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Math 10560, Practice Final Exam: Instructor: May 1, 2012
- Be sure that you have all 15 pages of the test.
- No calculators are to be used.
- The exam lasts for two hours.
- When told to begin, remove this answer sheet and keep it under the rest of your test. When told to stop, hand in just this one page.
- The Honor Code is in effect for this examination, including keeping your answer sheet under cover. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
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Instructor:
Multiple Choice
1.(6 pts.) Let f (x) = ex^ −1 and let f −^1 denote the inverse function. Then (f −^1 )′(e^2 −1) = is
(a) e−^1 (b)
e^2 − 1 (c) e
(d) e^2 (e) e−^2
2.(6 pts.) Solve the following equation for x:
ln(x + 4) − ln x = 1.
(a) x =
1 − e (b) x =
e − 1 and x =
e + 1
(c) There is no solution. (d) x = e + 2 and x = e − 2
(e) x =
e − 1
Instructor:
5.(6 pts.) The integral (^) ∫ π/ 2 0
x cos(x)dx
is
(a) divergent (b)
(c) π 2
(d) 0 (e) 1 − π 2
6.(6 pts.) Evaluate (^) ∫ x^2 √ 9 − x^2
dx.
(a)
[
arcsin(x/3) − x 3
]
x
9 − x^2 + C
(c) 9 arcsin(x/3) + C (d)
[
arcsin(x/3) − x
9 − x^2 9
]
+ C
(e)
[
arcsin(x/3) − x^2 9
]
+ C
Instructor:
7.(6 pts.) If you expand 2 x + 1 x^3 + x as a partial fraction, which expression below would you
get?
(a)
x
−x + 2 x^2 + 1 (b)
x^2
x + 1
(c)
x
x^2 + 1 (d)
x
x x^2 + 1
(e)
x
x^2 + 1
8.(6 pts.) The integral (^) ∫ 2 0
1 − x dx
is
(a) π √ 2
(b) 0 (c) divergent
(d) π 6 (e) ln 2
Instructor:
11.(6 pts.) If x dy dx
x
, and y(1) = 10, find y(2).
(a) 2 (b)
(c) 7 (d)
(e) 0
12.(6 pts.) The solution to the initial value problem
y′^ = x cos^2 y y(2) = 0
satisfies the implicit equation
(a) cos y = x − 1
(b) ey 2 = ecos^ x^ − ecos 2
(c) tan(y) = x^2 2
(d) cos(y) = x + cos(2)
(e) e^2 y+1^ = arcsin(x − 2) + e
Instructor:
13.(6 pts.) The general solution to the differential equation
d^2 y dt^2
dy dt
is given by
(a) y(t) = c 1 e^2 t^ + c 2 et^ (b) y(t) = c 1 et^ + c 2 te−t
(c) y(t) = c 1 cos(2t) + c 2 sin(t) (d) y(t) = c 1 e^2 t^ + c 2 tet
(e) y(t) = c 1 e−^2 t^ + c 2 et
14.(6 pts.) Find
∑^ ∞
n=
22 n 3 · 5 n−^1
(a)
(b)
(c)
(d)
(e)
Instructor:
17.(6 pts.) The interval of convergence of the series ∑^ ∞
n=
(x + 3)n √ n
is
(a) [2, 4] (b) [− 4 , −2) (c) (− 4 , −2)
(d) (2, 4) (e) (− 1 , 1)
18.(6 pts.) If f (x) =
∑^ ∞
n=
(−1)n^ (x − 2)n (2n + 1)! , find the power series centered at 2 for the
function
∫ (^) x
2
f (t) dt.
(a) The given function can not be represented by a power series centered at 2.
(b)
∑^ ∞
n=
(−1)n^ (x − 2)n+ (n + 1)!
(c)
∑^ ∞
n=
(−1)n^ (x − 2)n+ (n + 1)(2n + 1)!
(d)
∑^ ∞
n=
(−1)n^ (x − 2)^2 n+ (n + 1)(2n)!
(e)
∑^ ∞
n=
(−1)n^ (x − 2)n+ (n^2 )(2n + 1)!
Instructor:
19.(6 pts.) Which series below is the MacLaurin series (Taylor series centered at 0) for x^2 1 + x
(a)
∑^ ∞
n=
(−1)nx^2 n^ (b)
∑^ ∞
n=
(−1)nx^2 n−^2 n! (c)
∑^ ∞
n=
x^2 n+
(d)
∑^ ∞
n=
(−1)n^ xn+2^ (e)
∑^ ∞
n=
xn+ n + 2
20.(6 pts.) Find the degree 3 MacLaurin polynomial (Taylor polynomial centered at 0) for the function ex 1 − x^2
(a) 1 + x − x^3 6 (b) 1 − x^2 2
x^3 5
(c) 1 + x − 5 x^3 3
(d) 1 + x + 3 x^2 2
7 x^3 6
(e) 1 + x + x^2 6
Instructor:
23.(6 pts.) The point (2, 11 π 3 ) in polar coordinates corresponds to which point below in Cartesian coordinates?
(a) (1, −
(b) (
(c) (−
(d) (− 1 ,
(e) Since 11 π 3
2 π, there is no such point.
24.(6 pts.) Find the equation for the tangent line to the curve with polar equation: r = 2 − 2 cos θ at the point θ = π/2.
(a) y = 2 − π + 2x (b) y = 2 + π 2 − x (c) y = 2 − x
(d) y = 0 (e) y = 2 + 2x
Instructor:
25.(6 pts.) Find the length of the polar curve between θ = 0 and θ = 2π
r = e−θ.
(a) 2 e−^4 π^ (b) 14 (1 − e−^4 π) (c) 2 − e−^2 π
(d) 2 π(1 + e−^2 π) (e)
2(1 − e−^2 π)
Math 10560, Practice Final Exam: Instructor: ANSWERS May 1, 2012
- Be sure that you have all 15 pages of the test.
- No calculators are to be used.
- The exam lasts for two hours.
- When told to begin, remove this answer sheet and keep it under the rest of your test. When told to stop, hand in just this one page.
- The Honor Code is in effect for this examination, including keeping your answer sheet under cover. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
- (a) (b) (c) (d) (•)
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- (a) (•) (c) (d) (e)
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