AP Calculus Review Day 7
C 1. If y  5( x 2) , then
3
dy

dx
( A) ( x3  2)5( x 2)
3
( B) 3x2 (ln 5)5( x 2)
3
NC 2. If f ( x)  3  x  2 , then f ′(2) is
(A) 3
(B) 1
(C) (3x2 )5( x 2)
( D) (ln 5)5( x 2)
(C) -1
(D) 2
3
3
( E) x3 (ln 5)5( x 2)
(E) nonexistent
NC 3. Let f and g be differentiable function such that f (1)=4, g(1)=3, f ′(3)=-5, f ′(1)=-4, g′(1)=-3, g′(3)=2.
If h(x)=f(g(x)), then h′(1)=
(A) –9
(B) 15
(C) 0
(D) -5
(E) -12
f ( x)  f ( a )
 0, which of the following must be true?
xa
( A) lim f ( x) does not exist (B) f (a) does not exist (C) f '(a)  0 ( D) f (a)  0 ( E) f ( x)is continuous at x=0
C 4. If f is a function such that lim
x a
x a
tan 2( x  h)  tan(2 x)
is
h 0
h
(A) 0
(B) 2cot(2x)
C 5. lim
(C) sec2(2x)
C 6. If f(x)=(2+3x)4, then the fourth derivative of f is
(A) 0
(B) 4!3
(C) 4!(34)
C 7.
(D) 2sec2(2x)
(E) nonexistent
(D) 4!(35)
(E) 4!(2+3x)
d 3ln x
e  
dx
(A) e3ln x
(B)
e3ln x
x
(C) x3
d2y
NC 8. If x y  yx  6, then 2 at the point (1,3) is
dx
(A) –18
(B) –6
(C) 6
2
(D) 3x2
(E) 3
2
(D) 12
(E) 18
C 9. Let f ( x)  x3  7 x 2  25x  39 and let g be the inverse function of f. What is the value of g′(0)?
(A) –1/25
(B) 1/25
(C) 1/10
(D) 10
(E) 25
3
1
C 10. If f ( x)  Arc tan   , then f ′(x)=
 x
1
x
x2
( A) 2
( B)
(C ) 2
x x
x 1
x2 1
( D)
1
x 1
( E)
2
1
x 1
2
e x  2, x  0
NC 11. If f ( x)  
is differentiable at x=0, then a+b=
ax

b
,
x

0

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
dy

dx
( B) (ln x)(2 x  ln x)
C 12. If y  x(ln x)2 , then
( A) 3(ln x)2
(C) (ln x)(2  ln x)
( D) (ln x)(2  x ln x)
( E ) (ln x)(1  ln x)
NC 13. Let f(x) be a continuous and differentiable function. The table below gives the value of f(x) and f ′(x),
1
the derivative of f(x), at several values. If g ( x) 
, what is the value of g′(2)?
f ( x)
1
1
1
x
1
2
3
4
( A) 
(C )
( D)
( E ) 16
 B 0
f(x)
-3
-8
-9
0
8
16
64
f ′(x)
3
-5
-4
3
16
2 2
NC 14. The slope of the tangent to the curve y x + y x = 6 at (2, 1) is
3
5
3
(A) 
(B) -1
(C) 
(D) 
(E) 0
2
14
14
NC 15. The maximum value of f ( x)  2 x 3  9 x 2  12 x  1on [-1,2] is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
NC 16. The graph of y  3x 2  x 3 has a relative maximum at
(A) (0,0) only
(B) (1,2) only
(C) (2,4) only
(E) (0,0) and (2,4)
C 17. Let f be a function whose derivative is given by f ' ( x) 
points does f(x) have in the interval 0  x  4 ?
(A) None
(B) One
(C) Two
(D) (4,-16) only
x
 sin(e 0.2 x ). How many relative maximum
15
(D) Three
8x  k
have a relative maximum at x=4?
x2
(B) –16
(C) 0
(D) 16
(E) More than three
C 18. For what value of k will
(A) –32
(E) 32
NC 19. The amount A(t) of a certain product is given by A(t )  4000  48(t  3)  4(t  3) 3 where t is in hours
since the beginning of the workday at 8am. At what time is the production increasing most rapidly?
(A) 8am
(B) 10am
(C) 11am
(D) Noon
(E) 1pm
C 19. A differentiable function f has the property that f (5) = 3 and f ′(5) = 4. What is the estimate for f (4.8)
using the local linear approximation for f at x = 5?
(A) 2.2
(B) 2.8
(C) 3.4
(D) 3.8
(E) 4.6
C 20. The graph of f ( x)  2 x 2 
(A) 1
(B) -1
k
has a point of inflection at x=-1, then the value of k is
x
(C) 2
(D) -2
(E) 0
C 21. The function f ( x)  x 4  bx 2  8x  1 has a horizontal tangent and a point of inflection for the same value
of x. What must be the value of x?
(A) -1
(B) 4
(C) 1
(D) 6
(E) -6
C 22. If the radius of a sphere is increasing at the rate of 2 inches per second, how fast, in cubic inches per
second, is the volume increasing when the radius is 10 inches?
(A) 800
(B)800
(C) 3200
(D) 40
(E) 80
C 23. Two cars start at the same place and at the same time. One car travels west at a constant speed of 50mph
and a second car travels south at a constant speed of 60mph. Approximately how fast is the distance between
them changing one-half hour later?
(A) 72mph
(B) 74mph
(C) 76mph
(D) 78mph
(E) 80mph
C 24. The volume of a cube is increasing at the rate of 20 cubic cm per second. How fast, in square cm per
second, is the surface area of the cube increasing at the instant when each edge of the cube is 10cm long?
(A) 1.333
(B) 2
(C) 4
(D) 6