Module 1: Review and Exploration of Rates of Change

Module 1 Objectives

1. Given a set of data construct a graph, an approximate trendline, a selected set of points and an appropriate mathematical model of the data.

2. 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.
3. Apply the difference quotient and limits to get the rate-of-change function for a given quadratic function.
4. Solve problems requiring the application of properties of basic functions.  (Basic functions include quadratic, exponential, power, trigonometric, polynomial and rational functions).
5. Graph generalized basic functions using transformations and symmetry.
6. 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.
7. Apply proportionality and variation to solve problems.
8. Construct combinations of basic functions, including sums, products, quotients and composites.
9. Test a given basic function on a given interval for invertibility, and if possible find its inverse.
10. Solve algebraic, trigonometric, exponential and logarithmic equations.
11. Construct a numerical solution of a given rate-of-change equation.
12. Find the given limit of a given expression.
13. Determine intervals of continuity for a given function.

Detailed assignment-level objectives for Module 1 are included below.

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.

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}

3.  Where the sequence of average slopes of a y vs. t graph, over a series of intervals, has an identifiable pattern, identify and con   bntinue the pattern and use to project new graph points.

4.  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.

5.  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

6.  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

7.  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}

8. 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. 

9. 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.

10. 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.

11.  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

12.  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 }

 13.  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

14.  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

15.  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

16.  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

17.  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

18.  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)}

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

20.  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

21.  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)

22.  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.

23.  Test functions for symmetry about the x or y axis.

24.  Given the graphs of two functions construct the graph of their sum, product, quotient or specified composite.

25.  Given a function determine whether it has an inverse and if so find the inverse.

26.  Given the graph of a function determine whether it has an inverse and if so find the inverse.

27.  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

28.  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

29.  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

30.  Solve exponential and logarithmic equations.

31.  Analyze exponential and logarithmic functions graphically and analytically.

32.  Use exponential and logarithmic functions to model real-world phenomena.

33.  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.

23.  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.

35.  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

36.  Solve trigonometric equations.

37.  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)

38.  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

39.  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 |

40.  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 |

41.  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

42.  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

43.  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

44.  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)

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

Module-level Objectives

1. Apply the difference quotient and limits to obtain formulas for derivatives of power functions.
2. Apply the rules of differentiation to calculate derivatives of various combinations of basic functions.
3. Construct graphs relating given functions to their first and second derivatives and antiderivatives.
4. Interpret the behavior of a given graph, given the meanings of the dependent and independent variables.
5. Using graphical, numerical, verbal and symbolic representations, explain the Fundamental Theorem of Calculus
6. Construct integrals of sums and differences of functions using Riemann Sums.
7. Solve problems by applying the tangent line and the differential.
8. Interpretation the indefinite and definite integrals of a function, given the meanings of the dependent and independent variables.

Detailed assignment-level objectives for Module 2 are included below.

Assignments 12 - 17

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, procedures and insights 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, procedures and insights 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, procedures and insights 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

7.  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, procedures and insights 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

8.  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, procedures and insights 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)

9.  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)

10.  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, procedures and insights 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)

10.  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, procedures and insights 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

12.  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, procedures and insights 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)

13.  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, procedures and insights to Relate

• average rate of change on interval

• instantaneous rates at endpoints and midpoint

• approximation error and `dt

14.  Given the y = f(x), defined analytically, find the functions f ' (x) and f '' (x)

15.  Explain the interpretation of f ''(x) as a rate at which a rate changes.

16.  Explain the effect of the second derivative of f(x) on the concavity of the graph of y = f(x).

17.  Given the position function for a particle find its velocity and acceleration functions.

18.  Interpret the motion of a particle analytically, numerically and graphically in terms of its position, velocity and acceleration functions.

19.  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, procedures and insights 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

20.  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, procedures and insights 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

21. 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, procedures and insights 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

 

22. 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, procedures and insights 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]

23.  Using product and quotient rules and combinations of these rules, calculate and simplify derivatives, and apply the results to real-world situations.

24.  Analytically determine the equation of the tangent line to a given curve at a given point using implicit differentiation.

25.  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

26.  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

27.  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.

28.  Explain the meaning of the antiderivative function of f(x) as change-in-quantity function for the quantity represented by f(x).

29. 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.

30. 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.

31. 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.

32. Apply the formulas for derivatives of power and exponential functions to problems involving functions build using sums, differences and constant multiples of these functions.

 

Module-level Objectives

1. Apply the difference quotient and limits to derive formulas for derivatives of exponential, sine and cosine functions.
2. Explain l'Hopital's Rule in terms of local linearizations, and apply to find limits and dominance.

3. Apply the tangent line approximation and local linearization to mathematical and real-world applications.

4. Apply implicit differentiation to obtain information about implicitly defined functions.
5. Apply the chain rule to find the derivative of the inverse of a given function.
6. Apply derivatives and other techniques to graph a curve given its parameterization, and investigate its properties.
7. Apply the derivative to solve problems involving optimization, graphing and rates.
8. Construct mathematical models of mathematical and real-world phenomena.
9. Approximate derivatives and integrals, and estimate approximation errors.

Detailed assignment-level objectives for Module 3 are included below.

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.

3.  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.

4.  Apply the chain rule to find derivatives of composite functions.

5.  Apply all rules of differentiation to expressions involving trigonometric functions.

6.  Apply trigonometric functions and derivatives of the resulting expressions to various mathematical and real-world applications.

7.  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

8.  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)

9.  Use the chain rule to find expressions for the derivatives of the inverses of given functions.

10.  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.

11.  Apply the definitions of the hyperbolic sine, hyperbolic cosine and hyperbolic tangent functions in order to find the derivatives of these functions.

12.  Apply the tangent line approximation and local linearization to mathematical and real-world applications.

13.  Estimate for function f(x) the error at point x of the local linearization of f(x) about (a, f(a)).

14.  State and apply l'Hopital's rule.

15.  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.

16.  Use l'Hopital's rule to establish dominance between two functions with infinite limits.

17.  Given the parameterization x(t), y(t) of a curve in two dimensions, sketch the curve.

18.  Given an ellipse or a straight line in the plane, analytically determine a parameterization.

29.  Given a geometric definition of a curve in the plane, analytically determine a parameterization.

20.  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.

21.  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, procedures and insights

Relate

• function y = f(x)

• function y = f ' (x)

• equation f ' (x) = 0

• points of horizontal tangency

• graph of y = f(x)

22. 

Relate

• function y = f(x)

• function y = f ' (x)

• equation f ' (x) = 0

• points of horizontal tangency

• graph of y = f(x)

22.  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

23.  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.

24.  Within the context of a mathematical or real-world application, relate the behavior of a function family to its parameters.

Module-level Objectives

1. Apply optimization techniques to price, cost and demand functions to optimize revenue and/or profit.

2. Apply the derivative and limits to the graphing of functions with asymptotes, including but not limited to rational functions.

3. Construct proofs of mathematical statements involving limits, differentiability and continuity.

4. Construct proof of mathematical statements involving the intermediate value theorem and/or the nested interval theorem.

5. Solve real-world optimization problems.

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, procedures and insights

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

2.  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, procedures and insights to

Relate

• y = f(x) rational function

• vertical asymptotes

• limits at vertical asymptotes

• horizontal asymptotes

• slant asymptotes

• limits at +-infinity

• intercepts

• graph

• interpretation

3.  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.

4.  Solve optimization problems related to mathematical or real-world applications.

5.  Find upper and lower bounds on a given function on a given interval.

6. 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)

7.  Use to rigorously prove statements about limits, continuity and differentiability:

• definition and properties of a limit

• definition and properties of continuity

• definition of differentiability

8.  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.

9.  Solve optimization problems related to mathematical or real-world applications.

10.  Find upper and lower bounds on a given function on a given interval.

11.  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, procedures and insights 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