Student: Phone: ***

1 Calculus
x2 − 7x + 12 x2 − 7x + 12
Exercise 1. Let a = lim− and b = lim+ . Find S = 2a + 3b?
x→3 |x − 3| x→3 |x − 3|

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Exercise 2. Let


 1 if x ≤ 0

f (x) = 5 − 2x if 0 < x < 4
 1

if x ≥ 4.

5 − 2x
(a) Where is f discontinuous?

(b) Where is f not differentiable?

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dy dx
Exercise 3. Find when x = 1 if = 2 and x3 + 3x2 + y + 1 = 0.
dt dt

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0
3x2 + 5x − 1
Z
2
Exercise 4. Suppose that dx = a ln + b, where a, b are real constants.
−1 x−2 3
Then, evaluate S = a + 2b?

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Z 0.4
Exercise 5. Let I = sin(x2 )dx. Use the following methods to approximate the inte-
0
gral I with n = 4.

(a) Midpoint method;

(b) Trapezoidal method;

(c) Simpson method.

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Exercise 6. Check the following integrals, which one converges?

Z +∞ √ 3
x +x+1
(a) I1 = dx?
0 x2 + 1
Z +∞ √ 3
x +x+1
(b) I2 = dx?
0 x3 + x − 1
Z +∞ 4
x + 3x3 − 1
(c) I3 = √ ?
1 e− x

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√
Question 1. State the domain of f (x) = ln x2 − 7x + 12?
A. (−∞, 3). B. (−∞, 3] ∪ [4, +∞).
C. (−∞, 3) ∪ (4, +∞). D. (3, 4).
Question 2. Given lim f (x) = a and lim (5f (x) + 3g(x)) = 2, evaluate lim g(x)?
x→3 x→3 x→3
2 − 5a 2 + 5a 3 + 2a 3 − 5a
A. . B. . C. . D. .
3 3 3 3
Question 3. Find A = lim− f (x) + lim+ f (x) if
x→3 x→3
(
x2 if x ≤ 3
f (x) =
x + 4 if x > 3.

A. A = 9. B. A = 7. C. A = 16. D. A = 2.
x2 − 16
Question 4. Let f (x) = . Assume that lim + f (x) = a and lim − f (x) = b,
|x + 4| x→(−4) x→(−4)
evaluate a + b?
A. 8. B. −8. C. −16. D. 0.
dy
Question 5. Let xy − cos(xy) = 1. Find ?
dx
dy y dy y dy x dy x
A. =− . B. = . C. = . D. =− .
dx x dx x dx y dx y
Question 6. Let


 0 if x ≤ 0

f (x) = 3 − x if 0 < x < 2
 1

if x ≥ 2.

3−x
Find the points at which the given function is discontinuous?
A. x = 3. B. x = 2.
C. x = 0. D. None of the other choices are correct.
1 2 dy
Question 7. Let 30x 3 y 3 = 360. Find (27, 8)?
dx
4 4 27 27
A. . B. − . C. . D. − .
27 27 4 4
3 2
Question 8. Let y = x −3x +1. Find the tangent line of the given curve at (1, −1)?
A. y = −3x + 4. B. y = −3x + 2. C. y = −3x − 2. D. y = 3x + 2.
Question 9. Let f (x) = x2 e3x . Compute f ′′ (0)?
A. f ′′ (0) = 0. B. f ′′ (0) = 3. C. f ′′ (0) = 2. D. f ′′ (0) = 6.
Question 10. If a ball is thrown vertically upward with a velocity of 80 ft/s, then its
height after t seconds is s = 80t − 16t2 . What is the maximum height reached by the
ball?
A. 100f t. B. 120f t. C. 90f t. D. 93f t.
Question 11. A particle moves with position function s = t − 4t − 20t2 + 20t. At what
4 3

time does the particle have a velocity of 20m/s?

A. t = 15.16. B. t = 15.72. C. t = 14.72. D. t = 13.61.
Question 12. Find the inflection points for the function y = x4 − 6x2 + 1?
A. F1 = (1, −4). B. F2 = (−1, −4).
C. None of the other choices are correct. D. Both F1 and F2 .

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√
Question 13. Find √ the linear approximation of f (x) = x at x = 9 and use the approx-
imation to estimate 9,1?
A. 3,0167. B. 3,02. C. 3,11. D. 3,05.
Question 14. Find the point on the line y = 4x − 5 that is the closest to the point
(0, 2)?       
28 27 27 28 27 29 25 28
A. , . B. , . C. , . D. , .
17 17 17 17 17 17 17 17
Question 15. Use Newton’s method to solve x2 −2 = 0, starting x0 = 1. After two steps,
we obtain
A. x2 = 1.4142. B. x2 = 1.4167. C. x2 = 1.4176. D. x2 = 1.4184.
Question 16. Use Newton’s method to solve x2 − 3x + 1 = 0, starting x0 = 3. After two
steps, we obtain
A. x2 = 2.6191. B. x2 = 1.5144. C. x2 = 2.6180. D. x2 = 2.6181.
Question 17. Find the general antiderivative of f (x) = cos x + 6x?
A. sin x + 3x2 + C. B. − sin x + 3x2 + C.
C. sin x + 6x2 + C. D. − sin x + C.
√
Question 18. Find the general antiderivative of f (x) = 2x − 1.
2 √ 2 √
A. (2x − 1) 2x − 1 + C. B. (2x − 1) 2x − 1 + C.
3 3
1√ 1√
C. − 2x − 1 + C. D. 2x − 1 + C.
3 R 2
Question 19. Compute (x − sin x)dx
x2 x2
A. + sin x + C. B. + cos 2x + C.
2 2
cos 2x x2 cos 2x
C. x2 + + C. D. + + C.
2 2 2
1
Question 20. Find the general antiderivative of f (x) = .
2x + 3
1
A. ln |2x + 3| + C. B. ln |2x + 3| + C.
2
1 1
C. ln |2x + 3| + C. D. log(2x + 3) + C.
ln 2 2  
1 2
Question 21. Assume that the function f (x) defined on R \ and f ′ (x) = ,
2 2x − 1
f (0) = 1, f (1) = 2. Compute f (−1) + f (3)?
A. 2 + ln 15. B. 3 + ln 15. C. ln 15. D. 4 + ln 15.
1
Question 22. Let F (x) be an antiderivative of f (x) = on (1, +∞) and F (e + 1) =
x−1
4. Find F (x).
A. 2 ln(x − 1) + 2. B. ln(x − 1) + 3. C. 4 ln(x − 1). D. ln(x − 1) − 3.
x 2
Question
R 23. Suppose that F (x) = e + x is an antiderivative of f (x) on R. Compute
f (2x)dx
1 1
A. 2ex + 2x2 + C. B. e2x + x2 + C. C. e2x + 2x2 + C. D. e2x + 4x2 + C.
2 2
x 2
Question
R 24. Suppose that F (x) = e − 2x is an antiderivative of f (x) on R. Find
f (2x)dx
1 1
A. 2ex − 4x2 + C. B. e2x − 4x2 + C. C. e2x − 8x2 + C. D. e2x − 2x2 + C.
2 2

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x−1
Question 25. Find the area of the region which is bounded by y = and x-axis,
x+2
y-axis?
A. 3 + ln 2. B. 3 − ln 2. C. 3 + 2 ln 2. D. 3 − 2 ln 2.
Question 26. Find the area of the region between the following curves y = x2 − 4 and
y = 2x − 4?
4 4π
A. 36. B. . C. . D. 36π.
3 3
Question 27. Find the volume generated by rotating the regions between the given
curves and y = 0 around the x-axis
√
y = 1 − x2 , x = 0, x = 1.

2 π 2π 2π
A. V = . B. V = . C. V = . D. V = .
3 3 5 3
Question 28. Find the volume V generated by rotating the regions between the given
1
curve y = , y = 0, x = 1, x = a, a > 1 around the x-axis.
x
       
1 1 1 1
A. V = 1 − . B. V = 1 − π. C. V = 1 + π. D. V = 1 + .
a a a a
Question 29. Find
2
√ the volume V generated by rotating the regions between the given
curve y = x , y = x around the x-axis.
3π 9π π 7π
A. V = . B. V = . C. V = . D. V = .
10 Z 1 10 10 10
xdx
Question 30. Let 2
= a + b ln 2 + c ln 3 where a, b, c are rational numbers.
0 (x + 2)
Compute T = 3a + b + c?
A. T = 2. B. T = 1. C. T = −2. D. T = −1.
Question 31. Use the trapezoidal method to approximate the following integral with
two steps
Z 0.4
2
I= ex dx, h = ∆x = 0,2.
0

A. I = 0, 4255. B. I = 0,4223. C. I = 0,4299. D. I = 0,4371.

Question 32. Find the following limit
12 + 22 + · · · + n2 − n(1 + 2 + · · · + n)
A = lim .
n→∞ n3
1 1 1 1
A. A = . B. A = − . C. A = . D. A = − .
6 Z ∞ 6 Z ∞ 3 12
dx sin xdx
Question 33. Let A = and B = . Which of the following statement
1 x 1 x
is true?
A. A diverges and B converges. B. A converges and B diverges.
C. Both converges. Z D. Both diverges.
∞
Question 34. Let I = xα e−x dx, where α ∈ R is a parameter. Which of the following
1
statement is true?
A. I converges for all α. B. I diverges for all α.
C. I only converges for α > 0. D. I only converges for α < 0.