MATH 140/140E
FINAL EXAM SAMPLE A
1. Compute
5. Compute
x−1
.
(x + 1)2
lim
x→−1+
d
dx
a) ∞
a)
1
cos(x) − 1
b)
cos(x) − cos(2x)
(1 − cos(x))2
c)
1 − cos(x)
sin2 (x)
sin(x)
1 − cos(x)
.
b) −∞
c) 0
d) 1
e) −1
d) cot(x) +
2. Compute
lim
x→3
x |3 − x|
.
x−3
e)
a) −3
1
sin(x)
1
1 − cos(x)
q
6. Compute the derivative of the function f (x) =
cos
√
2x.
b) 3
c) 0
a)
d) 1
1
√ p
√
2 2x cos 2x
√
− sin 2x
c) √ p
√
2x cos 2x
√
− sin 2x
d) √ p
√
2 x cos 2x
√
sin 2x
√
e)
2x
b)
e) The limit does not exist
3. Compute
lim
x→0
√
− sin 2x
√ p
√
2 2x cos 2x
2 sec(x)
.
1 − tan(x)
a) 0
b) 1
c) 2
d) −2
e) The limit does not exist
7. Find the equation of the tangent line to the curve y 2 +4xy +x2 = 13
at the point (2, 1).
4. Compute
6x3
lim √
x→∞ 3 13
+x
+ x9
.
a) y = x − 11
a) −∞
b) y = x + 13
6
b)
13
c)
c) y = −x + 15
7
5
d) y = − x +
4
2
6
√
3
13
4
13
e) y = − x +
5
5
d) 6
e) +∞
1
MATH 140/140E
FINAL EXAM SAMPLE A
Z
p
8. Suppose oil spills from a ruptured tanker and spreads in a circular
3 3
4 − x2 dx
1 12. By using an appropriate substitution, the indefinite integral x
pattern. If the radius of the oil increases at a constant rate of
can be transformed to which one of the following integrals?
2
m/s, how fast is the area of the spill increasing when the radius is
Z
30 meters?
4
1
1
(u 3 − 4u 3 )du
a)
2
Z
a) 30π m2 /s
1
4
b)
(4u 3 − u 3 )du
2
b) 2π m /s
Z
4
1
c) 60π m2 /s
c) −
u 3 du
2
Z
d) 60 m2 /s
4
d) 2 u 3 du
2
e) 90π m /s
Z
4
1
e) 2 (u 3 − 4u 3 )du
2
9. Find the x-values at which all local extrema occur for f (x) = 3x 3 −
x.
13. Compute
Z
√
a) minimum at x = 2 only
1
b) maximum at x = 8 only
1
dx.
√
x(1 + x)2
a) 6
c) minimum at x = 0 and maximum at x = 2
b) −
d) maximum at x = 2 only
c)
e) minimum at x = 0 and maximum at x = 8
10. How many asymptotes does the graph of the following function have?
(13x2 − 6x)x
(13x − 6)(x − 1)
f (x) =
4
3
4
d) −
e)
1
6
1
3
1
3
14. Evaluate the integral
a) One vertical asymptote and one slant asymptote
2
Z
b) Two vertical asymptotes and one slant asymptote
1
c) One horizontal asymptote and one slant asymptote
a)
7
4
b)
15
4
d) Two vertical asymptotes only
e) One slant asymptote only
2x4 + 2
dx.
x3
c) 0
11. A particle moves in a straight line and has acceleration given by
a(t)
= cos t. Find
position of the particle, s(t), given that
π
π the π
v
= 2 and s
= .
2
3
3
a) s(t) = cos t + t −
1
4
e) −
1
2
√
15
4
15. Find the derivative of the function
Z sin(x)
F (x) =
sec t dt.
3
2
b) s(t) = − cos t + t +
c) s(t) = − cos t + t +
d)
π
1
2
a) sec(sin x)
d) s(t) = − cos t + t
e) s(t) = − cos t + 2t +
b) 1
1
π
−
2
3
c) sec(x) sin(x)
d) sec(sin(x)) + 1
e) sec(sin x) cos(x)
2
MATH 140/140E
FINAL EXAM SAMPLE A
2
Z
18. Which one of the following curves is the graph of the function f (x) =
x3
?
2
x −4
(x + |x|) dx.
16. Compute
−1
a) 4
b) 3
c) 8
d)
3
2
e) 5
17. Find the volume of the solid obtained
√ by rotating about the x-axis
the region bounded by the curve y = x − 1 and the x-axis between
x = 2 and x = 5 .
a)
5π
2
b)
7π
2
c)
11π
2
d)
15π
2
e)
19π
2
(b)
(a)
(d)
(c)
(e)
3
MATH 140/140E
FINAL EXAM SAMPLE A
Questions 19 through 21 are true/false type. On your scantron
mark A for true, B for false. Each true/false question is worth
4 points.
19. If
a.
lim f (x) = lim f (x), then f is a continuous function at
x→a−
x→a+
a) True
b) False
b
Z
20. If a function f is continuous on [a, b], then
Z b
2
f (x) dx .
f 2 (x) dx =
a
a
a) True
b) False
21. If f is a continuous function on [a, b], then
d
dx
Z
b
f (t) dt
= 0.
a
a) True
b) False
4
22. (8 points) For which x-value(s) on the curve of y = 1+40x3 −3x5
does the tangent line (not y) have the largest slope?
MATH 140/140E
FINAL EXAM SAMPLE A
23. (10 points) Find the area between the curves y = x and y = x2
for 0 ≤ x ≤ 2.
24. (12 pts total) A region R in the first quadrant is bounded by
8
, y = x2 , and x = 1.
x
y=
a) (4 pts) Sketch the region R. You must find and label any points
of intersection.
b) (4 pts) Write an integral expression for the volume when R is
revolved around the x-axis. (Do not evaluate the integral. Just set
it up.)
c) (4 pts) Write an integral expression for the volume when R is
revolved around the y-axis. (Do not evaluate the integral. Just set
it up.)
FINAL EXAM- VERSION A
1. B 2. E 3. C 4. D 5. A 6. A 7. E 8. A 9. E 10. A 11. C 12. A 13.
E 14. B 15. E 16. A 17. D 18. A 19. B 20. B 21. A 22. x = −2, 2;
23. 1; 24. a) pic with intersection points: (1, 8), (2, 4), (1, 1); b)
Z 2
Z 2
8
8
V =
π[( )2 − (x2 )2 ]dx; c) V =
2πx( − x2 )dx
x
x
1
1
5