Download Ch5Review - AP Calculus AB/BC Overview

5–1
Chapter 5
Review/Test Problems
1.
Use the first derivative test to classify the critical points of f (x) = x 2 – 4x + 5 .
2.
Suppose f (x) = 2x3 – 9x2 + 12x + 2 .
Find the global maximum and minimum of f (x) on the interval [0, 3].
3.
Suppose f (x) = x 2/ 3(x 2  25) .
a) Find f (x) .
b) Find all critical numbers of f .
c) Locate any local maximum(s) and minimum(s) of f .
d) On what intervals is f decreasing?
4.
Assume that p is a positive constant and that f (x) = x 3/ 7 – px10/ 7 .
a) Find f (x) .
b) Find all critical numbers of f (your answers may involve the constant p ).
c) Classify each critical number as leading to a local maximum, local minimum, or neither,
and say how you reach each conclusion.
d) Does f have a global maximum on its domain? Use derivative information to justify
your answer.
5.
Using this graph of f , answer, if
possible, the questions below.
graph of f
-3
3
a) On what interval(s) is f increasing?
b) On what interval(s) is f concave
down?
c) At what x does f have a local
maximum?
6.
2
1
-4
-3
-2
-1
1
2
3
4
-1
-2
4
2
Use the second derivative test to classify the critical points of f (x) = x  4x .
In problems 7–9, the derivative of a function is given. Use f  to determine
a) the interval(s) on which f is increasing;
b) the interval(s) on which f is concave up;
c) the inputs x (if any) at which f has a local maximum;
d) the inputs x (if any) at which f has a point of inflection.
7.
f (x) = (x + 3)(x – 2)
8.
f (x) = (x + 3) 2 (x – 2)
9.
f (x) = (x + 3) 2 (x – 2)2
5–2
In problems 10–13, the second derivative of a function is given. Use f  to determine
a) the interval(s) on which f is concave up;
b) the inputs x (if any) at which f has a point of inflection.
10.
f (x) = 2 – x
11.
f (x) = (2 – x)(x + 1)
12.
f (x)
 = (2 – x) (x + 1)
13.
f (x)
 = (2 – x) (x + 1)
14.
Let f (x) =
2
2
2
3
2x
x .
e
a) Find f  and f  .
b) Find all critical numbers for f .
c) Classify each critical number in part b) as a local maximum, local minimum, or
neither. Justify your answers.
d) Locate all inflection points of f . Justify your answers.
e) For what values of x is f concave down and increasing?
15.
Evaluate the following limits.
5
2
3x + 2x + 5
5
4
3
x 7x – 5x + 9x
a) lim
c) lim
x 
3x
x 2 – 5x  6
2
x – 5x
3
x x – 25x
b) lim
5x 2 + cos x
2
x 2x + sin x
d) lim
16.
Find the horizontal and vertical asymptotes of f (x) =
17.
Let g(x) =
a)
b)
c)
d)
18.
2
x – 5x
.
2
x – 25
2
(x – 4x + 3)( x – 2)
.
3
x –x
Determine the domain of g .
Find all zeros of g .
Locate all vertical asymptotes of g .
Locate all horizontal asymptotes of g .
A tank with a rectangular base and rectangular sides is to be open at the top. It is to be
constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the
tank costs $10 per square meter for the base and $5 per square meter for the sides, what is
the cost of the least expensive tank? Justify your answer.
19.
5–3
A person has 340 yards of fencing for enclosing two separate fields, one of which is to be
a rectangle twice as long as it is wide and the other a square. The square field must contain at
least 100 square yards and the rectangular one must contain at least 800 square yards.
a) If x is the width of the rectangular field, what are the maximum and minimum
possible values of x ?
b) Set up a function A(x) that represents the total area enclosed by the two fields.
c) What is the greatest number of square yards that can be enclosed in the two fields?
Justify your answer.
20.
The top and bottom margins of a page
1.5
1
are each 1 2 inches and the side margins
are each 1 inch. The area of the printed
material is 30 square inches. What are
the dimensions of the page of smallest
possible area?
1
Printed Material
21.
Determine the total number of local maximum and minimum points of the function
whose derivative is given by f (x) = x(x – 3)2 (x + 1)4 .
22.
Let f (x) 
2
a + 2x
2
2
x –a
about f ?
2
where a > 0 is a constant. Which of the following is/are TRUE
a) The graph of f has two vertical asymptotes.
b) lim f (x) = – 1 .
x
c) lim f (x) = 2 .
x
2
23.
What point on the curve y = x + 4x + 3 is closest to the origin?
24.
5
2
Suppose f (x) = x – 3x + 4 and a = –2 and b = 2 . Find all numbers c such that
a < c < b and f (c) =
f (b) – f (a)
.
b –a
Approximate your answers to two decimal places.
In problems 25–28, determine whether f satisfies the hypotheses of the Mean Value Theorem on
the given interval [a, b] . If it does, find all numbers c in (a, b) such that
f (c) =
f (b) – f (a)
.
b–a
2
25.
f (x) = x + 2x – 2 ; [-2, 2]
26.
f (x) =
27.
f (x) =
28.
f (x) = x – 3x + 1 ; [-1, 2]
2
x –1
x–2
2
x –1
x –2
3
; [-2, 1]
; [0, 4]
5–4
29.
30.
Suppose that a function f has a derivative at all x and that f (a) = f (b), where a < b.
Show that f (c)  0 for at least one number c in the open interval (a,b).
A car is moving along a track in such a way that is velocity is given by the equation
2
V(t) = 21 + 18t – 3t centimeters per second (t  0).
a) During what time intervals is the car moving to the left?
b) If P(0) = 0 , write an expression for the position, P(t), of the car at any time, t, (t  0).
c) Determine the change in position of the car between t = 0 and t = 9.
d) At what time t between t  0 and t  9 does the instantaneous velocity of the car equal
the average velocity of the car on the same time interval?
In problems 31–36, find the general antiderivative of each function.
5
32.
f (x) = x – 3
5
34.
f (x) = 1+ e
36.
f (x) = sin x + sec x
31.
f (x) = x
33.
f (x) = 2
35.
f (x) =
37.
A ball is thrown upward from a height of 64 feet above the ground with an initial velocity
of 128 ft/sec. Determine when and with what speed the ball hits the ground.
38.
3
Investigate the family of curves y = x + ax + 1 for a = –3, –1, 0, 1, 3. Explain the
effect of changes in the parameter a .
39.
At each point P(x, f (x)) on the graph of a function f , the slope of the tangent line is
3
x +
given by 8x 3 +
3
1
x
1
2
x
x
2
. If the point A(1, 10) lies on the graph of f , determine f (x) ,
the given function.
40.
2
t seconds after the brakes are applied, the acceleration of a car is a(t) = – 6t – 4 ft/ sec .
If the car was moving at the speed of 32 ft/sec when the brakes were applied, how far
does it travel before coming to a stop?