is in fact proportional to distance fallen, we see that the proportionality v = k x gives different values of k for different (x, v) points, so that the proportionality fails. It turns out that the proportionality v = k `sqrt(x) works, as can be checked in the same way. #08: Text problems; First Introduction to Differential Equations Text Problems Given a table of y vs. x data in which the x values are evenly spaced, in order to determine whether the set is exponential or not we need only look at the ratios of successive y values. If the ratio is constant, then the data indicates an exponential function. We see why this is so by looking at the form y = A b^x of an exponential function. Given a proportionality y = k x^2 and values of y and x, we determine k. From the resulting y = k x^2 relationship we can determine y for any given x or x from any given y. Introductory Example of a Differential Equation When an object cools in a constant-temperature room, the rate at which its temperature changes is proportional to the difference T -Troom between its temperature T and that of the room: rate = k (T - Troom). From a given rate and a given temperature we can evaluate k. The rate of temperature change is denoted dT / dt, so we have the proportionality dT / dt = k (T -Troom). This sort of equation, in which a derivative is treated as a variable, is called a differential equation.
Objectives: 1. Given two graphs find their coordinate-axis intercepts and their intersections points. 2. Relate depth function rate function derivative of depth function antiderivative of rate function derivative of rate function family of antiderivatives uniqueness of derivative non-uniqueness of antiderivative 3. Relate linear or quadratic function f(x) F(x), an antiderivative of f(x) points x_1 and x_2 the change in F(x) corrresponding to the interval x_1 <= x <= x_2 the average value of f(x) the function F ' (x) Alternatively Relate {F(x) \| F(x) is antiderivative of linear or quadratic f(x)} U {x_1, x_2, change in F(x), ave value of f(x), F ‘ (x) 4. Apply shifting and stretching transformations to functions given analytically, graphically or numerically; specifically be able to determine from given information for a function f(x) the same information for the function A f(x - h) + k. 5. Test functions for symmetry about the x or y axis. 6. Given the graphs of two functions construct the graph of their sum, product, quotient or specified composite. 7. Given a function determine whether it has an inverse and if so find the inverse. 8. Given the graph of a function determine whether it has an inverse and if so find the inverse. 9. Relate y = f(x) domain range difference quotient invertibility of f(x) graph of f(x) table of f(x) construction of table of inverse function construction of graph of inverse function 10. Relate functions f(x), g(x) with compatible domains and ranges graphs of f and g linear combination of f and g construction of graph of linear combination domain and range of linear combination product function construction of graph of product function domain and range of product function quotient function construction of graph of quotient function domain and range of quotient function composite function construction of graph of composite function domain and range of composite function effect of `dx on combined functions
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Asst 08	qa_08	Ch 1.4	Week 3 Quiz #3 Week 4 Quiz #2 Week 4 Quiz #3		text_08		Class Notes #09		query_08
Class Notes Summarized: #09: Text problems; Introduction to Natural Logarithms and Composite Functions; Numerical Solution of Differential Equations Quiz Questions A proportionality statement involving the rate of change of a quantity y, the quantity y itself and the independent variable x can be interpreted as a differential equation. We look at some examples of such situations. Questions on text problems To find the value of an inverse function g^-1(x) at a given value of x we can look at the graph of g(x). Locating the specified x on the y axis, we project over to the graph of g(x) and the up or down to the x axis; the value we obtain on this axis is the value of g^-1(x). Alternatively we can evaluate g(x) at different values of the independent variable until our result is sufficiently close to the specified x. We construct the graph of the natural log function y = ln(x) by first constructing the graph of the exponential function y = e^x, then reflecting through the y = x line to get the graph of the inverse function, which is y = ln(x). The forms y = A b^t and y = A e^(kt) are equivalent, with b = e^k or equivalently with k = ln(b). For exponential growth k is positive and b is greater than 1; for exponential decay k is negative and b is less than 1. Composite Functions Most of the functions we use in calculus and in modeling the real world are composite functions of the form f(g(x)), with f and g usually being the simple power functions, exponential functions, logarithmic functions, polynomial functions, etc.. The ability to decompose a given composite function f(g(x)) into its constituent functions f and g is essential in later applications. It is to learn to do this now so that the skill is available when it is needed later. Don't wait to develop the skill until you need to apply to learn something else. Numerical Solution of a Differential Equation Given a differential equation of the form dT / dt = k (T - Troom), and given a value of T at a clock time t, we can determine the approximate value of T at clock time t + `dt by using the fact that `dT = dT / dt * `dt. If `dt is small enough that dT / dt doesn't change by much between t and t + `dt, the approximation will be a good one. The process can be continued for successive intervals to determine approximations that t + 2 `dt, t + 3 `dt, etc.. The accuracy of the approximation decreases more and more rapidly with succeeding intervals.
Objectives: 1. Relate r(t) given either algebraically or graphically, r ' (t) regarded as rate of change function y ' (t) for depth function y(t) time interval t_0 <= t <= t_f value y(t_0) increment `dt partition t_0 < t_0 + `dt < t_0 + 2 `dt < ... < t_0 + n `dt = t_f number n of increments required to partition time interval [ t_0, t_f ] approximate values of y at partition points behavior of graph of y(t) { r(t) \| r(t) = y ‘ (t), y depth function} U {t_0, y(t_0)} U {`dt, n, t_f} U {y_approx(t_0 + i * `dt), 1 <= i <= n} U {behavior of graph of y(t)} Approximation based on rate at initial point of each interval 2. Solve exponential and logarithmic equations. 3. Analyze exponential and logarithmic functions graphically and analytically. 4. Use exponential and logarithmic functions to model real-world phenomena.
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Asst 09	qa_09	Ch 1.5			text_09		Class Notes #10		query_09
Objectives: 1. For the slope function y ' = f(x, y), given point (x_0, y_0) and increment `dx, estimate the coordinates of the point (x_1, y_1), where x_1 = x_0 + `dx. 2. For a quadratic function y = f(x) determine the equation of the tangent line at a given point and quantify the accuracy of the tangent line as an approximation of the original function in a given neighborhood of the point. 3. Relate for a quantity y modeled by a sinusoidal function of the form A sin(k (t - t_0)) + c value of A value of k value of t_0 value of c phase shift period amplitude horizontal shift vertical shift vertical stretch horizontal compression angular frequency real-world behaviors of oscillatory systems 4. Solve trigonometric equations. 5. Use a predictor-corrector approximation to approximate values of y based on and initial point and values of y '. Technically: Relate based for each interval `dt on rate at initial point averaged with rate at predicted final point r(t) = y ‘ (t), where y(t) is a depth function t_0, y(t_0) `dt, n, t_f {y_approx(t_0 + i * `dt), 1 <= i <= n} behavior of graph of y(t)
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Asst 10	qa_10	Ch 1.6		Brief Synopsis: The_Idea_of_the_Integral	text_10	Focus Questions	Class Notes #11		query_10
Complete the Major Quiz as part of Assignment 10
Class Notes Summarized: #11: Trigonometric Functions; Brief Intro. to Polynomials · modeling by trigonometric functions · polynomials
Objectives: 1. Relate income stream f(t) growth rate r (constant) interval 0 <= t <= t_f increment `dt, partition arbitrary subinterval sample t value during subinterval income during subinterval time span from subinterval to t_f value obtained by income during subinterval rate of change of final value during subinterval rate of change of final value as function of t antiderivative of rate function definite integral 2. Relate factored form of a polynomial zeros of polynomial x intercepts of graph of polynomial y intercept of graph of polynomial degree of polynomial behavior of graph of polynomial for large \| x \| 3. Relate for rational function f(x) = p(x) / q(x) zeros of f(x) zeros of p(x) zeros of q(x) degree of p(x) degree of q(x) graph of q(x) graph of p(x) factored form of q(x) factored form of p(x) existence of horizontal asymptotes existence of vertical asymptotes location and nature of vertical asymptotes behavior for large \| x \|
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Asst 11	qa_11	Ch 1.7, 1.8		PHeT Exercise 3	text_11		Class Notes #12		query_11
Class Notes Summarized: #12: Rational Functions, Continuity · graphing rational functions · surface area of constant-volume cylinder as radius approaches zero · continuity of 1 / (x – 3), 1 / sin(x), x / sin(x)
Objectives: 1. Relate y = f(t) given either graphically or algebraically c limit[t -> c, +] f(t) limit[t -> c, -] f(t) limit[t -> c] f(t) existence of limit[t -> c] f(t) graphical representation of limit at c numerical approximation of limit at c algebraic determination of limit at c 2. Relate the power function y = x^p the exponential function y = e^x the natural logarithm function y = ln(x) the sine and cosine functions y = sin(x) and y = cos(x) formulas for derivatives of function derivative of constant multiple of function derivative of linear combination of two or more functions derivative of product of two functions derivative of quotient of two functions 3. Relate using definition of continuity f(t) a linear, quadratic, polynomial, exponential, power, sine, cosine, tangent, or a function built from these by a combination of sums, products, quotients, or composites points of discontinuity intervals of discontinuity intervals of continuity continuity and discontinuity the real world applications 4. Relate function f(t) increment `dt difference quotient at t_0 for increment `dt limiting value of difference quotient at t_0 as `dt -> 0 (Restricted to cases where the difference quotient can be simplified algebraically) difference quotient at t derivative of f(t) 5. State, evaluate and demonstrate examples showing the relationship between differentiability and continuity
Module 2: The Derivative and the Integral Test 1 should be taken within a week of completing Module 2 Assignments 12 - 17
Module-level Objectives Apply the difference quotient and limits to obtain formulas for derivatives of power functions. Apply the rules of differentiation to calculate derivatives of various combinations of basic functions. Construct graphs relating given functions to their first and second derivatives and antiderivatives. Interpret the behavior of a given graph, given the meanings of the dependent and independent variables. Using graphical, numerical, verbal and symbolic representations, explain the Fundamental Theorem of Calculus Construct integrals of sums and differences of functions using Riemann Sums. Interpretation the indefinite and definite integrals of a function, given the meanings of the dependent and independent variables. Connection between Integral and Derivative If a function y(t) represents some quantity vs. clock time, then its derivative dy / dt represents the time rate of change of that quantity. We can therefore write r(t) = dy / dt. We can find the change in y(t) between two given clock times either by evaluating y(t) at these clock times, or by integrating r(t) between these clock times. If we integrate this derivative function r(t) between clock times t = a and t = b, we will obtain the change in the quantity between these clock times. We can also find the change in the quantity by evaluating y(t) at t = a and at t = b, obtaining quantities y(a) and y(b). The change in the quantity is thus y(b) - y(a). We thus see that the definite integral of r(t) = dy / dt between t = a and t = b is simply equal to y(b) - y(a). A more familiar way of stating this is to say that is the following: If f(t) and F(t) are functions such that f(t) = dF(t) / dt (i.e., f(t) is the derivative of F(t)), then the definite integral of f(t) between t = a and t = b is simply equal to F(b) - F(a). The above statement is the First Fundamental Theorem of Calculus. This Theorem tells us that to integrate a function f(t), which for now we can think of as a rate function, we need only find some function F(t) whose derivative is equal to f(t). In the context of rates, F(t) will be a quantity function whose rate function is f(t). The function F(t) is called an antiderivative of f(t). For a linear rate of depth change function r(t) = m t + b, a corresponding quantity function is the quadratic function y(t) = 1/2 m t^2 + b t (it is easy to verify that the derivative of y(t) is r(t), so y(t) is an antiderivative of r(t)). Other quantity functions are possible. For example the derivative of 1/ 2 m t^2 + b t + 12 is also equal to m t + b. In fact any function of the form 1/2 m t^2 + b t + c, where c can be any constant number, is an antiderivative of r(t) = m t + b. It doesn't matter which antiderivative we use to determine the change in depth between to clock times, since the constant c will cancel when we calculate the change y(b) - y(a). This means that when we integrate a rate function, we do not find the quantity function, we rather find a quantity function. Any quantity function we find for a given rate function will differ from any other quantity function by at most a constant. More generally, we can say that if F(t) is an antiderivative of a function f(t), then so is F(t) + c, where c is any constant number. We can furthermore say that the set of all such functions F(t) + c, where c can be any constant number, includes all possible antiderivatives of f(t). The graphs of a function f(t) and an antiderivative function F(t) are related. The 'heights' of the f(t) graph are the slopes of the F(t) graph. To obtain the f(t) graph from the given F(t) graph we plot the slopes of the given graph as the heights of our f(t) graph. To obtain an F(t) graph from the given f(t) graph we start at an arbitrary point and begin constructing slopes which are equal to the heights of the f(t) graph. Since the starting point for constructing the F(t) graph is arbitrary, we see that there are infinitely many possible F(t) graphs. However, since the slopes of these graphs are all determined by f(t), the F(t) graphs will all be congruent in that one graph will differ from the others only by its vertical location.
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Asst 12	qa_12	Ch 2.1, 2.2			text_12		Class Notes #13, 14		query_12
Class Notes Summarized: #13: The Derivative (Text 2.1, 2.2) · quiz: algebraically find derivatives of x^2 and 1 / x^2 · expansion of a binomial and derivatives of power functions · sequencing set of slope and average slopes for a given curve · how much do we have to squeeze x values to confine y values in specified manner · finding the equation of a tangent line #14: The Derivative Function · constructing graph of f ‘ from graph of f, interpretation for depth model · graphing a f given characteristics of its derivative f ‘ · interpreting slopes, interpreting the derivative function
Objectives: 1. Apply the sum, power, quotient and chain rules to sums, products, quotients composites of linear, quadratic, polynomial, exponential, power, sine, or cosine functions to find the derivatives of those functions. More generally, use these skills to Relate {f(t), g(t) where each is a linear, quadratic, polynomial, exponential, power, sine, or cosine function derivative of f(t) derivative of g(t) derivative of linear combination of f(t) and g(t) derivative of (f * g) (t) derivative of (f / g) (t) derivative of f(g(t)) 2. Evaluate, graph and solve problems involving the greatest-integer function. 3. Solve problems involving compound interest. 4. Given a function y = f(t) and a specific value t = t_0, find f ' (t), f ' (t_0), the slope of the graph of y vs. t at the point corresponding to t_0, the equation of the tangent line at the point corresponding to t_0, and the deviation of the graph from the tangent line at a given t. Apply these skills to Relate y = f(t) t_0 f ‘ (t) f ‘ (t_0) slope of graph at t_0 tangent line at t_0 deviation of tangent line from graph for increasing \| t - t_0 \| 5. Given position as a function of clock time, apply the definition of average rate of change to obtain an expression for average velocity and by taking appropriate limits obtain an expression for instantaneous velocity. 6. For a given function y = f(x) or its graph near x = a, value x = a and interval width h, write the expression for the average rate of change of f with respect to x over the interval from x = a to x = a + h, take the limit to find the instantaneous rate of change of f at x = a, write the expression for the difference quotient at x = a, determine the existence or nonexistence of the limit of the difference quotient at x = a, when the limit exists calculate the limiting value of difference quotient at x = a, test the differentiability of f(x) at x = a, find the slope of tangent line at x = a, construct the equation for and graph of the tangent line to graph of f(x) at x = a, construct and graph the secant line of graph between (a, f(a)) and (a + h, f(a + h)), find the slope of this secant line, find the change in f between x = a and x = a + h, determine the derivative of f(x) at x = a and apply these skills to Relate for a function f(x) average rate of change of f with respect to x over the interval from x = a to x = a + h instantaneous rate of change of f at x = a difference quotient at x = a existence of limit of difference quotient at x = a limiting value of difference quotient at x = a differentiability of f(x) at x = a graph of f(x) near x = a tangent line to graph of f(x) at x = a slope of tangent line secant line of graph between (a, f(a)) and (a + h, f(a + h)) slope of secant lint change in f between x = a and x = a + h numerical behavior of f(x) in the vicinity of x = a the derivative of f(x) at x = a specific value of a symbolic value of a
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Asst 13	qa_13	Ch 2.3, 2.4			text_13		Class Notes #15		query_13
Class Notes Summarized: #15: Interpretation of the Derivative; the Second Derivative · interpreting derivative function, statements about derivative function · second derivative · approximating f ‘’ given f · rate at which rate changes (depth model and others) · trapezoidal approximation graph: rate of slope change · second derivative and trend of first derivative; second derivative and concavity · area under a curve (example for v vs. t) · area under trapezoidal approximation graph approaches actual area for strictly increasing function (upper sum and lower sum, difference between upper and lower sum)
Objectives: 1. Given a function f(x), defined graphically and/or algebraically, construct the derivative function graphically, if possible apply the rules of differentiation to find the derivative function in algebraic form, and interpret in a real-world context. More generally, apply these skills to Relate function f(x) given and/or constructed algebraically and/or graphically derivative function f ' (x) given and/or constructed algebraically and/or graphically interpretation and application in a real-world context 2. Given a graphical representation of f(x) or a table of values for f(x), calculate a series of values for f ' (x) and construct a graph of f ' (x). Use these skills to Relate for a function f(x) graphical representation of f(x) numerical values of f(x) series of estimated or approximate values of f ' (x) graphical representation of f ' (x) 3. Apply the difference quotient to obtain the expression for the derivative of a given power function. More generally, Relate for a power function f(x) = x^p difference quotient (f(x + h) - f(x) ) / h derivative function f ' (x) 4. Given the meanings of the x and y quantities for a function y = f(x) and the graph of this function, interpret the meaning of the 'rise', 'run' and slope between two points, the meaning of the change in the x value, the meaning of the change in the y value, the meaning of the average rate of change of y with respect to x, the approximate value and the meaning of f ' (x). Using these skills Relate meaning of y meaning of x meaning of 'rise' between two points of graph meaning of 'run' between two points of graph meaning of slope between two points of graph meaning of change in x value meaning of change in y value meaning of average rate of change of y with respect to x approximate value of f ' (x) meaning of f ' (x)
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Asst 14	qa_14	Ch. 2.5, 2.6, 5.1		Brief Synopsis: Tangent-Line_Approximation (Not Currently Assigned): Collaborative Investigation 2	text_14		Class Notes #16, 17		query_14
#16: Second Derivative; Definite Integral · is the function increasing or decreasing, and is it doing so at a increasing or decreasing rate · text problem: values of f, f ‘, f ‘’ estimated from table of f values (includes clarification by trapezoidal approximation graph) · characteristics of f , f ‘’ given graph of f ‘ · from velocity function approximate displacement on given interval (analysis includes trapezoidal approximation graph, upper and lower estimates) · integral of sin(t^2) as area beneath curve #17: Using DERIVE for integrals and Riemann sums · relevant DERIVE commands
Objectives: 1. Where f(t) is a squaring, square root or reciprocal function and t_0 a value of t close but not equal to t = 1, analytically construct the tangent-line approximation of the function in the vicinity of t = 1 and use to approximate the value of the function at a given t, find the differential of the function and use to approximate the differential change in its value between t = 1 and a given t as well as the differential estimate of its value at t. Apply these skills to Relate in the context of square roots and squares and reciprocals of numbers close to 1 f(t) t_0 tangent line differential change differential estimate differential 2. Given a graph of f(x), f ' (x) or f ''(x) on an interval, subdivide the interval to obtain a sequence of values for the given function, derivative, or second derivative, and a sequence of approximated numerical values of the other two. Apply these skills to Relate graph of f(x) graph of f ' (x) graph of f '' ( x) sequence of actual or estimated numerical values of f(x) sequence of actual or estimated numerical values of f ' (x) sequence of actual or estimated numerical values of f '' (x) 3. Given the average rate of change of a function on an interval, the instantaneous rates at which the function changes at the beginning and end of the interval, and the duration of the interval make a valid estimate of the approximation error of a trapezoidal approximation to the change in the value of the function on the interval. Apply these skills to Relate average rate of change on interval instantaneous rates at endpoints and midpoint approximation error and `dt 4. Given the y = f(x), defined analytically, find the functions f ' (x) and f '' (x) 5. Explain the interpretation of f ''(x) as a rate at which a rate changes. 6. Explain the effect of the second derivative of f(x) on the concavity of the graph of y = f(x). 7. Given the position function for a particle find its velocity and acceleration functions. 8. Interpret the motion of a particle analytically, numerically and graphically in terms of its position, velocity and acceleration functions. 9. Given a function y = f(x) and a value of a, write the expression for the limit of the difference quotient at x = a, test for the existence of this limit, test for continuity and differentiability of f(x) at x = a. Apply these skills to Relate for a function y = f(x) expression for limit{h -> 0} ( f(a + h) - f(a) ) / h ) existence of limit{h -> 0} ( f(a + h) - f(a) ) / h ) differentiability of f(x) at x = a continuity of f(x) at x = a 10. For a velocity function v(t) and a time interval and a series of subintervals, given the initial and final velocities on each subinterval calculate the approximate average velocity on the entire interval, the displacement associated with each subinterval and the entire interval, right- and left-hand estimates of the displacement and the difference and perent difference between these estimates, the area of the associated region beneath the graph of v vs. t, the trapezoidal approximation of the area, and the behavior of the associated graph of position vs. t. Apply these skills to Relate for a velocity function v(t), a time interval and a series of subintervals initial and final velocity of an object on an interval approximate average velocity on that interval duration `dt of each interval displacement of object during the interval graph of velocity vs. clock time corresponding to the interval displacement during interval based on left-hand estimate displacement during interval based on right-hand estimate difference between left- and right-hand estimates percent difference between left- and right-hand estimates shrinking values of `dt area of region beneath graph trapezoidal approximation of area beneath graph behavior of position vs. t graph
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Asst 15	qa_15	Ch 5.2, 5.3		Brief Synopsis: Connection_between_Integral_and_Derivative	text_15		Class Notes #18		query_15
Class Notes Summarized: #18: The Fundamental Theorem of Calculus · quiz: difference between left- and right-hand sums for given function on given interval, different increments · text problem: given graph of car velocity compare positions with those of truck moving at given constant speed (trapezoidal approximation graph used in solution; graphical representation of two v vs. t graphs, approx points of equal accumulated areas; graphs of position functions) · average value of a function · Fundamental Theorem · example with linear rate function: compare are beneath graph to change in antiderivative
Objectives: 1. Develop the left- and right-hand Riemann sum for a positive monotone function on an interval, explain the relationship of the sums to the area beneath the curve, explain the significance of the difference between the sums and the behavior of this differences as the number of subintervals approaches zero. Apply these skills more generally to Relate the following for a function y = f(x) which is monotone and positive on an interval a <= x < = b number n of subintervals subinterval length `dt left-hand sum right-hand sum magnitude of difference between left- and right-hand sums partition of the interval [a, b] 0 = t_0 < t_1 < t_2 < ... < t_(i-1) < t_i < ... < t_(n-1) < t_n = b Riemann sums area beneath graph corresponding to each Riemann sum area beneath graph of f(x) on [a, b] definite integral of f(x) with respect to x on the interval [a, b] (denoted integral(f(x), x, a, b)) sample points c_i general Riemann sum 2. Where f(t) or f ' (t) is a linear or quadratic function, integrate f '(t) on a given interval, determine the derivative of the indefinite integral of f(t), find the derivative and an antiderivative of f(t), calculate the average value of f(t) on an given interval, interpret each of these quantities in the context where f(t) is a quantity that changes with respect to t, and solve problems in which these operations and interpretations are applied to population function, cost, demand, marginal cost, profit and marginal profit. Apply these skills more generally to Relate in the context of linear and quadratic functions f(t) f ‘ (t) integral(f ‘ (t), t, a, b ) (integral(f(t), t) ) ‘ antiderivative change in value of antiderivative rate and quantity interpretation applications to population, cost, demand, marginal cost, profit, marginal profit average value of f(t) on interval [a, b]
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Asst 16	qa_16	Ch 5.4			text_16		Class Notes #19		query_16
Class Notes Summarized: #19: Review Notes for Test · (pictures only, no text)
Objectives: 1. Using product and quotient rules and combinations of these rules, calculate and simplify derivatives, and apply the results to real-world situations. 2. Analytically determine the equation of the tangent line to a given curve at a given point using implicit differentiation. 3. Apply to the solution of problems properties of integrals related to the integral of f(x) on an interval [a, b] with respect to the following: interchange of limits of integration splitting the interval [a, b] into [a, c] U [c, b], a < c < b integral of a constant multiple of f(x) use of symmetry comparison with other integrals 4. Apply the properties of integrals related to two functions f(x) and g(x) on an interval [a, b], with respect to problems related to the following: integral of the sum or difference function area between curves 5. Explain why the definite integral of a product or quotient of two functions is not generally equal to the product or quotient of the integrals. 6. Explain the meaning of the antiderivative function of f(x) as change-in-quantity function for the quantity represented by f(x).
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Asst 17		Ch 3.1, 3.2		Brief Synopsis: Implicit_Differentiation Riemann_Sums	text_17		Class Notes #20, 21		query_17
Class Notes Summarized: #20: The Fundamental Theorem of Calculus; Derivatives of Polynomials · properties of definite integrals (reversing limits, splitting interval of integration, integral of sum or difference of two functions) · f(x) > m on interval [a, b] => integral >= m ( b – a) · we didn’t say that the integral of f / g is the integral of f divided by the integral of g; unless g is consanat it almost certainly isn’t so #21: The Fundamental Theorem of Calculus; Derivatives of Polynomials · {F(t_i), F(0)} for f(t) piecewise linear · where is derivative of cubic polynomial greater than given value? · altitude y(t) quadratic, find velocity and acceleration functions, questions about position, velocity, acceleration · dV/dr for volume of sphere · , questions about position, velocity, acceleration · dV/dr for volume of sphere · derivative of exponential function from definition of e · slope = altitude for y = e^x · derivative of y = a^x
Objectives: 1. Derive the formula for the derivative of y = x^2 or y = x^3 based on the definition of the derivative and expansion of (x + h) to the appropriate power. 2. Derive the formula for the derivative of the general power function y = x^n, based on the definition of the derivative and the binomial expansion of (x + h)^n. 3. Derive the formula for the derivative of the exponential function y = e^x, based on the fact that limit{h -> 0} ( (e^h - 1) / h) = 1. 4. Apply the formulas for derivatives of power and exponential functions to problems involving functions build using sums, differences and constant multiples of these functions. (nothing new) Applications
Module 3: Finding and applying derivatives Test 2 should be taken within a week of completing Module 3
Module-level Objectives Apply the difference quotient and limits to derive formulas for derivatives of exponential, sine and cosine functions. Explain l'Hopital's Rule in terms of local linearizations, and apply to find limits and dominance. Apply the tangent line approximation and local linearization to mathematical and real-world applications. Apply implicit differentiation to obtain information about implicitly defined functions. Apply the chain rule to find the derivative of the inverse of a given function. Apply derivatives and other techniques to graph a curve given its parameterization, and investigate its properties. Apply the derivative to solve problems involving optimization, graphing and rates. Construct mathematical models of mathematical and real-world phenomena. Approximate derivatives and integrals, and estimate approximation errors. Derivatives of Basic Functions The derivatives of the basic functions are as follows: d / dx ( x^n) = n x^(n-1) for positive integers n, obtained using the Binomial Theorem. d / dx ( e^x ) = e^x, obtained from the definition of the derivative and the fact that as x -> 0, the difference between e^x and x approaches 0 faster than does x. d / dx (sin(x)) = cos(x), obtained by a geometrical argument showing that as x -> 0 the ratio between the arc length on the unit circle corresponding to angle x and the y coordinate of the unit circle point defining sin(x) approaches 1. d / dx ( cos(x) ) = - sin(x), obtained by a geometric argument similar to that used for sin(x) (also derivable using the chain rule (see below) and the Pythagorean identity). The Chain Rule The derivative of f(g(x)) is the product of the rate at which g(x) changes at x, and the rate at which f(z) changes at z = g(x). When x changes it causes a change in the value of z = g(x). A change in z results in a change in the value of f(z). If x changes by `dx then z = g(x) changes by approximately `dz = g ' (x) `dx. If z changes by `dz then f(z) changes by approximately `df = f ' (z) `dz. If `dz = g ' (x) `dx then `df = f ' (z) * [ g ' (x) `dx ], approximately. Thus in the limit df / dx = f ' (z) * g ' (x) = g ' (x) * f ' (g(x)). If a function f(z) depends on the value of z(w), which in turn depends on the value of w(v), which in turn depends on v(x), then a change in x causes a change in v which causes a change in w which causes a change in z which causes a change in f. The changes in successive variables form a 'chain', so that we have the approximations (which become precise as `dx -> 0): `dv = v ' (x) * `dx, `dw = w ' (v) * `dv = w ' (v) * v ' (x) * `dx, or `dw / `dx = (`dw / `dv) * (`dv / `dx) `dz = z ' (w) * `dw = z ' (w) * w ' (v) * v ' (x) * 'dx, or `dz / `dx = (`dz / `dw ) * (`dw / `dv) * (`dv / `dx). The chain keeps growing until we have the approximation `df / `dx = ( `df / `dz) * (`dz / `dw ) * (`dw / `dv) * (`dv / `dx). In the limit as `dx -> 0 we have df / dx = df / dz * dz / dw * dw / dv * dv / dx. Derivative of Inverse Functions If g(x) = f^-1(x), then f(g(x)) = x and f ' (g(x)) = 1. However, we also have f ' (g(x)) = g ' (x) * f ' (g(x)), so that g ' (x) = 1 / [ f ' ( g ( x) ) ]. Provided we can obtain an expression for f ' (g(x) ) we can thus find the formula for the derivative g ' (x) of the inverse function g(x) = f^-1(x). Using this technique we find that d / dx ( x^p) = p x^(p-1) for any rational number p not equal to -1. d / dx (ln(x)) = 1 / x, using f(x) = e^x and g(x) = ln(x). d / dx (arcsin(x)) = 1 / `sqrt(1-x^2), using f(x) = sin(x) and g(x) = arcsin(x) d / dx (arctan(x) ) = 1 / (x^2 + 1), using f(x) = tan(x) and g(x) = arctan(x). Implicit Differentiation If y is a function of x and g is a function of y, then dg / dx is the derivative with respect to x of the composite function g ( y(x) ). This derivative, by the Chain rule, is *dg / dx = y ' (x) g ' ( y (x) ). g ( y(x) ) is often written just g(y) and we have to remember that if the derivative is with respect to x, this is a composite. Instead of writing y ' (x) and g ' ( y(x) ) we might just write y and g ' (y), or even y and dg / dy. When we do so we have to remember that y is a function of x and g ' means dg / dy. Thus we might write dg / dx = y ' dg / dy. A product function** of the form f(x) * g(y) can be differentiated with respect to x (recalling that dg / dx = dg / dy * dy / dx, or dg / dx = dg / dy * y ' ) using the product rule. Thus *d / dx ( f(x) g(y) ) = df / dx + dg / dx = df / dx + dg / dy * y ', where y ' = dy / dx.** For example d / dx ( x^2 y^3) = 2x y^3 + x^2 * 3y^2 y ' d / dx ( sin (x^2 y^3) ) = [ 2x y^3 + x^2 * 3y^2 y' ] cos (x^2 y^3) (note the use of the Chain Rule) An equation f(x, y) = 0 will often be satisfied by an infinite number of order pairs or points (x, y). These points will typically form a curve or a set of curves in the x-y plane. Usually it is impossible to find the equation of the curve(s) because we can't solve the equation explicitly for y. However, the curve still exists and at a point such a curve will typically have a slope. If we know the coordinates of a point on the curve and have a formula for y ' in terms of x and y we can simply insert the coordinates into the formula to find the slope at that point. Given the point and the slope we can find the equation of the tangent line at the point. Given an equation of the form f(x, y) = 0, where f(x, y) denotes any expression involving x and y, we can differentiate the equation with respect to x. The resulting equation will involve the variables x and y and the derivative y ' = dy / dx. The equation can often be solved for y ' in terms of x and y, especially when y ' appears only as a linear factor of one or more terms of the equation. Having obtained the solution y ' = dy / dx in terms of x and y we can find y ' at any point (x, y) for which the original equation f(x, y) = 0 is true. Thus given any point (x, y) on the curve implicitly defined by f(x, y) = 0 we can find the equation of the tangent line to the curve at that point. Optimization A function f(x) which is continuous and twice-differentiable near x = a will have a maximum at x = a under the following conditions: f ' (a) = 0 and f ' (x) changes from positive to negative at x = a (the first-derivative test), or f ' (a) = 0 and f '' (a) is negative (the second-derivative test). A function f(x) which is continuous and twice-differentiable near x = a will have a minimum at x = a under the following conditions: f ' (a) = 0 and f ' (x) changes from negative to positive at x = a (the first-derivative test), or f ' (a) = 0 and f '' (a) is positive (the second-derivative test).
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Asst 18		Ch 3.3		Brief Synopsis: Derivatives_of_Basic_Functions	text_18		Class Notes #22		query_18
Class Notes Summarized: #22: Product and Quotient Rules · Can we take the derivatives of some given functions? (yes for most given quiz questions, but no if product, quotient or composite) · Derivative of e^x based on e^(`dx) close to 1 + `dx · line tangent to y = 1 – e^x · product rule, geometric proof · quotient rule
Objectives: 1. Explain the product rule in terms of the area of the rectangle whose dimensions are f(x) by g(x). 2. Apply the product and quotient rules as needed to find the derivatives of product and quotient functions. In the case of real-world applications interpret the results.
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Asst 19		Ch 3.4, 3.5			text_19		Class Notes #23, 24, 25		query_19
#23: The Chain Rule · quiz: limit definition of (f g ) ‘ · quiz: derivative of given product function · text problem: derivative of x^(1/2) using product rule · text problem: when is x e^-x concave down · chain rule as ‘correction’ for rate of change of ‘inner function’ · examples of chain rule · ( (e^x)^2 + sqrt(e^x) ) as composite of z^2 + sqrt(z) with z = e^x #24: The John Glenn Launch: A review of proportionality and integration; Trigonometric Functions · how much work is required to raise 75 kg to orbital height (or to distance of Moon); use of proportionality and definition of work, integrate · derivatives of trigonometric functions asserted · chain rule with trigonometric functions · unit circle geometry required to prove derivative of sine function #25: Trigonometric Functions; Applications of Chain Rule; Inverse Functions · examples: finding derivatives of composites involving trigonometric functions · tangent line · population function · projection of uniform, nonuniform circular motion · derivatives of inverse functions using chain rule · derivative of sqrt(t) using chain rule · derivative of ln(x) using chain rule
Objectives: 1. Given an application where the meanings of f(x), g(x) and their derivatives are known, explain the meaning of the chain rule in the context of the application. 2. Apply the chain rule to find derivatives of composite functions. 3. Apply all rules of differentiation to expressions involving trigonometric functions. 4. Apply trigonometric functions and derivatives of the resulting expressions to various mathematical and real-world applications.
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Asst 20		Ch 3.6, 3.7, 3.8		Brief Synopsis: The_Chain_Rule Derivative_of_Inverse_Functions (Not Currently Assigned): Collaborative Investigation 3	text_20		Class Notes #26, 27, 28		query_20
Class Notes Summarized: #26: Inverse Functions; Chain Rule; Implicit Differentiation · quiz problem: derivative of a composite function involving ln(z), also derivative of e^(ln(x) + 1) · derivative of arcsin(x) · dF/dt rocket moving away from Earth at given velocity, rate of change of power required · implicit differentiation, example #27: Implicit Differentiation; Tangent Line Approximation; l'Hopital's Rules · implicit differentiation example · tangent line, use for approximation · l’Hopital’s rules and tangent line approximations #28: Implicit Differentiation; Maxima, Minima and Inflection Points · quiz: implicit differentiation to get tangent line at given point, approximate new value · l’Hopital’s rules, equations of tangent lines · relative maxima and minima · first derivative test · second derivative and rate of change of first derivative · second derivative test
Objectives: 1. Given an equation which implicitly defines y as a function of x, perform the operations necessary to find the derivative y ' = dy/dx as a function of x. More generally: Relate explicit definition of a function y(x) implicit definition of a function y(x) derivative of y(x) in terms of x and y 2. Given an equation which implicitly defines y as a function of x, perform the operations necessary to find the derivative y ' = dy/dx as a function of x, and where possible solve the equation for y as a function of x and calculate the necessary derivative to verify the result. More generally: Relate equation f(x, y) = 0 equation (d / dx) f(x, y) = 0 solution for dy/dx confirmation that given x and y values satisfy f(x, y) = 0 values of dy/dx for given x and y explicit solution for y of f(x, y) = 0 where possible derivative of explicit solution for given x derivative of explicit solution reconciled with dy/dx for given x and y graph of y = f(x) 3. Use the chain rule to find expressions for the derivatives of the inverses of given functions. 4. Apply the rules of differentiation to obtain the derivatives of expressions which include the natural log, arcsine, arccosine and arctangent functions based on knowledge of the derivatives of these functions. 5. Apply the definitions of the hyperbolic sine, hyperbolic cosine and hyperbolic tangent functions in order to find the derivatives of these functions. Applications
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Asst 21		Ch 3.9, 4.7, 4.8			text_21				query_21
Objectives: 1. Apply the tangent line approximation and local linearization to mathematical and real-world applications. 2. Estimate for function f(x) the error at point x of the local linearization of f(x) about (a, f(a)). 3. State and apply l'Hopital's rule. 4. Explain how the local linearizations of f(x) and g(x) are related, and a point a where the limiting values of f(x) and g(x) are both 0, to l'Hopital's rule. 5. Use l'Hopital's rule to establish dominance between two functions with infinite limits. 6. Given the parameterization x(t), y(t) of a curve in two dimensions, sketch the curve. 7. Given an ellipse or a straight line in the plane, analytically determine a parameterization. 8. Given a geometric definition of a curve in the plane, analytically determine a parameterization. 9. Given a parameterization x(t), y(t) of a curve in two dimensions, obtain the expressions for the instantaneous speed of a point moving along the curve, the slope of the curve as a function of t and the concavity of the curve as a function of t.
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Asst 22		Ch 4.1, 4.2			text_22		Class Notes #29, 30		query_22
#29: Local Maxima and Minima; Families of Functions · quiz: max/min of given quadratic function · hourglass container, water flowing in at constant rate, depth vs. clock time; motivates point of inflection · family of functions modeling y(t) for free fall (variable parameters are v_0 and y_0) · function family: shape of a standing wave · families e^(-(x-a)^2), e^(x^2 / b); probability #30: Families of Functions; Optimization · max and min for family 3 t^2 + b t + 8 (relative extrema lie on parabola -3 t^2 + 8) · family x – k sqrt(x); critical point ¼ of way between zeros · DERIVE to illustrate · family a x e^(- b x) · relative and local maxima and minima
Objectives: 1. For a given function y = f(x) analytically determine the derivative of the function and the points where the derivative is zero, and apply this knowledge to the construction of the graph of the functions. Using these skills Relate function y = f(x) function y = f ' (x) equation f ' (x) = 0 points of horizontal tangency graph of y = f(x) 2. For a function y = f(x) analytically determine the following and represent the analytically determined behavior graphically: df/dx critical values x intervals where f(x) is increasing x intervals where f(x) is decreasing graph of y = f(x) x intervals on which f ‘ (x) is positive x intervals on which f ‘ (x) is negative x intervals on which f ‘’(x) is positive x intervals on which f ‘’(x) is negative x intervals on which f(x) is concave up x intervals on which f(x) is concave down values of x at which the sign of f ‘’ ( x) changes critical values at which f ‘ changes from negative to positive critical values at which f ‘ changes from positive to negative critical values at which f ‘ does not change sign critical values at which f ‘’(x) is positive critical values at which f ‘’(x) is negative critical values at which f ‘’(x) is zero critical values at which f(x) is concave up critical values at which f(x) is concave down relative maxima of f relative minima of f inflection points of f interval a <= x <= b absolute extrema of f on interval a <= x <= b Using the ability to do each of these things, Relate y = f(x) df/dx critical values x intervals where f(x) is increasing x intervals where f(x) is decreasing graph of y = f(x) x intervals on which f ‘ (x) is positive x intervals on which f ‘ (x) is negative x intervals on which f ‘’(x) is positive x intervals on which f ‘’(x) is negative x intervals on which f(x) is concave up x intervals on which f(x) is concave down values of x at which the sign of f ‘’ ( x) changes critical values at which f ‘ changes from negative to positive critical values at which f ‘ changes from positive to negative critical values at which f ‘ does not change sign critical values at which f ‘’(x) is positive critical values at which f ‘’(x) is negative critical values at which f ‘’(x) is zero critical values at which f(x) is concave up critical values at which f(x) is concave down relative maxima of f relative minima of f inflection points of f interval a <= x <= b absolute extrema of f on interval a <= x <= b first-derivative test second-derivative test
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Asst 23		Ch 4.3		Brief Synopsis: Optimization	text_23		Class Notes #31		query_23
Complete Test #2
#31: Optimization · graph of horizontal range of ball rolling down incline, off edge, falling to floor, with respect to elevation h of high end of incline · maximizing x(h) = k sqrt(L^2 h^2 – h^3) (-k h^(3/2) + sqrt(K^2 H^3 + 19 600) ) / 980 with respect to h · expected zeros of x(h) based on the physical system · approximate graph of expected behavior of x(h) vs. h between zeros · optimizing m(x) = ½ W L x – ½ m x^2 · maximizing f(g(theta)) with g(theta) = sin (theta) + mu cos(theta), f(x) = 1/x
Objectives: 1. Given a one-parameter family of functions determine the effects of the parameter on relative maxima and minima, points of inflections, intervals of concavity, intervals of monotonicity. Sketch a graph showing the influence of the parameter on the function family. 2. Within the context of a mathematical or real-world application, relate the behavior of a function family to its parameters.
Module 4: Further Applications of the Derivative; Theory The Final Exam should be taken within a week of completing Module 4
Module-level Objectives Apply optimization techniques to price, cost and demand functions to optimize revenue and/or profit. Apply the derivative and limits to the graphing of functions with asymptotes, including but not limited to rational functions. Construct proofs of mathematical statements involving limits, differentiability and continuity. Construct proof of mathematical statements involving the intermediate value theorem and/or the nested interval theorem. Solve real-world optimization problems. L'Hopital's Rule Often we need to evaluate the limiting value of an expression of the form f(x) / g(x) where both f(x) and g(x) have limiting values 0. Since 0 / 0 is undefined (i.e., it could mean 0, it could mean an infinite quantity, it could mean any finite quantity), we can't get the desired limit from the quotient of the limits. However if f(x) and g(x) have tangent lines at the limit point, f(x) / g(x) will at that point approach the ratio of the slopes of these tangent lines. Thus we see that If lim {x -> a} f(x) = 0 and lim {x -> b} g(x) = 0, then near x = a we have the approximations *f(x) = f(a) + f ' (a) ( x - a ) = 0 + f ' (a) * (x - a) = f ' (a) * (x - a) g(x) = g(a) + g ' (a) * ( x - a ) = 0 + g ' (a) * (x - a) = g ' (a) * (x - a) both of which become accurate as x -> a. Thus the approximation f(x) / g(x) = f ' (a) * (x - a) / [ g ' (a) * (x - a) ] = f ' (a) / g ' (a) becomes accurate in the limit as x -> a. Thus when f(x) and g(x) both approach 0 as x -> a, and when f(x) and g(x) have derivatives at x = a, lim{x -> a} [ f(x) / g(x) ] = lim {x -> a} f(x) / [ lim {x -> a} g(x) ] . This is known as l'Hopital's Rule. This rule is easily adapted to the case where f(x) and g(x) both have infinite limits at x = a (instead of f(x) / g(x) we use [ 1 / f(x) ] / [ 1 / g(x) ], where the numerator and denominator both have limit 0. It is also easy to adapt to the case where the limits of the two functions are both zero as x -> infinity**.
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Asst 24		Focus on Theory assts 1 and 2			text_24		Class Notes #32		query_24
#32: Optimization; Completeness Axiom; Limits · maximizing the range of a projectile · existence of zeros of a polynomial and the completeness axiom (any set of numbers with an upper bound has a least upper bound) · nested interval theorem (the intersections of any infinite set of nested intervals is not empty) · intermediate value theorem · definition of a limit
Objectives: 1. Use the following to prove selected properties of the real numbers: *the completeness axiom (any nonempty set of real numbers which has an upper bound has a least* upper bound). the nested interval theorem (any intersection of nested intervals is nonempty) the intermediate value theorem (all intermediate values for are taken by a continuous function) 2. Use to rigorously prove statements about limits, continuity and differentiability: definition and properties of a limit definition and properties of continuity definition of differentiability**
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Asst 25		Focus on Theory asst 2 (2d part)					Class Notes #33, 34, 35		query_25
#33: Limits, Differentiability, Review · quiz: squeezing y by squeezing x · definition of a limit in terms of epsilons and deltas; right- and left-hand limits · examples: (x – 2 ) / \| x – 2 \|, sin(1/x) · definition of continuity · continuity of composite functions · definition of differentiability · error of the tangent-line approximation Derivatives of the basic functions Be able to use the definition of the derivative to get the formula for at least a quadratic function. Know the formulas for the derivatives of power, exponential, sine, cosine and tangent functions and their inverse functions. Derivatives of composite functions Know and be able to use the chain rule in both the g'(x) f'(g(x)) and df /dz * dz / dx notations. Applications Be able to recognize when derivatives are appropriate for interpreting a situation, and be able to take the derivatives and interpret them. Implicit differentiation Given an equation in x and y, with y an implicit function of x, the able to perform the implicit differentiation and arrive at a formula for y' in terms of x and y. The able to use this formula to obtain a tangent-line approximation at a given point. Linear approximation and l'Hopital's Rule Be able to construct the tangent-line approximation to any given function at a specified point. Know the conditions for and be able to apply l'Hopital's Rule to obtain the limiting value of the quotient f(x) / g(x) of two functions whose values at the limiting point are both zero. Be able to find and apply knowledge of the local and global maximum and minimum points of a function. Know how to find critical points and apply the first and second derivative tests to determine whether a function has a relative maximum or relative minimum at a critical point. Be able to identify inflection points using the behavior of the second derivative. Be able to identify the function in an application which must be maximized or minimized, and to interpret your results. #34: Test Review (includes practice test) · practice test with notes Class Notes #35-#41: Review and Final Topics #35: Class Review (continued) · continued notes on practice test
Objectives: 1. Given demand as a function of price, and cost as a function of price: Analytically determine revenue as a function of quantity produced. Analytically determine profit as a function of quantity produced. Analytically determine the maximum profit and the quantity to be produced to maximize profit. Analytically determine marginal profit. Using these skills Relate price function demand function cost function profit function interval of definition for profit function maximum of profit function price for which profit function is maximized marginal profit
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Asst 26		Focus on Theory asst 3		Brief Synopsis: LHopitals_Rule	text_26		Class Notes		query_26
Objectives: 1. For a rational function y = f(x) Find analytically any and all vertical asymptotes of its graph, and whether the function approaches each asymptote through positive or negative values on either side near the asymptote. Find analytically any and all horizontal asymptotes of its graph, and whether the function approaches each such asymptote from above, below or alternately one then the other. Find analytically any and all slant asymptotes of its graph, and whether the function approaches each such asymptote through positive or negative values near the asymptote. Find analytically the limits of the function at +infinity and at -infinity. Find all x and y intercepts of the function. Based on analytical results graph the function. When the function represents some real-world phenomenon, based on analytical results interpret in the real-world context the above behaviors of the function. Ultimately use these skills to Relate y = f(x) rational function vertical asymptotes limits at vertical asymptotes horizontal asymptotes slant asymptotes limits at +-infinity intercepts graph interpretation 2. Use the definition of the definite integral in terms of the least upper bound of lower sums and the greatest lower bound of upper sums to rigorously prove statements about integrals.
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Asst 27		Ch 4.4			text_27		Class Notes #36		query_27
#36: Marginality and modeling · marginality of revenue, cost, profit is rate at which that quantity changes with respect to production · differential estimates of pendulum behavior · lifeguard problem (optimization)
Objectives: 1. Solve optimization problems related to mathematical or real-world applications. 2. Find upper and lower bounds on a given function on a given interval.
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Asst 28		Ch 4.5, 4.6			text_28		Class Notes #37, 38		query_28
#37: Modeling · modeling pendulum velocity as a function of position, change in velocity expected for given short interval in vicinity of given position · marginal revenue, marginal cost, optimal profit when equal · minimize surface area of can with given volume · distance for best view of Statue of Liberty · minimize diagonal of rectangle with given area #38: Modeling; Introduction to Hyperbolic Functions · maximize volume of box with given surface area · minimize time for light ray to travel from point A to point B (Snell’s Law) · hyperbolic sine and cosine, graphs, derivatives Class Notes Summarized: #36: Marginality and modeling · marginality of revenue, cost, profit is rate at which that quantity changes with respect to production · differential estimates of pendulum behavior · lifeguard problem (optimization)
Objectives: 1. Solve optimization problems related to mathematical or real-world applications. 2. Find upper and lower bounds on a given function on a given interval.
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Asst 29		Ch 3.10			text_29		Class Notes #39, 40, 41		query_29
#39: Hyperbolic Functions, Differentiability · shape of a chain · differentiability: a function which is continuous on a closed interval has a global maximum on that interval · if f(x) has a local maximum and is differentiable at that point then its derivative at that point is zero #40: Review I. Click also for · This document consists of an extensive list of key terms, concepts and procedures. Differentiability II. · Mean Value Theorem · a function with a positive derivative on an interval is strictly increasing on that interval · racetrack principle: if f(t) and g(t) start with the same value and f ‘ > g ‘, then f > g #41: Review II · The Fundamental Theorem of Calculus · The Chain Rule · Derivatives of Sine and Exponential Functions · Tangent line approximation · Proof of product rule · Implicit differentiation · Marginality · Modeling · Pendulum · `epsilon, `delta · Theorems important to the foundations of the calculus
Objectives: 1. For a function F(q) and some specific value q_0 of q, where q is the number of items produced and F(q) could represent revenue, profit, or cost as a function of q: Calculate the derivative dF/dq. Calculate the differential dF. Explain the relationship among dF, `dF and `dq Construct the tangent line to the graph of F vs. q at the point corresponding to q_0. Calculate the expression for the marginal value of F at arbitrary q, and at q_0. Calculate the estimated change in F near q_0 based on the value of dF/dq at q = q_0. Calculate the estimated change in F near q_0 based on the differential dF. Calculate the estimated change in F near q_0 based on the equation of the tangent line. Interpret any or all of these quantities for the case where F is interpreted as a cost function, a revenue function or a profit function. Apply any or all of these procedures to problems involving real-world applications. Ultimately, use these skills to Relate F(q) q_0 F ‘ (q) dF `dq `dF tangent line equation of tangent line profit function cost function revenue function marginal F change in F near q_0 estimated by rate of change at q_0 change in F near q_0 estimated by differential change in F near q_0 estimated by tangent line marginal analysis where F is typically profit or cost or revenue real-world applications

By the end of Module 1 you will have

solidified and perhaps expanded your understanding, obtained in prerequisite courses, of linear, quadratic, power, exponential, logarithmic and trigonometric functions and the analytical tools required to work with these functions

explored the idea of the rate of change of one quantity with respect to the other

developed the idea of rate of change into the idea of the derivative and integral and applied this insight in the case of linear and quadratic functions.

Trapezoidal Approximation Graphs

The Idea of the Derivative

The Idea of the Integral

Class Notes Summarized:

#01: Quadratic Model of Depth vs. Time for water flowing from a uniform cylinder.

#02: Rates of change for the depth vs. time model.

#03: The Rate of Change of a Quadratic Depth Function: Differentiation and Integration

#04: The Concepts of Differentiation and Integration in the Context of Rate Functions

#05: Growth of an Exponential Function; Trapezoidal Representation of Approximate Derivatives and Integrals

#06: Project #3; Derivative of y = a x^3; The Differential

Linked Outline of Introductory Topics through 9/04/98: Study these for the upcoming 9/13/98 major quiz

#07: The Differential; Tangent Line Approximation to Differential

#08: Text problems; First Introduction to Differential Equations

#09: Text problems; Introduction to Natural Logarithms and Composite Functions; Numerical Solution of Differential Equations

#11: Trigonometric Functions; Brief Intro. to Polynomials

· modeling by trigonometric functions

· polynomials

#12: Rational Functions, Continuity

· graphing rational functions

· surface area of constant-volume cylinder as radius approaches zero

· continuity of 1 / (x – 3), 1 / sin(x), x / sin(x)

Connection between Integral and Derivative

#13: The Derivative (Text 2.1, 2.2)

· quiz: algebraically find derivatives of x^2 and 1 / x^2

· expansion of a binomial and derivatives of power functions

· sequencing set of slope and average slopes for a given curve

· how much do we have to squeeze x values to confine y values in specified manner

· finding the equation of a tangent line

#14: The Derivative Function

· constructing graph of f ‘ from graph of f, interpretation for depth model

· graphing a f given characteristics of its derivative f ‘

· interpreting slopes, interpreting the derivative function

#15: Interpretation of the Derivative; the Second Derivative

· interpreting derivative function, statements about derivative function

· second derivative

· approximating f ‘’ given f

· rate at which rate changes (depth model and others)

· trapezoidal approximation graph: rate of slope change

· second derivative and trend of first derivative; second derivative and concavity

· area under a curve (example for v vs. t)

· area under trapezoidal approximation graph approaches actual area for strictly increasing function (upper sum and lower sum, difference between upper and lower sum)

#16: Second Derivative; Definite Integral

· is the function increasing or decreasing, and is it doing so at a increasing or decreasing rate

· text problem: values of f, f ‘, f ‘’ estimated from table of f values (includes clarification by trapezoidal approximation graph)

· characteristics of f , f ‘’ given graph of f ‘

· from velocity function approximate displacement on given interval (analysis includes trapezoidal approximation graph, upper and lower estimates)

· integral of sin(t^2) as area beneath curve

#17: Using DERIVE for integrals and Riemann sums

· relevant DERIVE commands

#18: The Fundamental Theorem of Calculus

· quiz: difference between left- and right-hand sums for given function on given interval, different increments

· text problem: given graph of car velocity compare positions with those of truck moving at given constant speed (trapezoidal approximation graph used in solution; graphical representation of two v vs. t graphs, approx points of equal accumulated areas; graphs of position functions)

· average value of a function

· Fundamental Theorem

· example with linear rate function: compare are beneath graph to change in antiderivative

#19: Review Notes for Test

· (pictures only, no text)

#20: The Fundamental Theorem of Calculus; Derivatives of Polynomials

· properties of definite integrals (reversing limits, splitting interval of integration, integral of sum or difference of two functions)

· f(x) > m on interval [a, b] => integral >= m ( b – a)

· we didn’t say that the integral of f / g is the integral of f divided by the integral of g; unless g is consanat it almost certainly isn’t so

#21: The Fundamental Theorem of Calculus; Derivatives of Polynomials

· {F(t_i), F(0)} for f(t) piecewise linear

· where is derivative of cubic polynomial greater than given value?

· altitude y(t) quadratic, find velocity and acceleration functions, questions about position, velocity, acceleration

· dV/dr for volume of sphere

· , questions about position, velocity, acceleration

· dV/dr for volume of sphere

· derivative of exponential function from definition of e

· slope = altitude for y = e^x

· derivative of y = a^x

Derivatives of Basic Functions

The Chain Rule

Derivative of Inverse Functions

Implicit Differentiation

Optimization