MATH 148, SPRING 2016
COMMON EXAM I (PART 1) - VERSION A
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Question
1–10
Points Awarded
Points
50
1
PART I: Multiple Choice (5 points each)
Z
1. Evaluate
x sin(5x) dx.
(a) −5x cos(5x) + 25 sin(5x) + C
x
1
(b) − cos(5x) +
sin(5x) + C
5
25
(c) None of these.
x
1
(d) cos(5x) +
sin(5x) + C
5
25
(e) 5x cos(5x) + 25 sin(5x) + C
2. Find the third-degree Taylor polynomial for f (x) =
1
(a) T3 (x) = 1 + (x − 1) −
2
1
(b) T3 (x) = 1 + (x − 1) −
2
1
(c) T3 (x) = 1 + (x − 1) −
2
1
(d) T3 (x) = 1 + (x − 1) −
2
1
(e) T3 (x) = 1 + (x − 1) −
2
1
(x − 1)2 +
4
1
(x − 1)2 +
8
1
(x − 1)2 +
8
1
(x − 1)2 +
4
1
(x − 1)2 +
8
√
x at a = 1.
3
(x − 1)3
8
3
(x − 1)3
16
1
(x − 1)3
8
1
(x − 1)3
8
1
(x − 1)3
16
2
2
Z
x3 ln x dx.
3. Evaluate
1
3
4
15
4 ln 2 −
16
4 ln 2
3
4 ln 2 +
4
15
4 ln 2 +
16
(a) 4 ln 2 −
(b)
(c)
(d)
(e)
4. Solve the differential equation
dy
= xy with initial condition y(0) = 1.
dx
(a) y = ex
(b) y = ex
2
(c) y = e2x
(d) y = ex
2
/2
(e) y = e2x
2
3
Z
4
5. Evaluate
2
x2 + 1
dx. (Hint: Use long-division to simplify the integrand.)
x−1
(a) 4
(b) 8 + 2 ln 3
(c) 4 + 2 ln 3
(d) 6 + ln 3
(e) 2 + 2 ln 3
Z
6. Evaluate
0
1
√
x
x2
+1
dx.
(a) None of these.
1
(b)
2
√
(c) 2
√
√
(d) 5 − 2
√
(e) 4 2 − 4
4
Z
7. Evaluate
2
4
1
dx.
(x − 4)3
1
4
1
(b)
8
(a)
1
4
1
(d) −
8
(e) The integral diverges.
(c) −
Z
8. Which of the following statements is true about the integral
1
Z
∞
(c) The
(d) The
(e) The
4
√ dx?
x2 + x
4
dx.
2
x
1
Z ∞
4
dx.
integral converges by comparison with
2
x
1
Z ∞
4
√ dx.
integral diverges by comparison with
x
1
Z ∞
4
√ dx.
integral converges by comparison with
x
1
convergence/divergence of the integral cannot be determined.
(a) The integral diverges by comparison with
(b) The
∞
5
Z
9. Evaluate
∞
e−2x dx.
0
1
4
(b) 0
(a)
(c) 2
1
(d)
2
(e) The integral diverges.
Z
10. Evaluate
x+2
dx.
x2 (x − 1)
(a) −3 ln |x| +
2
+ 3 ln |x − 1| + C
x
2
+ 3 ln |x − 1| + C
x
2
(c) −3 ln |x| − + 3 ln |x − 1| + C
x
2
(d) 3 ln |x| + + 2 ln |x − 1| + C
x
2
(e)
+ 3 ln |x − 1| + C
x
(b) −
6