Find any discontinuities in the given functions

CALCULUS
EXAM REVIEW
SEME STER 1
NAME:
Evaluate the following limits:
3. lim
x 3
7. lim
 0
x 1  2
x3
1. lim
x 1
sin x
x 0
x
4. lim
cos  tan 
8. lim  sec 

 
x2  1
x 1
2. lim
x 0
1  cos x
x 0
x
sin x
x 0 6 x
5. lim
9. lim
t 0
x 2  3x
x
6. lim
sin 3t
t
Find any discontinuities in the given functions:
1
10. f ( x) 
x 5
x3
11. f ( x)  2
x 9
x
 1 x  2
12. f ( x)   2
3  x x  2
x2
13. Find intervals for which f ( x)  2
is continuous.
x 4
14. Find any vertical asymptotes for the graph of f ( x) 
15. Determine whether g ( x) 
x2  2
.
x2  x  2
x2  6 x  7
has a removable discontinuity or a vertical
x 1
asymptote at x = -1.
Find the indicated limits
 1
16. lim 1  
x 0 
x
17. lim
x 0
x2  2x
x3
18. lim
x 1
2  3x
1 x
19. use the definition of the derivative to find f '( x) for f ( x)  2 x 2  x  1 .
20. Find the equation of the tangent line to the graph of f ( x)  3x 2  4 at  2,8 .
21. A ball is thrown upward from the surface of the earth with an initial velocity of 110
feet per second. Find its velocity after 1 second and 2 seconds. When does the ball
reach its highest point? How high is the ball at its highest point? When does the ball
hit the ground? What is the ball’s velocity when it hits the ground?
Find derivatives for the following functions
7

22. f ( x)  3x  x 3  
x

23. g ( x)  sin x  cos x
25. f ( x)  3 x 2 cos x
26. f ( x)  sin x  cos x  tan x 
28. g ( x)   4  x3 
29. h( x) 
31. y  x 2 tan
Find
33.
2
3
1
x
2x 1
x3
24. h( x) 
3
 tan x
x3
27. f ( ) 
2
1  cos 
1
30. y  sec3  2 x 
4
32 y  sin x
dy
by implicit differentiation
dx
 x  y
3
 x3  y 3
36. x cos y  1
34. sin x  cos 2 y  2
35. x3  2 x 2 y  3xy 2  38
37. Find the derivative of y  a 2bc3  abc 2 with respect to c.
38. The radius of a circle is increasing at a rate of 3 inches per minute. Find the rate of
change of the area when the radius is a) 6 inches and b) 12 inches.
39. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a
rate of 10 cubic feet per minute. The diameter of the base of the cone is
approximately three times the altitude. At what rate is the height of the pile changing
when it is 15 feet high?
40. Locate the absolute extrema of the given function over the given interval:
t2
g (t )  2
1,1
t 3
41. Find all values of c in the interval  0,   such that f '(c) 
f (b)  f (a)
if
ba
f ( x)  2sin x  sin 2 x .
42. Find the open intervals on which f is increasing or decreasing and locate all relative
x
extrema of f ( x) 
.
x 1
43. Find intervals of concavity as well as any points of inflection of
44. Find any horizontal asymptotes of g ( x)  4 
 x  1
f ( x) 
x
2
.
1
x 1
2
Find the indicated limits
45. lim
x 
3x  1
x2  x
x2
x  2 x 2  1
46. lim
47. lim
x 
x4  x
x2  x
2x
.
1  x2
49. Sketch the graph of the function. Label the intercepts, relative extrema, points of
x3
infection, and asymptotes. f ( x)  2
.
x 1
48. Find any intercepts, symmetry, and asymptotes of f ( x) 
50. A rectangular plot that will contain a vineyard of once acre in area (43,560 sqft) is to
be laid out. The vineyard must have a boundary of 8 feet on all sides and an 8 foot
pathway down the middle for equipment to pass through. What is the minimal
acreage required for this setup?
51. A standard can contains a volume of 900 cubic centimeters. The can is in the shape
of a right circular cylinder with a top and a bottom. Find the dimensions of the can
that minimize the amount of material needed for construction.
52. The turning effect of a ship’s rudder is found to be T  k cos  sin 2  , where k is a
positive constant and  is the angle that the direction of the rudder makes with the
keel line of the ship  0    90  . For what value of  is the rudder most effective?
ANSWERS
1. 0
2. –3
3. ¼
4. 1
5. 1/6
6. 0
8. 
7. 1
9. 3
10. At x = 5
11. at x  3 12. At x  2
13. Discontinuities at x  2 so continuous on  , 2 2, 2 2,  
14. VA at x  1, 2 15. Removable discontinuity at x  1 16.  17. 
18.  19. 4x+1 20. y  12 x  16
21. v(1)  78 ft / sec, v(2)  46 ft / sec , reaches max height at t = 3.4375 sec, height at
this time is 189.0625 ft, ball hits ground at t = 6.875 sec, velocity at this time –110 ft/sec
9
22. f '( x)  12 x3 23. g '( x)  cos x  sin x 24. h '( x)  4  sec 2 x
x
3x sin x  2cos x
sin x
25. f '( x) 
26. f '( x)   sin 2 x 
 cos 2 x  sin x
2
3
cos x
3 x
27. f '( ) 
2  2 cos   2 sin 
1  cos  
2
2

34.
2 x 2
3
4 x
3
29. h '( x) 
1
1
31. y '   sec 2    2 x tan
x
 x
3
30. y '  sec3  2 x  tan  2 x 
2
dy   2 xy  y
33.

dx
x 2  2 xy
28. g '( x) 
dy
cos x

dx 2sin 2 y
dy cos y
2ab
37. y  3a 2bc 2  3

c
dx x sin y
dh
39.
 .0063 ft / min
dt
36.
35.
5
2  x  3
32. y ' 
2
cos x
2 sin x
dy 3x 2  4 xy  3 y 2

dx
2 x 2  6 xy
38. 36 in 2 / min , 72 in 2 / min
40. Abs min at (0, 0), abs max at (-1, ¼) and (1, ¼)
41. x 

3
,
42. Increasing everywhere, no extrema
43. Concave down on  , 0  , concave up on  0,   , no POI since x = 0 is a VA
44. Y = 4 45. –3 46. ½ 47. 1 48. HA at y = 0, VA at x = 1 , no SA
,  3  3, 1  1, 0  0,1 1, 3
3, 
49. CN at x  0, 1,  3


+
PPOI at x = -1, 0, 1

-

-
-

-

+
 , 1 1, 0  0,11,  
-
+
50. A  1.2acres or 52,117.44 sq feet
-
+
51. r  5.23cm, h  10.46cm
52. 54.7˚
x3
2x 1

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