MATH 147, SPRING 2016
COMMON EXAM III (PART 1) - VERSION A
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Question
1–13
Points Awarded
Points
52
1
PART I: Multiple Choice (4 points each)
1. The equilibria (fixed points) of the difference equation (recursion) xt+1 =
3xt
are x∗ = 0 and x∗ = 1.
xt + 2
Which of the following statements (if any) is true?
(a) The equilibrium x∗ = 0 is locally stable and approached without oscillations.
(b) The equilibrium x∗ = 0 is locally stable and approached with oscillations.
(c) The equilibrium x∗ = 1 is locally stable and approached without oscillations.
(d) The equilibrium x∗ = 1 is locally stable and approached with oscillations.
(e) None of the above.
2. Determine where f (x) = x2 ex has local extrema.
(a) Local minimum at x = 0 and no local maximum
(b) Local maximum at x = −2 and local minimum at x = 0
(c) Local maximum at x = 2 and local minimum at x = 0
(d) Local maximum at x = −2 and no local minimum
(e) Local maximum at x = 2 and no local minimum.
2
3. Find the most general antiderivative of f (x) = 2x3 − 5x +
(a)
(b)
(c)
(d)
(e)
3
+ sin x.
x
1 4 5 2
x − x + 3 ln |x| + cos x + C
2
2
3
6x2 − 5 − 2 + cos x
x
1 4 5 2
3
x − x + 2 + cos x + C
2
2
2x
1 4 5 2
x − x + 3 ln |x| − cos x + C
2
2
None of the above.
4. The curve below is the graph of f for a recursion (difference equation) xt+1 = f (xt ).
Which of the following statements is true?
(a) If x0 = 0.5, then lim xt = 2
t→∞
(b) If x0 = 0.5, then lim xt = 0
t→∞
(c) If x0 = 1.5, then lim xt = 2
t→∞
(d) If x0 = 1.5, then lim xt = 1
t→∞
(e) None of the above.
3
5. Find all vertical and horizontal asymptotes of f (x) =
3x2 + 6x
.
x2 − 2x − 8
(a) Vertical: x = −4 and x = 2
Horizontal: y = 3
(b) Vertical: x = 4 and x = −2
Horizontal y = 3
(c) Vertical: x = −4
Horizontal: None
(d) Vertical: x = −4 and x = 2
Horizontal: None
(e) Vertical: x = −4
Horizontal: y = 3
6. The graph of the first derivative f 0 of a function f is shown. Find the intervals on which f is concave upward.
(a) (2, 4) ∪ (6, 8)
(b) (1, 3) ∪ (5, 7) ∪ (8, 9)
(c) (2, 4) ∪ (6, 9)
(d) (0, 2) ∪ (4, 6)
(e) (0, 1) ∪ (3, 5) ∪ (7, 8)
4
7. Find the absolute (global) extrema of f (x) = 3x − x3 on the interval [0, 3].
(a) Absolute maximum: 2, Absolute minimum: −18
(b) Absolute maximum: 0, Absolute minimum: −18
(c) Absolute maximum: 2, Absolute minimum: −2
(d) Absolute maximum: 2, Absolute minimum: 0
(e) Absolute maximum: 4, Absolute minimum: −2
8. Suppose that f 0 (x) = 3 cos x + 5 sin x and f (0) = 4. Use an antiderivative to find f (π).
(a) 4
(b) −6
(c) 2
(d) 14
(e) 7
5
ex − cos x − 2x
.
x→0
x2 − 2x
9. Evaluate lim
(a) 1
(b) −
1
2
(c) 0
1
(d)
2
(e) Does not exist.
10. Evaluate lim
x→0
1
1
−
.
x cos x x
(a) −1
(b) 1
(c) 0
(d) Does not exist (∞).
(e) Does not exist (−∞).
6
ln n
11. Determine the limiting behavior of the sequence given by the formula an = √ .
n
(a) The sequence converges with limit 2.
1
(b) The sequence converges with limit .
2
(c) The sequence diverges.
(d) The sequence converges with limit 0.
(e) The sequence converges with limit 1.
12. A bacterial colony has an initial population size of 10 bacteria and triples every 30 minutes. Find a formula for the
population size, Nt , if one unit of time corresponds to one hour.
(a) Nt = 10 · 3t/3
(b) None of these.
(c) Nt = 10 · 3t
(d) Nt = 10 · 27t
(e) Nt = 10 · 9t
7
13. Consider the recursive sequence
an+1 = 8 −
Find the limit of the sequence.
(a) 4
(b) 8
(c) 2
(d) 6
(e) None of these.
8
12
,
an
a0 = 4.