MATH 147, SPRING 2016
COMMON EXAM II (PART 1) - VERSION A
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Question
1–10
Points Awarded
Points
50
1
PART I: Multiple Choice (5 points each)
1. Consider the function f (x) = loga (6x), where a is a positive real number. If f 0 (1) =
(a) a = 3
(b) a = e3
(c) a = 2
(d) a = e2
(e) None of these
2. Find the derivative of f (x) = e3x cos(2x).
(a) 6e3x sin(2x)
(b) None of these
(c) 3e3x cos(2x) + 2e3x sin(2x)
(d) −6e3x sin(2x)
(e) 3e3x cos(2x) − 2e3x sin(2x)
2
1
, what is the value of a?
3
3. Find the slope of the tangent line to the graph of f (x) = x3 + 3x at x = 1.
(a) 3
(b) 6
(c) 3 + ln 9
(d) 3 + ln 3
(e) None of these
4. Strontium-90 has a half-life of 25 years. Let W (t) denote the mass of a sample of Strontium-90 remaining after t
years. Find a differential equation for W (t).
(a)
(b)
(c)
(d)
(e)
ln 2
dW
=
W (t)
dt
25
dW
ln 2
=−
W (t)
dt
25
dW
25
=−
W (t)
dt
ln 2
25
dW
=
W (t)
dt
ln 2
None of these
3
5. Find the slope of the line tangent to the curve x3 + 3xy + y 3 = 15 at the point (2, 1).
(a) −
3
5
(b) 0
5
3
1
(d) −
2
(e) −2
(c) −
2
6. Let f (x) = ex + 4x + 1, where x ≥ 0. Evaluate
df −1
(2) = (f −1 )0 (2).
dx
(a) 5
1
(b)
5
(c) 4
1
(d)
4
(e) None of these
4
7. Find the SECOND derivative of f (x) = ln(2x3 − 1).
(a)
−1
(2x3 − 1)2
−12x(1 + x3 )
(2x3 − 1)2
1
(c)
(2x3 − 1)2
(b)
(d)
12x(1 + x3 )
(2x3 − 1)2
(e) None of these
8. Suppose that f is a differentiable function with f (1) = 1, f 0 (1) = 2, f (2) = 3, and f 0 (2) = 4. Let g(x) = 3x2 − 11.
d
Find
f (g(x)) at x = 2.
dx
(a) 24
(b) 4
(c) 2
(d) 48
(e) None of these
5
9. Find the derivative of f (x) = xln x .
2 ln x
ln x
(a) x
x
2
ln x
(b) xln x
x
1
(c) xln x
x2
2 ln x
(d)
x
(e) ln x(xln x−1 )
10. Find the derivative of f (x) =
x3 + 1
.
x2 + 1
x4 + 3x2 + 2x
x2 + 1
5x4 + 3x2 + 2x
(b)
x2 + 1
x4 + 3x2 − 2x
(c)
x2 + 1
x4 + 3x2 − 2x
(d)
(x2 + 1)2
(a)
(e)
5x4 + 3x2 + 2x
(x2 + 1)2
6