Final Exam
(Mock Test)
Math 12010-003
Answer the questions in the spaces provided. Show all of your work.
Name:
ID: U
Question:
1
2
3
4
5
6
7
8
9
10
11
Total
Points:
10
10
10
10
10
10
10
5
10
10
5
100
Score:
1. For each of the following problems sketch the region R, shade it and calculate the area.
√
(a) 5 points Region bounded by y = 3 x, y = 0, x = −8 and x = 8.
(b) 5 points Region bounded by x = 4y 4 and x = 8 − 4y 4 .
2. 10 points Find the volume of the solid generated by revolving the region R bounded
1
by y = , y = 0, x = 2 and x = 4, about the x-axis. (Sketch the region first and shade
x
it).
3. 10 points Find the volume of the solid generated by revolving the region R bounded
by y = 6x and y = 6x2 , about the x-axis. (Sketch the region first and shade it).
Page 2
4. 10 points Find the volume of the solid generated by revolving the region R in the first
quadrant bounded by y = x2 , y = 2 − x2 and x = 0, about the y-axis. (Sketch the region
first and shade it).
2
5. (a) 5 points Find the length of the arc y = (x2 + 1)3/2 between x = 1 and x = 2.
3
Page 3
(b) 5 points Fin the area of the surface generated by revolving the arc
x = t, y = t3 ; 0 ≤ t ≤ 1, about the x-axis.
6. Evaluate the following indefinite integrals.
√
Z
x sin( x2 + 4)
√
(a) 5 points
dx
x2 + 1
(b) 5 points
Z
(x4/3 − 2x1/3 ) dx
Page 4
7. Evaluate the following definite integrals.
Z π/4
(a) 5 points
(cos 2x + sin 2x) dx
0
(b) 5 points
Z
π/2
x2 sin2 (x3 ) cos(x3 ) dx
−π/2
8. (a) 3 points Evaluate
Z
0
4


1
f (x) dx where f (x) = x


4−x
Page 5
if 0 ≤ x < 1
if 1 ≤ x < 2
if 2 ≤ x ≤ 4
(b) 2 points Find the derivative f (x) where f (x) =
′
Z
x2
−x2
t2
dt.
1 + t2
2 1 − 2x
−2 1 + 4x
· 1/3 and f ′′ (x) =
· 4/3 .
3 x
9
x
(a) 5 points Find the points where f has local maxima and local minima.
9. Let f (x) = x2/3 (1 − x), where f ′ (x) =
(b) 5 points Find the intervals where f is concave up and concave down. Also, find
the points of inflection.
Page 6
10. (a) 5 points Find
(b) 5 points Find
dy
where cos(xy 2 ) = y 2 + x.
dx
d3 y
3x
, where y =
.
3
dx
1−x
 x3 −x

4x2 −4x


2
11. (a) 3 points Find the points of discontinuity of the function f (x) =

x3 + 1



2x − 1
Page 7
if x < −1
at x = −1
if − 1 < x ≤ 0
if x > 0
5x3 − 8x + 10
x→+∞ 2x3 + 4x2 − 5
(b) 2 points Evaluate lim
12. 10 points (Bonus Problem) Evaluate
Z
2
0
Page 8
|x − 1| dx