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Material Type: Exam; Professor: Jones; Class: Calculus; Subject: Mathematics; University: University of California - Berkeley; Term: Spring 2008;
Typology: Exams
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There are 500 points altogether.
The first 15 questions are multiple choice, each worth 15 points. Choose the most correct answer to each question and mark the corresponding box in the grid ON THE BACK OF THIS PAGE. Mark only one box per question. No partial credit.
TA use only:
Total
Question a b c d e
(4) Which of the following is MOST CORRECT for the complex numbers Z and W marked with x’s in the picture of the complex numbers below? (The dashed circle represents the unit circle - that is to say all complex numbers of modulus one.) (a)Z = W + 3i (b)Z = W 2 (c)W = Z^2 (d)Z = 1/W (e)Z = 2W
x
0 1
x i
Z W
(a) e
√ 5 (cos(tan−^1 (2)) + i sin(tan−^1 (2)))
(b) e
√ (^5) (cos(tan− (^1) (2)) − i sin(tan− (^1) (2)))
(c) e(cos 2 + i sin 2) (d) e^2 (cos 1 + i sin 1) (e) a horse
7)Which of the following is true for any sequence {an} with lim n→∞ an = ∞?
(a) There is an N > 0 for which an > 2 for all n ≤ N. (b) There is an N for which |an − 4 | < 1 for all n ≥ N. (c) lim n→∞ (an + an+1) = ∞.
(d) For no value of n is an smaller than 300. (e)For any ǫ > 0 there is an N with |an| < ǫ for all n ≥ N.
8)The integral
1
x^2 − 4 x + 4
is
(a) divergent (b) 1 (c) − 1 (d) ln(| 3 |) (e) ln(3)
9)The general solution to the differential equation y′′^ +4y′^ +5y = 0 is (where c 1 , c 2 , A and φ are arbitrary constants)
(a) c 1 e^2 x^ + c 2 ex
(b) c 1 e^2 x^ + c 2 e−x
(c) e
√ 2 x(c 1 cos √x 5 +^ c^2 sin^ √x 5 )
(d) Ae−^2 x^ sin(x + φ)
(e) Ae−x^ cos(4x + φ)
(a) 2
d^2 S dt^2
dS dt
(b)
d^2 S dt^2
dS dt
(c)
dS dt
100 + t
(d)
dS dt
100 + t
(e)
dS dt
(a)2π
1
(9 − x)
1 + (ln x)^2 dx
(b)2π
∫ (^) e 2
e
(ex^ − 9)
1 + e^2 x^ dx
(c)2π
∫ (^) e 2
e
(9 − ln(y))
1 + e^2 y^ dy
(d)2π
∫ (^) e^2
e
(ln(y) − 9)
y^2
dy
(e)2π
1
(9 − ey^ )
1 + e^2 y^ dy
ex 2 − cos x − 3 x
2 2 ln(1 − x^4 )
equals
(a) ∞ (b) − (^1124) (c) 1 (d) (^1532) (e) 0
The next six questions are not multiple choice. Show your reasoning and give your answers in the space provided.
1.(60 points)
Find two linearly independent solutions to the differential equation
y′′^ + xy = 0
. Don’t worry about expressing your answer in terms of factorials. If you write down the first four terms of each series correctly and the pattern is clear you will get full credit.
3.(40 points) Solve the initial value problem
y′′^ + 2y′^ + y = 0, y(0) = 0, y′(0) = 1
.
6)(45 points-15 each) Evaluate the following integrals:
(i)
0
4 − x^2 dx
(ii)
1
xe^
dx
(iii)
x(ln x)^2 dx
