Final Exam Review
1. Determine the derivative for each of the following:
6
dy 
a. y   2 x  3
dx
dy 
b. y  sec x
dx
dy

c. y  ln x
dx
dy 
d. y  23 x
dx
2
dy 
e. y  e x  x
dx
dy
f. y  sin  sin x 

dx
dy 
g. y  cos  x 2  3x  1
dx
dy 
h. y  e2
dx
dy 
i. y  log 2 x 2
dx
5
dy 
6

j. y   x  
dx
x

dy 
k. y  cosh 4 x
dx
dy 
1
l. y  sin 1  
dx
 x
2
dy 
m. y  ln  x 2  4 
dx
dy 
2
n. y  tan  
dx
x
dy 
o. y  arctan  e x 
dx
dy 
1
p. y 
dx
x2  4
dy 
q. y  3cos x
dx
dy 
r. y  1  cos x
dx
dy 
s. y  tan 1  tan x 
dx
dy 
t. y  ln  tan x 
dx
dy
2
2. Find
for y  x sin  x  1 .
dx
Math 131
1
2x
.
1  x2
3.
Find an equation for the tangent line at x  1 for the function f  x  
4.
2
Find d y
5.
Find dy
6.
Find dy
7.
At a distance of 6000 ft from the launch site, a spectator is observing a space shuttle being
launched. If the space shuttle lifts off vertically, at what rate is the distance between the spectator
and shuttle changing at the instant when the angle of elevation is  and the shuttle is traveling at
6
880 ft/sec?
dx 2
for y  xe2x .
for y   x  2 
1/ x
dx
dx
for xy 2  x2 y  2  0 .
2
1
dy
, use the definition of derivative to find
.
x
dx
8.
If y 
9.
Find the rectangle of largest area that can be inscribed in a semicircle of radius R, assuming one
side of the rectangle lies on the diameter of the semicircle as shown.
R
10. If f  x  
3 x
, find  f
x
f  x  . What is the domain of  f
f  x  ?
11. Are the lines 2 x  y  1 and 2 x  y  1 perpendicular?
12. True or False: If lim f ( x)  L then f  a   L .
x a
13. If f  x  and g  x  are differentiable, then
d
 f  x  g  x    _____________________.
dx
14. Sketch a function f where f is continuous at x  a but f is not differentiable at x  a .
3
15. If h( x)  f [ g ( x)], g(-3)  5, g'(-3)  2, f (5)  3, and f '(5)  -3 , find an equation of the tangent line
to the graph of h  x  at x  3 .
16. Find the maximum and minimum values of f  x   Ax  B where A  0 and B are constants on
[a, b].
sin 2 x
 ____ .
x 0
x
17. Find lim
18. Does the function g  x   x  2 satisfy the hypotheses of the Mean Value Theorem on [1,4].
19. By the ______________________theorem, if f is continuous on the interval [a, b] and K is between
f  a  and f  b  then K  f  c  for some c in (a, b).
ln x
. Calculate f "  x  and use it to find intervals where the graph of f  x  is concave
x
up and concave down. Find all inflection points.
20. Let f  x  
21. Find the equation of the tangent line to the graph of the equation tan( xy)  y 2 at (/4, 1).
4
22. Consider the function whose graph is
1
a. What is the value of  f ( x)dx
1
b.
f '  x   0 at x  _______ .
c.
f "  x   0 for ____ < x < ____
d.
f '  x  fails to exist at x  _______ .
e.
f  x  fails to be continuous at x  _______ .
f.
lim f ( x) fails to exist at x = ______.
x  xo
23. Evaluate the following limits
x3  7 x 2  12 x
a. lim
x 4
4 x
b. lim
x 0
ex 1
x2  x
sin  4 x 
x 0 ln 1  x 
c. lim
5
 
d. lim x sin  
x 
x
3 
 1
e. lim 
 2

x 3 x  3
x  3x 

f.
lim  sin x 
tan x
x  /2
24. For the graph of y  3 x , find the inflection point.
25. Does the graph of y 
2 x  sin x
have a vertical asymptote at x  0 ?
x
26. Verify that the function f ( x)  x satisfies the hypotheses of the Mean Value Theorem on [0, 2].
27. If f '( x) 
1 2
x then f ( x)  __________
2
28. If f  x   2 on [-3, 0], then the Riemann Sum for f (x) on the given interval is ______.
6
29. Evaluate the following:
2 1


a.   x 2/3  4    2  dx
x
x


b.
  sin 5x cos 2 x  dx
c.
 sec  2x  tan  2 x   sec 5x  dx
2


1
d.   e5 x 
 dx
2
1 9x 

e.
 x  x  1 dx
f.
x

dx

 1  x2
g.
  cosh x  sinh x  dx
2/3
1
h.
  3  2x 
5
dx
0
7
1
i.
2
 5 x dx
 3
 2x 1
0
8
j.
3
x
e
 2/3 dx
x
1
4

k. 


x
1

1

x 1
2
dx
3
4
30. Evaluate
 cos  2x  dx .
0
31. Find the area bounded between y  9  x 2 and y  x  3 .
8
32. A tree has been transplanted and after t years of growing at a rate of
dh
1
ft/yr. At two
 1
2
dt
 t  1
years it has reached a height of 5 ft. How tall was the tree when it was planted?
33. For f  x   x  sin x for [0, 2]
a. Find f '  x  . Use it to find critical values of f  x  and intervals where f  x  is increasing and
decreasing.
b. Find f "  x  . Use it to find inflection points of f  x  and intervals where f  x  is concave up
and concave down.
34. Sketch a possible graph of a function with the following properties.
Domain and Range
First Derivative
Second Derivative
f  0  0
f '  x   0 0  x  1, 1  x  2, x  2 f "  x   0 1  x  2
f 1  1
f  3  0
lim f  x   1
x 
lim f  x   
f ' x  0 x  0
f "  x   0 x  0, 0  x  1, x  2
lim f '  x   
x  0
lim f '  x   
x  0
x  2
lim f  x   
x  2
9
35. Sketch a possible graph of a continuous function y  f  x  using the graph of f '  x  shown below,
if f  0   f  3  0 .
10