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Module
1: Review and Exploration of Rates of Change
Module 1 Objectives
-
Given a set of data construct a graph, an
approximate trendline, a selected set of points and an
appropriate mathematical model of the data.
- Apply the rate of change definition to determine the
average rate of change of a quantity on an interval, or
given the average rate the change in the quantity on the
interval.
- Apply the difference quotient and limits to get the
rate-of-change function for a given quadratic function.
- Solve problems requiring the application of
properties of basic functions. (Basic functions
include quadratic, exponential, power, trigonometric,
polynomial and rational functions).
- Graph generalized basic functions using
transformations and symmetry.
- Using one or more trapezoids to approximate a
function determine the slopes and areas associated with
each, and the interpretation of each of these quantities.
- Apply proportionality and variation to solve
problems.
- Construct combinations of basic functions, including
sums, products, quotients and composites.
- Test a given basic function on a given interval for
invertibility, and if possible find its inverse.
- Solve algebraic, trigonometric, exponential and
logarithmic equations.
- Construct a numerical solution of a given
rate-of-change equation.
- Find the given limit of a given expression.
- Determine intervals of continuity for a given
function.
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 youre 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.
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:
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
-
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
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
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
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
45. State,
evaluate and demonstrate examples showing the relationship
between differentiability and continuity
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Module 2:
The Derivative and the Integral
Module-level Objectives
- Apply the difference quotient and limits to obtain
formulas for derivatives of power functions.
- Apply the rules of differentiation to calculate
derivatives of various combinations of basic functions.
- Construct graphs relating given functions to their
first and second derivatives and antiderivatives.
- Interpret the behavior of a given graph, given the
meanings of the dependent and independent variables.
- Using graphical, numerical, verbal and symbolic
representations, explain the Fundamental Theorem of Calculus
- Construct integrals of sums and differences of
functions using Riemann Sums.
- Solve problems by applying the tangent line and the
differential.
- Interpretation the indefinite and definite integrals
of a function, given the meanings of the dependent and
independent variables.
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
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
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:
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.
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