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Name KEY Instructor Section No.
Student Number
Math 113/113H – Winter 2006
Departmental Final Exam
Instructions:
- The time limit is 3 hours.
- Problems 1-6 short-answer questions, each worth 2 points.
- Problems 7 through 13 are multiple choice questions, each worth 3 points.
- For problems 14 through 24, give the best answer and justify it by giving suitable reasons and/or by showing relevant work.
- Work on scratch paper will not be graded.
- Please write neatly.
- Notes, books, and calculators are not allowed.
- Expressions such as ln(1), e^0 , sin(π/2), etc. must be simplified for full credit.
For administrative use only:
M.C. /
Total /
Short Answer.
ln x dx = x ln x − x + C
∫ (^) π/ 2 0 sin^ x dx^ =^ −^ cos^ x|
π/ 2 0 = 0^ −^ (−1) = 1
tan x sec^2 x dx = 12 tan^2 x + C or 12 sec^2 x + C
- The Taylor series for e^2 x, centered at 0, is 1 + 2x + (2x)
2 2! +^
(2x)^3 3! +^...^ =^
n=
2 nxn n!
∫ (^) x 2 √ x^2 + 4
dx can be integrated using the trigonometric substitution x = 2 tan θ
- The definite integral
0 2 πx
x^2 + 1 dx represents the volume of the solid of revolution generated
by revolving the curve y =
x^2 + 1 , x ∈ [ 0 , 3 ], about the y-axis.
Multiple Choice. Use the following chart to show your choices for Problems 7–13. Shade in your choice. Only this chart will be graded on these problems.
Problem 7 (a) (b) (d) (e) (f) (g) (h) (i) (j) Problem 8 (a) (b) (d) (e) (f) (g) (h) (i) (j) Problem 9 (a) (b) (c) (d) (f) (g) (h) (i) (j) Problem 10 (a) (b) (c) (d) (f) (g) (h) (i) (j) Problem 11 (b) (c) (d) (e) (f) (g) (h) (i) (j) Problem 12 (a) (b) (d) (e) (f) (g) (h) (i) (j) Problem 13 (a) (b) (c) (e) (f) (g) (h) (i) (j)
1
xp^
dx
(a) converges if 0 < p < 1 (b) converges if p = 1 (c) converges if p > 1
(d) diverges if p > 1 (e) diverges if p > 0 (f) diverges if p 6 = 1
(g) none of these
- By using two non-zero terms of the Taylor series, the integral
0 sin(x
(^2) ) dx is approximately
(to two digit accuracy)
(a) .96 (b) .43 (c) .31 (d) .29 (e) .12 (f).
ex^ cos x dx =
(a) ex^ sin x + C (b) ex^ cos x − ex^ sin x + C (c) ex^ sin x − ex^ sin 2x + C
(d) 12 ex^ sin x + 12 ex^ cos x + C (e) 2ex^ sin x − 2 ex^ cos x + C (f) none of the above
Answers to Problems 12 and 13 must be shaded in the chart on Page 1 to be counted.
Essay Problems Work Problems 14–24 as you would homework problems, showing your steps and justifying them.
- Find a formula for
1 − a^2 x^2 dx (a > 0). x = (^1) a sin θ, dx = (^1) a cos θ dθ,
1 − a^2 x^2 = cos θ =⇒ ∫ (^) √ 1 − a^2 x^2 dx =
cos θ · 1 a
cos θ dθ
=^1 a
cos^2 θ dθ
=
2 a
(1 + cos 2θ) dθ
=
2 a [θ^ +
2 sin 2θ] +^ C = 1 2 a
[θ + sin θ cos θ] + C
= 1 2 a
[sin−^1 (ax) + ax
1 − a^2 x^2 ] + C
- Find the length of the curve y = ln(sin x), x ∈ [ 16 π, 14 π]. y′^ = cos sin xx = cot x;
s =
∫ (^) π/ 4
π/ 6
1 + cot^2 x dx =
∫ (^) π/ 4
π/ 6
csc x dx
= ln | csc x − cot x|
∣∣π/ 4 π/ 6 = ln(
2 − 1) − ln(2 −
- Evaluate
∫ (^) π/ 2
0
sin^3 x cos^2 x dx.
∫ (^) π/ 2
0
sin^3 x cos^2 x dx =
∫ (^) π/ 2
0
sin^2 x cos^2 x sin x dx
∫ (^) π/ 2
0
(1 − cos^2 x) cos^2 x sin x dx
= −
1
(1 − u^2 )u^2 du [u = cos x, du = − sin x dx]
0
(u^2 − u^4 )du
=
3 u
5 u
5
1 0 =
3 −^
5 =^
- A swimming pool has a circular window of radius 1.2 meters in a side wall. When the water in the pool exactly covers the lower half of the window, what is the force of the water pressure on the window? (Assume that the side wall is vertical; give your answer in terms of the weight-density w of the water.)
Let r = 1.2 be the radius of the window. A horizontal strip a distance h below the diameter of the window has length l = 2
r^2 − h^2. Therefore the force on the window is
F =
∫ (^) r
0
wh · 2
r^2 − h^2 dh
= −w
r^2
u^1 /^2 du [u = r^2 − h^2 , du = − 2 hdh]
=^2 3
wu^3 /^2
r^2 0 =^2 3
wr^3 = 1. 152 w
- Find
∫ (^) x + 1 x^2 − 4 dx.
x + 1 x^2 − 4
= A
x + 2
+ B
x − 2
=⇒ x + 1 = A(x − 2) + B(x + 2) = (A + B)x + 2(B − A) =⇒ A = 14 , B = 34.
Therefore ∫ x + 1 x^2 − 4
dx =
x + 2
+ 3 /^4
x − 2
dx =^1 4
ln |x + 2| +^3 4
ln |x − 2 | + C
- For each of the following power series, determine what function it converges to, and find the interval of convergence (watch the end-points).
(a)
∑^ ∞
n=
nxn−^1
∑^ ∞
n=
nxn−^1 = d dx
n=
xn
= d dx
1 − x
(1 − x)^2
The geometric series converges on (− 1 , 1). At both –1 and 1, we get divergent series, so (− 1 , 1) is the interval of convergence.
(b)
∑^ ∞
n=
(−1)n^
x^3 n n!
∑^ ∞
n=
(−1)n^ x
3 n n!
∑^ ∞
n=
(−x^3 )n n!
= e−x^3.
This series converges for all x ∈ R.
- (a) Give the formal definition of the statement (^) nlim→∞ an = L.
nlim→∞ an^ =^ L^ ⇐⇒ ∀ >^0 ,^ ∃k^ ∈^ N^ :^ n > k^ =⇒ |an^ −^ L|^ < .
(b) Determine the limit (^) nlim→∞
1 + πn
)n (by any means).
nlim→∞
1 + π n
)n = exp
n^ lim→∞ n^ ln(1 +^ π n
= exp
n^ lim→∞
ln(1 + πn ) 1 /n
= exp
n^ lim→∞^ −(π/n
(^2) )/ ln(1 + π/n) − 1 /n^2
= exp
π (^) nlim→∞1 +^1 π/n
= eπ
Or use Theorem 117(5).
- (a) Sketch the graph of r = eθ/^2.
(b) Find the area inside the curve r = eθ/^2 and outside the circle r = 1 for 0 ≤ θ ≤ π.
A =
∫ (^) π 0
1 2 r^2 dθ^ −^
1 2 π^ =^
∫ (^) π 0
1 2 eθdθ^ −^
1 2 π^ =^
1 2 eθ|π 0 −^
1 2 π^ =^
1 2 eπ^ −^
1 2 −^
1 2 π.
(c) Find the slope of the polar curve r = eθ/^2 at the point [1, 0].
y = eθ/^2 sin θ =⇒ dy dθ
=^1
eθ/^2 sin θ + eθ/^2 cos θ =⇒ dy dθ
|θ=0 = 1;
x = eθ/^2 cos θ =⇒ dxdθ =^12 eθ/^2 cos θ − eθ/^2 sin θ =⇒ dxdθ |θ=0 =^12 ;
dy dx =^
