Module 1, Assignments 0 - 6

Major Quiz over Module 1 assigned in Assignment 09

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·         Calculus I Week I

·         Design of the Course

Understand selected subsets well enough to identify useful members of the power set

Define subsets

Identify subsets related to key situations

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Class Notes #01 - 03

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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:  Know and apply the following to answer questions and solve problems.

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.  Solve equations of the form f(x) = c, where f(x) is a linear, piecewise linear or quadratic expression.

Technical definition:

Relate{ x | x is a solution to f(x) = c } U {  f(x) a linear function, c constant, m slope, b the y intercept, f(c) }

Relate{ x | x is a solution to f(x) = c } U {  f(x) a piecewise linear function, c constant, f(c)}

Relate{ x | x is a solution to f(x) = c } U {  f(x) a quadratic function, c constant, f(c) }

3.  Solve inequalities of the form a <= f(x) = b, where f(x) is a linear, piecewise linear or quadratic expression.

Technical definition:

Relate { x | x is a solution to a < f(x) < b } U {  f(x) a linear function, a constant, b constant, f(a), f(c) }

Relate { x | x is a solution to a < f(x) < b } U {  f(x) a quadratic function, a constant, b constant }

4.  Find the equation of a straight line given two points on the line, or given intercepts, or given a single point and its slope.

Technical Definition:

{ (x_1, y_1), (x_2, y_2), slope, equation of linear function, equation of line, (x, y) on line, y intercept }

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Modeling Project #1

simulated flow data

View Calculus I clips on disk 1, listed in the HTML file as Disk 1 (Gen 1). 

think about questions posed in documentation

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Class Notes #04

#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:  Know and apply the following to answer questions and solve problems.

Identical to the goal stated for the Assignment 1, except for underlined additions.

1:  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}

 2:  Relate a linear absolute value equation or inequality to points and/or the intervals of solution and to graphed point an/or intervals of the x axis.

Technical definition: 

For linear f(x): Relate { | f(x) | = a, | f(x) | < a, | f(x) | > a, point(s) on x axis, interval(s) on x axis}

 3.  Find the midpoint between two points of the x axis, given numerically or symbolically.

Technically:

Relate { x_1, x_2, x_mid = midpoint between x_1 and x_2 } for numerical or symbolic x_1, x_2, midpoint

 

 

Applications

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Objectives:  Know and apply the following to answer questions and solve problems.

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

• erivative 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.

3.  Relate the following:

• an algebraic expression involving exponents and radicals

• the simplified form of expression

• the factored form of expression

• the expanded form of expression

• equivalent forms of expression

• typewriter form of expression (‘order-of-operations form’)

• domain of expression if of one variable

4.  Applications and interpretations of the three listed objectives.

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Week 2 Quiz #2

Modeling Project #2 including exercises 1-14

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Objectives:  Know and apply the following to answer questions and solve problems.

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

• polynomial f(x)

• factored form

• expanded form

• real zeros

• large-|x| behavior

• value for given x

• solutions of f(x) = c

• interval(s) of definition of sqrt(f(x))

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Week 2 Quiz #1

Modeling Project #2 completing all remaining exercises

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Class Notes #05

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#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:  Know and apply the following to answer questions and solve problems.

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} U

interpretation

2.  Simplify rational expressions.

Technically:

Relate the following:

{ rational expressions f(x), g(x) } U {addition or subtraction or multiplication or division} U {exponentiation operator and exponent) } U {simplified form}

Interpret for y = depth of water in a container, t = clock time.

Interpret for y = temperature of an object, t = clock time.

Interpret for y = illumination, t = distance from source.

Interpret for related y and t quantities as specified.

Identify and continue pattern of slopes, if identifiable pattern exists.

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.

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Week 3 Quiz #1

Week 3 Quiz #2

Modeling Project #3

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Linked Outline of Introductory Topics through Major Quiz

Class Notes #06

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

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

Objectives:  Know and apply the following to answer questions and solve problems.

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.

{p, y = x^p} U {basic points (x, y) | x = -1, 0, 1/2, 1, 2} U {graph of points} U {f(x) = y = 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)}

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.  provide link

Applications

Module 2, Assignments 7 - 12

Test #1 over Module 2 to be completed within about a week of completing Module 2.

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Week 3 Quiz #1

Week 3 Quiz #2

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Inverse Functions and Logarithms

Class Notes #07-08

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#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:  Know and apply the following to answer questions and solve problems.

1.  Given two graphs find their coordinate-axis intercepts and their intersections points.

2.  Given cost and revenue functions find the break-even point.

3.  Given supply and demand curves find the equilibrium point.

4.  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

5.  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)

6.  Relate

• equation of circle

• graph of circle

• center of circle

• radius of circle

• three points on circle

• two points on diameter of circle

Relate

• 7.  y = a x^2 + b x + c equation of parabola

• y = a (x-h)^2 + k equation of parabola

• graph of parabola

• vertex and zeros of parabola 

• quadratic mathematical models

8.  Relate

• set of data points

• hand-drawn scatter plot

• hand-drawn trendline

• coordinates x_1 and y_1 of

9.  Relate

• hand-sketched scatter plot of y vs. x data points

• hand-sketched linear trendline

• two selected points on the trendline

• equation of trendline

10.  Relate

• two points in the coordinate plane

• the slope of the straight line segment joining the points

• the equation of the straight line through the two points

• the general form of a point (x, y) on the line

• the y intercept of the line

Technically:  Relate { (x_1, y_1), (x_2, y_2), slope, equation of linear function, equation of line, (x, y) on line, y intercept }

11.  Relate

• equation y = f(x) of linear or quadratic function

• equation y = g(x) of linear or quadratic function

• graph of y = f(x)

• graph of y = g(x)

• points of intersection of two graphs, determined algebraically

• points of intersection of two graphs, determined graphically

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Week 3 Quiz #3

Week 4 Quiz #1

Week 4 Quiz #2

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Class Notes #09
Include Composite Functions; ignore Text problems, Natural Logarithms, Numerical Solutions to Differential Equations

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#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:  Know and apply the following to answer questions and solve problems.

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.  From the equation of a line, find the equation of the line through a given point which is perpendicular or parallel to the original line.

Technically:  Relate { (x_1, y_1), (x_2, y_2), slope, equation of linear function, equation of line, (x, y) on line, x intercept, y intercept, equation of perpendicular line, point of intersection of line and perpendicular line, equation of line in standard form }

3.  Relate in the context of the declining value of an object with respect to time

• original value

• useful life

• average rate of change of value

• value at given time

• linear function modeling value vs. time

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Week 4 Quiz #3

text_09

 Complete Major Quiz

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  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

2.  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

{ 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 averaged with rate at predicted final point

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See Tests to familiarize yourself with the Testing page.

Focus Questions  

Class Notes #11

ignore trigonometric functions--look at only the last paragraphs above the last figure and below the next-to-last figure

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#11:    Trigonometric Functions; Brief Intro. to Polynomials

·         modeling by trigonometric functions

·         polynomials

Objectives:  Know and apply the following to answer questions and solve problems.

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

• 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

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qa_11:  rules for basic derivatives

query_11: Rational Functions, Continuity; intervals of cont rat fn; fn split def on >=2 intervals; revenue for franchise;

1.7 continuity, limits etc.

Class Notes #12

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#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:  Know and apply the following to answer questions and solve problems.

1.  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

2.  Relate using definition of continuity

• f(t) a linear, quadratic, polynomial, exponential, power, sine, cosine, rational function

• points of discontinuity

• intervals of discontinuity

• intervals of continuity

• continuity and discontinuity the real world

• applications

3.  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)

4.  State, evaluate and demonstrate examples showing the relationship between differentiability and continuity

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 qa_12:  chain rule

query_12:  The Derivative; find deriv algebraically; binom exp (x + `dx); how deriv used to find tan line; find tan line; where is fn diff;   

sketch pos fn st vAve = slope(5); graph put properties at A, B, C, … in order; sin(3x) / x; interval st diff between estimate f(x) and actual is < .01; interpret f(4) / 4; roc of ln(cos(x)) near pi/6;

Class Notes #13

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#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

\

Objectives:  Know and apply the following to answer questions and solve problems.

1.  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 apply the greatest-integer function.

3.  Solve problems involving compound interest.

4.  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 from | t - t_0 |

Module 3, Assignments 13 - 21

Test #2 over Module 3 to be completed within about a week of completing Module 3.

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 qa_13:  013.  Applications of the Chain Rule  exp temp vs t; wt vs. t; sin ferris wheel, rational fn + sqrt gpa; roc of all, effect of `dt = ½ on last

query_13:  The Derivative Function; rules for deriv; appl (e.g. roc and value -> fn -> deriv = roc);

graph matching bus, car, etc.; price and quantity sold f(150) = 2000; v vs t with, without chute

Class Notes #14

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#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:  Know and apply the following to answer questions and solve problems.

1.  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

Applications including exponential temperature vs. clock time, exponential weight vs. clock time, sine-function model of height vs. clock time on a Ferris wheel, grade point average as a sum of a rational and power function of study time; once more interpretation of derivative as rate of change of function with respect to variable

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  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.  Calculate, apply and interpret derivatives in the context of motion and marginality

3.  Relate

• average rate of change on interval

• instantaneous rates at endpoints and midpoint

• approximation error and `dt

{

Marginal cost

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Class Notes #18

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#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:  Know and apply the following to answer questions and solve problems.

1. 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

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Class Notes #22

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#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:  Know and apply the following to answer questions and solve problems.

1.  Find and simplify derivatives using product and quotient rules and combinations of these rules, and apply the results to real-world situations.

2.  Find the equation of the tangent line to a given curve at a given point using implicit differentiation.

Applications

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Class Notes #19, 20, 21

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#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

Objectives:  Know and apply the following to answer questions and solve problems.

(nothing new)

Applications

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Class Notes #15, 16, 17, 23

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#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

#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

Objectives:  Know and apply the following to answer questions and solve problems.

(nothing new)

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Class Notes #15, 16

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#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

Objectives:  Know and apply the following to answer questions and solve problems.

1.  Relate  function f(x) and its higher derivatives

Technically:  Relate {f(x), f ‘ (x), f ‘’(x), … }

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query_20:  Implicit Differentiation;

Class Notes #26, 27, 28

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#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:  Know and apply the following to answer questions and solve problems.

1.  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

Applications

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  Relate

• equation f(x,y) = 0

• equation df/dt = 0

• values of x and y

• dx/dt

• dy/dt

• situation relating x(t) and y(t)

• interpretation of situation relating x(t) and y(t)

Module 4, Assignments 22 - 29

Test #3 over Module 4 to be completed within about a week of completing Module 4.

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  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)

2.  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)

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  Relate

• function y = f(x)

• function y = f ' (x)

• equation f ' (x) = 0

• points of horizontal tangency

• graph of y = f(x)

2.  Relate

• price function

• demand function

• profit function

• interval of definition for profit function

• maximum of profit function

• price for which profit function is maximized

y = f(x), df/dx, critical values, x intervals where f(x) is increasing, x intervals on which f(x) is decreasing, x intervals on which f ‘ (x) is positive, x intervals on which f ‘ (x) is negative, 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, relative maxima of f, relative minima of f, inflection points of f, interval a <= x <= b and absolute extrema of f on this interval}

{price function, demand function, profit function, interval of definition for profit function, maximum of profit function, price for which profit function is maximized}

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Objectives:  Know and apply the following to answer questions and solve problems.

Relate

• y = f(x)

• df/dx

• critical values

• x intervals on which f(x) is increasing

• x intervals on which f(x) is decreasing

• 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

• points of inflection of f(x)

• 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

• relative maxima of f, relative minima of f

• inflection points of f

• interval a <= x <= b and absolute extrema of f on this interval

• 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

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Class Notes #29, 30, 31

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#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

#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:  Know and apply the following to answer questions and solve problems.

1.  Solve optimization problems related to real-world situations and systems.

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  Relate

• price function

• demand function

• profit function

• interval of definition for profit function

• maximum of profit function

• price for which profit function is maximized

• marginal profit

• producer surplus

• price elasticity of demand

• , …

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  Relate

• y = f(x) rational function

• vertical asymptotes

• limits at vertical asymptotes

• horizontal asymptotes

• slant asymptotes

• limits at +-infinity

• intercepts

• graph

• interpretation

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Objectives:  Know and apply the following to answer questions and solve problems.

1.  Relate

• y = f(x)

• df/dx

• critical values

• x intervals on which f(x) is increasing

• x intervals on which f(x) is decreasing

• 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

• points of inflection of f(x)

• 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

• relative maxima of f

• relative minima of f

• inflection points of f

• interval a <= x <= b and absolute extrema of f on this interval

• 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

• vertical asymptotes

• horizontal asymptotes

• slant asymptotes

• graph of y = f(x)

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Class Notes #36

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#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:  Know and apply the following to answer questions and solve problems.

1.  Relate

• q = f(x)

• x_0

• f ‘ (x)

• df

• `dx

• `df

• tangent line

• equation of tangent line

• profit function

• cost function

• revenue function

• marginal q

• change in q near x_0 estimated by rate of change at x_0

• change in q near x_0 estimated by differential

• change in q near x_0 estimated by tangent line

• marginal analysis where q is typically profit or cost or revenue

• real-world applications