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Calculus AB
Fall Semester Review Sheet
Free Response
1) Find f   x  and f   2 for the following functions:
a) f  x   3x
b) f  x   sin 2  x  cos  x 
c) f  x    2 x 2  1
3
d) f  x  
x 3
x 1

e) f  x   5x3 6 x2  2

4
x
f) f  x   sin  
4
g) f  x   x2  5
2) The graph of the function f shown below has a vertical tangent at the point
(2, 0) and horizontal tangents at the points (1, -1) and (3, 1). For what values in
the interval [-2, 4] is f not differentiable?
3) Given: G ( x) 
table below.
x
-1
H(x)
20
1 3
x  x , and H(x) is a continuous function with the values in the
4
0
22
1
26
2
35
3
49
4
66
a) Find the average rate of change of both G and H over the interval [-1, 4].
b) Find the instantaneous rate of change of G at x = 3.
c) Approximate the instantaneous rate of change of H at x = 3.
d) Write the equation of the tangent line to G(x) at x = 1.
y
 x 2  1, x  2
4) If f  x   
2 x  1, x  2
a) Graph f  x 
b) lim f ( x)  ____
x
x 2
c) lim f ( x)  ____
x 2
d) Is f ( x) continuous at x = 2?_____
e) d) Is f ( x) differentiable at x = 2?_____
f ( x  h)  f ( x )
h 0
h
a) Using the difference quotient, find f ( x ) if f ( x)  2 x 2  5x . Show your work.
5) Given: f ( x)  lim
b) Recognize the following difference quotient: lim
h 0
tan( x  h)  tan( x)
 ________
h
Multiple Choice
20 x 2  13x  5
6) lim

x 
5  4 x3
(A) -5
(B) ∞
(C) 0
(D) 5
(E) 1
7) Let f ( x)  x  5 x 5  9  , find
(A) 6409
f '(4) .
(B) 30720
(C) 30729
(D) 25609
(E) 30756
8) If y is a differentiable function of x , then the slope of the curve xy 2  2 y  4 y 3  6 at
the point where y  1 is
(A) 1
18
(B) 1
9) lim x sin
x 
(B) ∞
10) If y 
(E)
(D) 11
18
(E) 2
(C) nonexistent
(D) -1
(E) 1
x3
dy
, then
=
2  5x
dx
17  10 x
 2  5x 
(B)
2
13
 2  5x 
11) If y  x 2  16 , then
(A)
18
1

x
(A) 0
(A)
(C) 5
26
1
3
(C)
x 3
 2  5x 
2
(D)
17
 2  5x 
(E)
2
13
 2  5x 
d2y

dx 2
(B) 4(3x  16)
2
4  x 2  16 
2
(C)
2
16
x 2  16
(D)
2 x 2  16
x
2
 16 
3
2
16
x
2
 16 
3
2
12) The maximum value of the function f ( x)  x 4  4 x3  6 on the closed interval [1,4] is
(A) 1
(B) 0
(E) none of these
(C) 3
(D) 6
2
dy

dx
5  y2
(E)
2 xy  4
13) If xy 2  3x  4 y  2  0 and y is a differentiable function of x , then
(A) 
1 y2
2 xy
(B)
3
2y  4
(C)
3
2 xy  4
3  y2
2 xy  4
(D)
14) If f ( x) is continuous at the point where x  a , which of the following statements may
be false?
(A) lim f ( x) exists
(B) lim f ( x)  f (a)
(D) f (a ) is defined
(E) lim f ( x)  lim f ( x)
(C) f   a  exists
x a
xa
x a
x a
15) Which of the following functions is not everywhere continuous?
2
x
(A) y  x
(B) y  2
(C) y  x 2  8
(D) y  x 3
x 1
(E) y 
4
 x  1
2
16) The equation of the tangent to the curve of y  x 2  4 x at the point where the curve
crosses the y-axis is
(A) y  8 x  4
(B) y  4 x
(C) y  4
(D) y  4 x
(E) y  4 x  8
17) On the graph of y  f ( x) below, f ( x ) and f ( x) are both positive on which interval?
y
x
c
(A) 0< x < a
d
b
(B) b < x < c
e
a
(C) c < x < d
 x 2 , for x  1
18) If f ( x)  
then
2 x  1, for x  1
(A) f ( x) is not continuous at x = 1
(B) f ( x) is continuous at x = 1 but f (1) does not exist
(C) f (1) exists and equals 1
(D) d < x < e
(E) x > e
(D) f (1)  2
(E) lim f ( x ) does not exist
x 1
19) The curve y 
(A) x > 3
1 x
is concave up when
x 3
(B) 1 < x < 3
(C) x > 1
(D) x < 1
(E) x < 3
2x2
has
4  x2
two horizontal asymptotes
two horizontal asymptotes and one vertical asymptote
two vertical but no horizontal asymptotes
one horizontal and one vertical asymptote
one horizontal and two vertical asymptotes
20) The curve y 
(A)
(B)
(C)
(D)
(E)
x2  x
for all x except x = 1. In order for the function to be
x 1
continuous at x = 1, the value of f (1) must be
21) A function f ( x) equals
(A) 0
(B) 1
(E) none of these
(C) 2
(D) ∞
 x  4 3
, if x  5

22) Let f ( x)   x  5
and let f be continuous at x = 5. Then c 
 c
, if x  5

1
1
(A)
(B) 0
(C)
(D) 1
(E) 6
6
6
23) The y-intercept of the line tangent to the graph of y  ln x  x at x = 1 is
(A) -1.0
(B) -1.25
(C) -1.50
(D) -1.75
(E) -2.0
 x 2  36
, if x  6

24) Let f ( x)   x  6
Which of the following statements is (are) true?
 12 , if x  6

I. f is defined at x = 6.
II.
lim f ( x ) exists
x 6
III. f is continuous at x = 6.
(A) I only
(B) II only
(C) I and II only
(D) I, II, and III
(E) none of the statements are true
25) The number c satisfying the Mean Value Theorem for f ( x)  sin x on the interval
[1, 1.5] is, correct to three decimal places,
(A) 0.995
(B) 1.058
(C) 1.239
(D) 1.253
(E) 1.399
26) If f   x  exists on the closed interval [a, b], then it follows that
(A) f ( x) is constant on [a, b]
(B) there exists a number c in (a, b) such that f   c   0
(C) the function has a maximum on the open interval (a, b)
(D) the function has a minimum on the open interval (a, b)
(E) the Mean Value Theorem applies
27) The height of a rectangular box is 10 inches. Its length increases at the rate of
2in/sec; its width decreases at the rate of 4 in/sec. When the length is 8 inches and the
width is 6 inches, the volume of the box is changing, in cubic inches per second, at the
rate of
(A) 200
(B) 80
(C) -80
(D) -200
(E) -20
28) If lim f ( x)  L , where L is a finite number, then it follows that
x c
(A) f   c  exists
(B) f  x  is continuous at x = c
(D) f  c  is defined
(E) none of the preceding is necessarily true
(C) f (c)  L
29) A 26-foot ladder leans against a building so that its foot moves away from the
building at the rate of 3 ft/sec. When the foot of the ladder is 10 feet from the
building, the top is moving down at the rate of r ft/sec, where r is
(A)
46
3
(B)
3
4
30) Find f   x  if f ( x)  sin 3 4 x
(C)
5
4
(D)
5
2
(E)
4
5
(A) 4 cos3 4x
(B) 3sin 2 4 x cos 4 x
(C) cos 3 4x
(D) 12sin 2 4 x cos 4 x
(E) none of these
31) The function f  x   5x4  x5 is increasing on the following intervals
(A) (-∞, 0)
(B) (4, ∞)
(E) (-∞, 0) and (4, ∞)
(C) (-1/4, 0)
(D) (0,4)
(C) 1
(D) 3
3  x  2 x2
32) lim

x 
4x2  9
(A) -1/2
(B) ½
33) If x and y are Real numbers, the domain of the function f ( x) 
(A) all x except x = 2 or x = -2
(C) x  1
(D) x  1 or x  1
(B) all x except x = 4
(E) all Reals
(E) does not exist
x2  1
is
4 x
34) Find f   x  if f ( x)  sin x 2
(A) 2(cos x 2  2 x 2 sin x 2 )
(B) 4 x sin x 2
(D) 2(cos x 2  x sin x 2 )
(E) none of these
35) lim
x 3
(C) 2 x cos x 2
x3

x2  9
(A) +∞
(B) 0
(C) 1/6
(D) -∞
(E) nonexistent
36) The only function that does not satisfy the Mean Value Theorem on the interval
specified is
1
(A) f ( x)  x 2  2 x on [-3, 1]
(B) f ( x )  on [1, 3]
x
3
2
1
x x
(C) f ( x)    x on [-1,2]
(D) f ( x)  x  on [-1, 1]
x
3 2
(E) f ( x)  x
37) lim
x 2
(A) -∞
2
3
x2
x2
on [ 12 , 32 ]

(B) -1
(C) 1
(D) ∞
(E) nonexistent
38) Which statement is true?
(A) If f ( x) is continuous at x = c, then f   c  exists.
(B) If f   c   0 , then f has a local maximum or minimum at  c, f  c   .
(C) If f   c   0 , then f has an inflection point at  c, f  c   .
(D) If f is differentiable at x = c, then f is continuous at x = c.
(E) If f is continuous on (a, b), then f attains a maximum value on (a, b).
39) The y-intercept of the line tangent to the graph of y  2 x at x = -1 is
(A) -2.44
(B) -.15
(C) .15
(D) .64
(E) .85
40) Find any critical numbers of the function g (t )  t 7  t , t  7 .
14
14
(A) 0
(B) 
(C)
(D) Both A & B
(E) Both A & C
3
3
41) Locate the absolute extrema of the function f ( x)  x3  3x on the interval 0,3 .
(A)
(B)
(C)
(D)
(E)
Absolute Max: (3, 18) ; Absolute Min: (1, - 2)
Absolute Max: (-1, 2) ; Absolute Min: (1, - 2)
Absolute Max: (3, 18) ; No Absolute Min
No Absolute Max ; Absolute Min: (0, 0)
No Absolute Max or Absolute Min
42) Find the dimensions of the rectangle of maximum area bounded by the x and y axis
8 x
and the graph of f ( x) 
.
2
(A) length 3 ; width 2.5
(B) length 4 ; width 2
(C) length 1 ; width 3.5
(D) length 2 ; width 3
(E) None of the above
43) Compare dy and y for y  2 x 4  1 at x  1 with dx  x  0.07 .
(A) dy  0.5400; y  0.6219
(B) dy  0.5700; y  0.6218
(C) dy  0.5900; y  0.6217
(D) dy  0.5600; y  0.6216
44) Evaluate
d 2 x 1
3
dx
47) f(x) = 3x  7 find
45)
d
ln( x 2  x)
dx
46)
d
log 7 (5 x)
dx
d 1
f ( x)
dx
NOTE:
Given f   x  and f   x  , be sure you can sketch a graph of f  x  and indicate where the
function is increasing or decreasing and over what intervals it is concave up or concave
down.
STUDYING PAYS OFF!! Good Luck on the Exam! Happy Holidays!!!