MATH 140
FINAL EXAM SAMPLE C
1. Find the critical numbers of f (x) = x − 2 sin x on [0, 2π].
5. Find lim
x→1+
a) x = 0, 2π
a) 0
b) x = π/12, 7π/12
b) 1
c) x = π/6, 7π/6
c) −1
d) x = π/3, 5π/3
d) ∞
e) x = 2π/3, 4π/3
2. If f (x) =
√
3
x2 − 3x − 4
.
x2 + 3x − 4
x + 1 + cos
e) −∞
x
3
, find f 00 (0).
6. Find the horizontal and vertical asymptotes of f (x) =
x+2
.
x2 − 2x − 3
a) −
1
3
b) −
2
9
b) Horizontal asymptote y = 1, vertical asymptote x = 3.
c) −
1
9
c) Horizontal asymptote y = 0, vertical asymptotes x = 3 and
x = −1.
a) Horizontal asymptote y = 0, vertical asymptote x = 3.
d)
1
3
d) No horizontal asymptotes, vertical asymptotes x = 3 and
x = −1.
e)
1
9
e) Horizontal asymptote y = 1, vertical asymptote x = −3
3. Assume f is a continuous function on [1, 4] such that f (1) = 10,
f (3) = −2 and f (4) = 2. Which of the following statements must
be true?
7. Find the derivative f 0 (x) of f (x) =
0
a)
a) f has no zeros in [1, 4].
b)
b) f has exactly one zero in [1, 4].
c)
c) f has at least three zeros in [1, 4].
d)
d) f has at least two zeros in [1, 4].
e)
√
2x x
3
3√
3x x
√
x
√
x x
√
x4 x
e) f has exactly two zeros in [1, 4].
Z
4. Evaluate
√
x2
1 − 2x3
8. If xy 2 = π + y, find
dx:
√
1 − 2x3
a) −
+C
3
√
x
b) −
+C
3
a)
y2 − 1
1 − 2xy
b)
y2 + 1
1 + 2xy
c) 2xy = 1
c) 2x3 − 1 + C
√
1 − 2x3
d) −
+C
2
d)
y2
1 − 2xy
6
e)
1 − 2xy
y2
e) − √
+C
1 − 2x3
1
x3
Z
dy
.
dx
√
t dt:
MATH 140
FINAL EXAM SAMPLE C
π/2
Z
9. Find
π/4
cos x
dx.
sin2 x
12. If f (x) =
√
a)
b)
c)
d)
a)
2
−1
2
√
2−1
√
2
1−
2
√
2
2
1
Z
1
p
2 −2x sin(x2 )
p
− sin(2x)
e) p
cos(x2 ) + 1
d)
x2 (3 + x) dx.
√
13. Evaluate lim
x→5
−1
a)
5
a)
b) 1
c)
√
28
.
28
5
5
√
2 7
d) −∞
5
4


x2 − 1



3
11. If f (x) =
3x − 3



x2
ments is true?
x2 + 3 −
x−5
b) 0
1
5
d) 0
e)
p
− sin(2x)
x sin x
c) p
cos(x2 ) + 1
0
c)
cos(x2 ) + 1, find f 0 (x).
−x sin(x2 )
b) p
cos(x2 ) + 1
e) 0
10. Evaluate
q
e) ∞
x<1
14. Find the interval(s) on which the function f (x) = x3 − 108x + 10 is
x=1
, which one of the following stateincreasing.
1<x<3
x≥3
a) (−∞, −18), (18, ∞)
b) (−∞, 6)
a) f has a removable discontinuity at x = 3.
c) (−6, 6)
b) f has a jump discontinuity at x = 1 and a removable discontinuity at x = 3.
d) (−6, ∞)
c) f has a jump discontinuity at both x = 1 and x = 3.
e) (−∞, −6), (6, ∞)
d) f has a removable discontinuity at x = 1 and a jump dis15. If the sides of a square are growing at a constant rate of 5 cm/min,
continuity at x = 3.
how fast is the area increasing when the sides are 20 cm long?
e) f is continuous at x = 1 and x = 3.
a) 100 cm2 /min
b) 200 cm2 /min
c) 300 cm2 /min
d) 400 cm2 /min
e) 500 cm2 /min
2
MATH 140
FINAL EXAM SAMPLE C
16. Find the area of the region enclosed by the curves y = |x| + 1 and
y = x2 − 1.
a)
20
3
21. There is a point c in the interval (a, b) such that f 0 (c) =
f (b) − f (a)
.
b−a
a) True
b) 10
b) False
c) 20
d)
10
3
e) 6
17. Set up, but do not evaluate an integral for the volume of the solid
generated by rotating the region bounded by y = cos x, y = 1 about
y = 4.
0 ≤ x ≤ 2π.
Z
2π
a) π
[(4 + cos x)2 − 9] dx
0
Z
2π
b) π
[(5 − cos2 x)] dx
0
Z
2π
c) π
[(4 − cos x)2 − 9] dx
0
2π
Z
[(4 − x) cos x] dx
d) 2π
0
2π
Z
[(x − 4) cos x] dx
e) 2π
0
Questions 18 through 21 are true/false type. On your scantron
mark A for true, B for false. Each true/false question is worth
3 points.
For questions 18 – 21, assume f (x) is a continuous function on
[a, b].
18. The derivative f 0 (x) must exist on (a, b).
a) True
b) False
Z
x
19. If F (x) =
f (t)dt, then F 0 (x) is continuous on (a, b).
a
a) True
b) False
20. The function f has an absolute maximum on [a, b].
a) True
b) False
3
MATH 140
FINAL EXAM SAMPLE C
22. (12 pts total) A region R in the first quadrant is bounded by
y = 8 − x2 , y = 2x, and the y-axis.
23. (12 pts. total) Maximize the area of a rectangle inscribed in a
circle of radius 5.
a) (4 pts) Sketch the region R. You must find and label any points
of intersection.
a) (2 pts)Write the total area as a function.
b) (4 pts) Write the total area as a function of one variable.
b) (4pts) 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) Find the dimensions of the rectangle of maximum area
described above.
c) (4pts) Write an integral expression for the volume when R is
revolved around the y-axis. (Do not evaluate the integral! Just set
it up.)
d) (2 pts) Justify that your answer is a maximum.
4
MATH 140
FINAL EXAM SAMPLE C
24. (12 pts.) A light on the ground shines on a wall 20 meters away.
If a man 2 meters tall walks from the light toward the wall at a rate
of 2.5 m/s, how fast is the length of his shadow decreasing when he’s
10 meters away from the wall?
FINAL EXAM- FORM A
1. D 2. A 3. D 4. A 5. E 6. C 7. B 8. D 9. B 10. E 11. D 12. B
13. C 14. E 15. B 16. A 17. C 18. B 19. A 20. A 21. B
22a. intersection points: (0,0), (0,8), (2,4)
2
Z
22b. V =
π[(8 − x2 )2 − (2x)2 ] dx
0
Z
2
22c. V =
2πx(8 − x2 − 2x) dx
0
23a. A = xy
23b. A(x) = x
p
100 − x2
√
23c. x = y = 5 2
23d. EVT works on Domain [0, 10]; or 1st der test; or 2nd der test
24. 1 m/s
5