Final Exam of Calculus I(MAS101)
2010. 5. 18(Tue)
7:00 PM ∼ 10:00 PM
Spring 2010
MAS101
Full Name :
Student Number:
Seat Number :
Do not write in this box.
Directions
• Turn off your mobile phone. No items other than pen, pencil, eraser and your id are allowed.
• Before you start, fill out the identification section on the title page and on the header of each page
with an inerasable pen.
• Show your work for full credit and write the final answer to each problem in the box provided.
You are encouraged to write in English but some Korean is acceptable. You can use the back of
each page as scratch spaces.
• The exam is for three hours. Ask permission by raising your hand if you have any question or
need to go to the toilet.
• Any attempt to cheat in the exam leads to serious disciplinary actions not limited to failing the
exam or the course.
Do not write in this table.
Problem
Score
Problem
Score
Problem
Score
1(a)
/ 10
1(b)
/ 10
2(a)
/ 10
2(b)
/ 15
3(a)
/ 10
3(b)
/ 10
4
/ 15
5(a)
/ 10
5(b)
/ 10
5(c)
/ 10
6(a)
/ 10
6(b)
/ 15
7
/ 15
8
/ 15
9(a)
/ 10
9(b)
/ 15
10
/ 10
Total
/ 200
–1–
MAS101
Name:
1. We are approximating the irrational number π.
∫ 1
1
dx = π/4, approximate π by evaluating the integral with Simpson’s rule for n = 4. (Write your final
(a) Using the fact
2
0 1+x
answer as a sum of fractions. In fact, it should be 3.141568.... on a hand-held calculator)
Ans:
(b) If you approximate π by using the Taylor series
tan−1 x =
what is the smallest possible integer n such that 4
∞
∑
(−1)k 2k+1
x
,
(2k
+
1)
k=0
|x| ≤ 1,
n
∑
(−1)k
gives an estimate of π that is correct up to the 4th decimal
(2k
+
1)
k=0
place.
Ans:
–2–
MAS101
Name:
2. Consider the region enclosed by the cycloid
x(θ) = θ − sin θ,
y(θ) = 1 − cos θ,
0 ≤ θ ≤ 2π
and the x-axis.
(a) Find the area or the region.
Ans:
(b) Find the centroid of the region.
Ans:
–3–
MAS101
Name:
3. Determine whether the following improper integrals converge:
∫ ∞ −t
e
√ dt.
(a)
t
0
Ans:
∫
(b)
0
1
1
dx.
sin x − x2
Ans:
–4–
MAS101
Name:
4. Find the function y = f (x) satisfying
y ′ = cos 2x + y 2 cos 2x,
and f (0) = 1.
Ans:
5. Determine whether the following series converge absolutely, converge conditionally, or diverge. (Specify the test used for each series.)
(a)
∞
∑
(−1)n an , where a1 = 1, an+1 = 12 (an + 2) for n ≥ 2.
n=1
Ans:
–5–
MAS101
Name:
 2

n ,
n = prime,
(b)
bn , where bn = 3n 1

− , otherwise.
n=1
3n
∞
∑
Ans:
(c)
∞
∑
n=2
(−1)n
(ln n)2
n
Ans:
–6–
MAS101
Name:
6. Find all x that make the following power series converge.
(a)
∞
∑
n=2
xn
√
n ln n
Ans:
(b)
∞
∑
(x2 − 1)2n+1
n=1
n
–7–
MAS101
Name:
Ans:
√
7. Find the Maclaurin series of f (x) = ln( 1 + x2 + x) and find its radius of convergence.
Ans:
–8–
MAS101
Name:
8. Using a Taylor series, give an estimate of
∫
1
e−x dx
2
0
−3
with an error ≤ 10 . (Write your final answer as a sum of fractions.)
Ans:
9. For the conic section with one focus at the origin, the eccentricity 3/2, and the corresponding directrix x = 34 , answer the following:
(a) Find its polar equation and its Cartesian equation.
Ans:
–9–
MAS101
Name:
(b) Find the coordinates of the center, the vertices and the foci and the equations of the directrices and sketch the conic section
together with all essentials obtained.
Ans:
10. The following is a maple code to draw the graph of r = f (theta) and calculate the length of the graph for theta ∈ [0, 2π]. Assuming
that f is already defined, fill the blanks.(Caution: Do not use plot to draw the graph!!)
with(____________);
______(___________________);
int(____________________);
Ans:
– 10 –