1. Which expression is the limit definition for the derivative of
1
1

a ) lim a  h a
h 0
h
ln( a  h )  ln a
h 0
h
b) lim
f ( x)  ln x at x = a?
ln x  ln a
x a
xa
c) lim
ln x  ln a
h 0
h
d ) both b and c are correct
e) lim
2. Which is the equation of the tangent line to
f ( x)  5e x  2( x  4) 3 2 at x  0 ?
a)
y  11x  131
d)
y  5 x  21 e) none of these
b)
y  11x  21
c)
y  6 x  16
3. The tangent line to f(x) at ( 2, f(2)) is y = 5x – 4. Which can be concluded about f(x)?
a) The tangent line is horizontal at x = 4/5.
c) f(x) is increasing and concave up at x=1
e) all of these are true
b) f(1)=1 and f '(1)=5
d) f(2)=6 and f '(2)=5
4.
f ( x )  5e x sin x .
Which is equal to
f ' ( x) ?
a ) 5xex 1 cos x b) 5e x sin x  5e x cos x
d )  5xex 1 cos x
c) 5 xex 1 sin x  5e x cos x
e) none of these
The derivative of f(x) is shown. Use this for questions 5 and 6.
5. Which of the following is true?
a) f(x) has a local minimum at x=0 and x=4 and f(x) has a local maximum at x=1.
b) f(x) has a local maximum at x=2 and a local minimum at x=5.
c) f(x) has extreme points at 0, 1 and 4.
d) f(x) is increasing on (0, 1) and on (4, 6).
e) none of these
6. Give the best answer. f(x) has inflection point(s) at x-value(s)
a) 2 and 5
b) 1 and 4
c) 0, 2 and 5
d) 0, 1, and 4
e) none of these
4
3
7. The derivative of f(x) is given by f ' ( x )  x  x . The function f(x) has
(an) inflection point(s) at x value(s):
a) 0 and 1
d) 3/4
b) 0 and 3/4
c) 0 and 1/2
e) There are no inflection points.
8. An object travels east and west in a straight line. The position relative to home is given by
s(t )  4t 3  t 4
0  t  4 . Which of the following is/are true?
a) The distance from home is increasing due east fastest at 2 seconds.
b) The object is furthest from home at 3 seconds.
c) both a and b are true.
d) The distance from home is increasing fastest due east at 4 seconds.
e) The distance from home is increasing fastest due east at 3 seconds.
x
9. The chain rule applied to f ( x)  ln[ g ( x)  e ] results in
a)
g ' ( x)
f ' ( x) 
1
g ( x)
b)
g ' ( x)  e x
f ' ( x) 
g ( x)  e x
d)
f ' ( x) 
g ' ( x)
g ( x)  e x
e)
f ' ( x) 
g ' ( x)  1
g ( x)  e x
c)
g ' ( x )  xex
f ' ( x) 
g ( x)  e x
10. The tangent line to f(x) at (2, f(2)) is
y  5x  3. Then the tangent line to
b)  30
a) 6
y  6 x  5 . The tangent line to g(x) at (3, g(3)) is
g(f(x)) at x = 2 has slope
d)  5
c) 1
e) There is not enough informatio n.
f ( x )  cos ( x ) at
2
11. The derivative of
a) 
1
2
b)  2
c)
1
2
x

4
is
d ) 1
e) none of these
12.
x 2  5x
f ( x)  3
x 4
a)
( 2 x  5)( 3x 2 )
( x 3  4) 2
c)
2x  5
3x 2
e)
( 2 x  5)( x 3  4)  ( x 2  5 x )( 3x 2 )
( x 3  4) 2
13.
d)
f ( x )  sec 3 x
a ) 3 sec 2 x tan x
d ) tan 3 x
Which of the following is equal to f '(x)?
b)
3x 2 ( x 2  5 x )  ( x 3  4)( 2 x  5)
( x 3  4) 2
( 2 x  5)( x 3  4)  ( x 2  5 x )( 3x 2  4)
x 6  16
Which of the following is equal to f '(x)?
b) sec 3 x tan 3 x
e) none of these
c) 3 sec 3 x tan x
Show all work in this section.
14. Work out the derivative of
derivative.
g ( x )  3x 2  1 at x = 1 using only the limit definition of the
i) Find the slope of the secant line from (1, g(1)) to (x, g(x)).
ii) Do the algebra so that the denominator can be canceled.
iii) Find the limit as x approaches 1.
15.
f ( x)  4 x
a) Find the equation of the tangent line to f(x) at (16, f(16)).
b) Approximate
4
16.2
using either the tangent line from part a or differentials.
c) Approximate
4
15.8
using either the tangent line or differentials.
16. Find the derivative of each function of x.
a)
f ( x)  ( x 2  1)7 (4 x  1)3
b)
 x 

f ( x )  ln  6
 x  4
c)
x2  9
f ( x)  2 x
e  5x 2
17. An object begins at home and travels back and forth in a straight line due east and due west.
The velocity at t seconds is given by
v(t )  t 3  12t 2  36t  16 meters per second.
a) Find the acceleration at time t.
b) For the time interval 0 < t < 5, when is the distance traveling east of home increasing the
fastest?
c) When is the distance traveling west of home increasing the fastest? (Negative velocity )
d) If s(t) is the position of the object at t seconds, what occurs in the graph of s(t) at the t values
found in b and c above?