MATH 160 Chapter 4, 5.1-5.3
Spring Semester, 2015
Chapter 4 Applications of Derivatives
(Section 4.1 was on Exam 2. Since later sections rely so much on the material in Sec. 4.1,
this section has content related to Exam 3.)
Sec. 4.1 Extreme Values of Functions (Repeat of material from Exam 2, necessary for Sec
4.3 and later)
___ • Write complete definitions of the following words and phrases:
(a) absolute maximum (aka global maximum) of a function (pg 185)
(b) absolute minimum (aka global minimum) of a function (pg 185)
(c) local maximum (aka relative maximum) (pg 187)
(e) local minimum (aka relative minimum) (pg 187)
(f) critical point (of a function) (pg 189).
Illustrate each of the above by sketching a graph.
Given a sketch of the graph of a function, identify and label all of the above.
(Notice that whether a function has absolute and local extrema depends on the domain of
the function
as well as the equation/formula for the function.)
Study Suggestion: Study Sec. 4.1.
Representative homework problems; Page 190-191; #1 – 10. Page 192 #77 – 80.
___ • Write a complete statement of the First Derivative Theorem for Local Extreme Values
(Theorem 2, pg 188).
Use a graph to explain in terms of secant and tangent lines how to see that this theorem is
plausible
(short of a formal proof).
Give examples to show that a function might or might not to have a local extremum at a
point where its derivative is 0.
Study Suggestion: Study Theorem 2 and the discussion of Finding Extrema in Sec. 4.1.
___ • Explain the connection between critical points and local extrema of a function.
Find the critical points, domain endpoints, and absolute and local extrema of a given
funtion.
Representative homework problems: Page 191; #11 – 14. Page 192; 49 – 66.
___ • Describe a step-by-step procedure for finding the absolute extrema of a continuous function
on a closed interval of finite length (summarized in the box below the middle of page 188).
Use this procedure to find the absolute maximum and minimum values of a given function
and the points where these extreme values are attained.
Representative homework problems: Page 191; #15 – 40.
___ • Give examples to show that a function can fail to have a local extremum at a critical point
and at an end point of its domain.
Representative homework problems: Pages 192; #67, 68, 69.
Sec. 4.3 Monotonic Functions and the First Derivative Test
___ • Write a complete definition of what it means for a function to be increasing and for a
function to be decreasing on an interval (Chapter 1, page 6).
Explain how to use the derivative to determine the intervals on which a function is
increasing and on which it is decreasing (pg 199, Corollary 3 sometimes called the First
Derivative Test for Monotonic Functions).
Explain how to use the first derivative to determine whether a function has a local
maximum, a local minimum, or neither at its critical points and the end points of its
domain. Illustrate your explanations with graphs (pg 201 First Derivative Test for Local
Extrema).
(Study Suggestion: Study Sec. 4.3.)
Representative homework problems: Pages 203; #15 – 18. Page 204; #63 – 66.
___ • Given the derivative f ′ ( x ) of a function f ( x ) (which is not given), find the critical points
of f ( x ), the intervals where f ( x ) is increasing, the intervals where f ( x ) is decreasing, and
the points where f ( x ) assumes local minimum or local maximum values. Explain how you
found these intervals and points from f ′ ( x ). Sketch a possible graph of the function y =
f(x) .
Representative homework problems: Pages 203; #1 – 14.
___ • Given a function defined by an expression y = f(x), calculate and analyze its first
derivatives (using algebra, not a calculator) to find the critical points of the function.
Knowing the critical points of a function, use the First Derivative Test for Monotonic
Functions (page 199, Corollary 3,) to determine the intervals where the function is
increasing and the intervals where the function is decreasing.
Use the First Derivative Test for Local Extrema (page 201) to determine whether the
function has a local maximum, a local minimum, or neither at each critical point and, if
applicable, at the end point of its domain.
Representative homework problems: Pages 203; #19 – 52, Page 204; #61, 62, 68, 69.
Sec. 4.4 Concavity and Curve Sketching
___ • Given the graph of a function, identify and indicate
(a) the intervals where the function is increasing,
(b) the intervals where the function is decreasing,
(c) the critical points,
(d) the intervals where the function is concave up,
(e) the intervals where the function is concave down, and
(f) the inflection points.
Study Suggestion: Read the examples in Section 4.4.
Representative homework problems: Page 212; #1 – 8. Page 214; #97, 98.
___ • Given a function defined by an expression y = f(x) (which may involve constants
designated by letters), analyze its first and second derivatives to determine
(a) the intervals where the function is increasing,
(b) the intervals where the function is decreasing,
(c) the critical points,
(d) the intervals where the function is concave up,
(e) the intervals where the function is concave down, and
(f) the inflection points.
If possible, use this information to sketch an accurate graph of the function. Then graph the
function on your calculator and verify that your conclusion in (a) – (f) and your graph are
correct.
Representative homework problems: Select from Page 211; #9 – 48. Page 215; 105 – 107,
111, 112.
___ • State the Second Derivative Test for Local Extrema.
Explain how to see that the sign of the second derivative reliably indicates whether a
function has a local maximum or local minimum at a point where the first derivative is 0.
Give examples showing that if the first and second derivative are both zero at a point x = c,
then the function may or may not have a local extremum at the point x = c.
___ • Sketch the graph of a function whose first and/or second derivatives have certain properties.
Properties may be given symbolically, algebraically, or graphically or you may have to
figure them out from an expression y = f(x) for the function or for its derivative.
Representative homework problems: Pages 213; #71 – 74, 101 – 104.
Sec. 4.5 Applied Optimization Problems
___ 1. Solve optimization problems similar to the exercises in this section of the textbook.
Representative exercises: Pages 221 - 225; #1 – 42. Pages 244 – 245; #51– 60.
NOTE: You are required to know the Pythagorean Theorem; properties of similar
triangles; formulas for the perimeters and areas of simple plane figures; formulas for the
volumes of spheres, cylinders, and a rectangular box; and the definitions of the
trigonometric functions in terms of right triangles and in terms of the unit circle.
Sec. 4.7 Antiderivatives
___1. Explain what is meant by an antiderivative and by the indefinite integral for a function.
What is the difference, if any, between an antiderivative and the indefinite integral of a
function?
Explain why an arbitrary constant is added in the indefinite integral.
___2. Determine whether a given function y = g(x) or family of functions y = g(x) + C is an
antiderivative or the indefinite integral for a given function y = f(x).
Representative exercises: Page 239; #57 – 62 and #63 – 68
___3. Find the indefinite integral of functions that can be written as sums/differences of constant
multiples of functions whose antiderivatives are known (i.e. functions listed in Table 4.2,
pg 234) by algebraic manipulations, perhaps using trigonometric identities. (You are
expected to know the basic trigonometric identities, how to represent all six trig functions
in terms of the sine and cosine, the Pythagorean identity that relates sine and cosine, and
the double angle formula for the sine.)
Representative exercises: Pages 238 - 239; #1 – 16 and #17 – 56. Page 245; #63 – 78.
___4. Interpret the general solution to a first order initial value problem
dy
= g(x) graphically.
dx
Explain graphically how an initial value selects one solution from the infinite number of
possibilities.
Representative exercises: Pages 240 – 241 ; #69 & 70 and #93 – 96.
___ 5.Solve initial value problems
dny
dx n
= g(x), 1 < n < 3 where g(x) can easily be
antidifferentiated.
Representative exercises: Page 240; #71 – 90, #97 – 101, #103, #104. Page 245; #79 –
82.
Chapter 5 Integration
Sec 5.1 Area and Estimating with Finite Sums
___1. Use finite sums to calculate reasonable estimates for
• the area of a region enclosed by the graph of a function y = f(x), the x-axis, and vertical
lines y = a and y = b, and
• the distance a body travels during a given time interval from a graph or table of its
velocity.
Representative exercises: Select from Pages 257 – 258; #1 – 8, #9 – 14. Page 301; #1, 2.
___2. • From a table giving the velocity of a moving body at various times (not necessarily equal
spaced), calculate a reasonable estimate (upper, lower, right, left, midpoint) for the
displacement (change in position) of the body over a given time interval.
(You are expected to use a calculator.)
• Given an equation v = f(t) for a velocity function (whose antiderivative can’t be found
easily), generate a table that gives the velocity of the moving body at various (equally
spaced) times, and use this table to calculate a reasonable estimate for the displacement
of the body over a given time interval.
(You are expected to use a calculator.)
• Explain why one might expect your calculations to produce a reasonable estimate for
the displacement.
Representative exercises: Select from Page 258; #15 – 18, #19 – 22.
Sec 5.2 Sigma Notation and Limits of Finite Sums
___1. A sum of the form Sn =
n
∑ f (ck )Δxk
is called a Riemann sum for f on the interval [a,
k =1
b].
Use words and pictures to explain how the interval [a, b] connects to the Riemann sum
and what each of the symbols n,
n
∑
, f , xk, Dxk, and ck in this expression means.
k =1
___2. Given a function y = f(x) defined on an interval [a, b] and given a partition a = x0 < x1
< … < xn-1 < xn = b of the interval, choose appropriate evaluation points c1, c2, c3, … , cn
, write an expression for the associated Riemann sum (in expanded form), and calculate
the numerical value of the Riemann sum.
Representative exercises: Select from Page 265; #33 – 36.
___3. Write an expression/equation that defines the Riemann integral in terms of Riemann sums.
Explain (using words and pictures) what this definition means and why it is necessary to
take a limit. (see HW10)
___4. Explain why the following integrals are equivalent:
b
!
𝑓(𝑢) 𝑑𝑢= f ( x)
!
a
∫
dx
(i.e. x is a dummy variable)
___5. Use a Riemann sum to calculate a reasonably accurate approximate value for a given
(definite) integral. (You are expected to use a calculator.)
Sec 5.3 The Definite Integral
___1. Use standard area formulas for triangles, rectangles, quarter circles, half circles, etc. to
evaluate definite integrals by identifying the integral with the area(s) of familiar shape(s).
(You must know these standard area formulas.)
___2. Use properties of the definite integral summarized in Table 5.6 (Page 270) entries #3 – #5
to evaluate definite integrals by rewriting the integral as a combination of integrals that
either have given values or can be evaluated by interpreting the integral as (positive or
negative) area.
Representative exercises: Page 275; select from problems #9 – 28.
___3. Estimate integrals by using inequalities #6 and #7 in Table 5.6 (Page 270).
Suggested problems: Select from Page 276; #71 – 80. (In #71 & 72 assume a < b.)