Sample Problems
1. lim √
x→0
x
√
=
2+x− 2−x
2. For what value(s) of c is the function
f (x) =
x2 + 3,
cx + c2 − c,
x≤1
x>1
continuous at x = 1?
3. A particle is moving with the given information a(t) = 6t + 2, v(0) = 2,
and s(0) = 5. The position equation of the particle is
1
1
1
0
4. If g(x) is a function such that g
= 2 and g
= √ , and
2
2
3
3
0 π
=
f (x) = [g(sin x)] then f
6
5. What asymptotes does the function f (x) =
8x2
have?
4 − x2
"
2 !#
n
P
1
i
6. Express lim
cos 1 + 4
as a definite integral.
n→∞ i=1 n
n
R sin x p
7. If f (x) = 0
(1 + t2 ) dt, then f 0 (x) =
50
R
8.
x x2 + 1
dx =
9. The graph of a function y = f (x) consists of a line segment between
the points (1, 3) and (3, 0) and then the lower half of a circle centered
at the point (5, 0) with a radius of 2. Find the value of the following
integral.
Z7
f (x) dx =
1
10. A spherical balloon is being inflated at the rate of 10cm3 per second.
At what rate is the radius of the balloon increasing when the radius of
the balloon is 5cm?
Volume:
V =
4 3
πr
3
(To get full credit, you must show all work.)
2
11. A cylindrical soup can (with top and bottom) is to be constructed
to have a volume of 54π cm3 . What is the radius of the can which uses
the least amount of material?
12. Set up, but DO NOT INTEGRATE the integral(s) that represent
the area bounded by the curves y = x3 + x2 and y = 6x.
13. Set up, but DO NOT INTEGRATE the integral(s) that represent
the volume of the solid of revolution obtained by rotating the region
bounded by y = x and y = x2 − 6x + 10 about the x-axis.
14. Find the volume of the solid of revolution that is generated by
rotating the region bounded by y = 2x and y = x2 − x about the y-axis.
3
n
2 P
2i
15. Evaluate
lim
in two different ways.
n→∞ n i=1
n
2
n
P
n(n
+
1)
A) Without using a definite integral:
HINT:
i3 =
2
i=1
B) By first converting to a definite integral and then evaluating the
integral:
16. Use the function f (x) = x3 + 3x2 − 24x + 1 to find the information
below
The critical numbers of f are
f is increasing on the interval
f is decreasing on the interval
f is concave up on the interval
f is concave down on the interval
The x-value(s) of the relative maximum(s) is(are)
The x-value(s) of the relative minimum(s) is(are)
There is an inflection point at x =
3
√
17. lim ( x2 + 6x + 2 − x) =
x→∞
18. The most general antiderivative of f (x) = sin(3x) + x1/2
" #
2
n
3i
3i
3 P
19. Express lim
sin
+
as a definite integral.
n→∞ n i=1
n
n
is
t
dt, then f 0 (x) =
0
2
1+t
R 3√
21. After substitution,
x x2 + 1 dx becomes
R
cos x
22.
dx =
(1 + sin x)2
20. If f (x) =
23.
R sin x
√
R 4 x2 + 2
dx =
2
x2
24. The graph below indicates the area of various regions between y =
f (x) and the x−axis.
Z11
f (x) dx =
1
25. A lighthouse is on a small island 3 km away from the nearest point P
on a straight shoreline and its light makes four revolutions per minute.
How fast is the beam of light moving along the shoreline when it is 1 km
from P ?
26. A rectangular storage container with an open top is to have a volume
of 12 m3 . The length of its base is three times the width. Material for
the base costs 10 dollars per square meter. Material for the sides cost
2 dollars per square meter. Find the dimensions of the box that will
minimize the cost.
27. Set up, but DO NOT INTEGRATE the integral(s) that represent
the area bounded by the curves y = x3 − x2 + 3x + 1 and y = 4x2 − 3x + 1.
28. Find the volume of the solid of revolution obtained by rotating the
region bounded by y = 0, x = 1, x = 2 and y = x2
... about the x-axis.
... about the y-axis.
4
29. Set up, but DO NOT INTEGRATE the integral(s) that represent
the volume of the solid of revolution obtained by rotating the region
bounded by the curves y = x2 + 2x and y = 6x
... about the x-axis.
... about the y-axis.
30. Evaluate lim
n→∞
n
4 P
n i=1
4i
n
2
in two different ways.
Without using a definite integral:
HINT:
n
P
i2 =
i=1
n(n + 1)(2n + 1)
6
By first converting to a definite integral and then evaluating the integral:
31. Use the function f (x) =
1
1 5
x − x3 + 4 to find the information below
5
3
The critical numbers of f are
f is increasing on the interval
f is decreasing on the interval
f is concave up on the interval
f is concave down on the interval
The x-value(s) of the relative maximum(s) is(are)
The x-value(s) of the relative minimum(s) is(are)
There is an inflection point at x =
32. Find real numbers a and b that maximize the value of
Z
b
(2 + x − x2 ) dx.
a
To get full credit you must explain why these particular values for a and
b will maximize the value of the definite integral.
What is the maximum value for the integral?
5
33.
R2
0
√
x2 x3 + 1 dx =
34. If h(x) =
R x3 sin(t)
dt, then h0 (x) =
0
t
35. What is the limit definition of the derivative?
36. Evaluate the following limits.
A) lim
x→5
x2 − 25
x−5
B) lim−
x−3
|x − 3|
C) lim
x−3
|x − 3|
x→3
x→3+
t−4
D) lim √
t→4
t−2
E) lim
(x + h)3 − x3
h
F) lim
x+1
x−2
G) lim
5 tan(3x) cos(2x)
sin(4x)
h→0
x→2
x→0
37. You are a member of a precision balloon inflation team. One event
requires that you inflate a shperical balloon from a radius of 2 inches to 8
inches in exactly 12 seconds where the radius of the balloon is increasing
at a CONSTANT rate.
A) At what rate must the radius increase?
B) At what rate is the volume of the balloon increasing when
4
the radius is 4 inches? V = πr3
3
38. Find the area between the curves y = x and y = x2 for 0 ≤ x ≤ 2.
39. Set up, but do not integrate, the integral representing the volume
of the solid obtained by rotating the region bounded by the x− axis and
y = x2 − x3 about
A) The x−axis.
B) The y−axis.
6
40.
R2
1
dx
=
(3x − 2)2
41. Find the area bounded by y = −x and y = 2 − x2 .
42. If y =
x2
, then y 0 =
2x + 1
43. The most general antiderivative of f (x) = sin(4x) + x4/3 + x3/2 is
R
44.
tan(3x) sec2 (3x) dx =
45. Find the second approximation to the root of the equation x5 −x2 −1 =
0 starting with the first approximation x1 = 1.
46. The area bounded by y = 2x + 3 and y = x2 is
R tan x √
1 + t3 dt, then f 0 (x) =
47. If f (x) = 0
48. A ladder 30 feet long, leaning against a wall, is slipping at a rate of 3
feet per second along the ground away from the wall. When the bottom
is 15 feet from the wall, how rapidly is the angle the ladder makes with
the ground changing?
49. A poster is to contain 50 square inches of writing. The top and
bottom margins must be 2 inches and the side margins must be 4 inches.
Find the dimensions of the poster that will minimize the area of the
poster.
Rπ
x
x
50. 0 2 sin4 cos dx =
2
2