Exam # 3
Spring 2005
MATH 1220-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) There are 10 problems, each worth 10 points.
(3) Justify all you answers. Correct answers with no justification will not be given any credit.
(4) No books, notes or graphing calculators are allowed.
(5) Calculators may be used only for the problems marked with a c in front.
Problem #
1
2
3
4
5
# Points
1
6
7
8
9
10
Total
2
Problem 1. Find the convergence set and the radius of convergence of the following power series:
∞
X
(−1)n (x + 2)n
n=1
n3
.
3
Problem 2. Find the power series representation for the following functions:
2
(a) et =
(b)
Z
0
x
2
et dt =
4
Problem 3. Find the terms through x3 in a Maclaurin series for the function
1
.
f (x) = √
x+1
5
Problem 4. Find the Taylor polynomial of order 3, P3 (x) of f (x) = e2x based at a = 1.
6
Problem 5. (a) Using the Trapezoidal Rule with n = 3, approximate the following integral:
Z π
2
sin2 x dx.
0
c (b) Determine an n such that the Trapezoidal rule approximate the integral from (a) with an
error En satisfying |En | ≤ 0.0001.
Hint: use that (sin2 x)′′ = 2 cos2 x − 2 sin2 x.
7
√
Problem 6. Using Newton’s Method to approximate π one obtains the following table:
n
xn
1
2
2 1.785398
3 1.772501
4 1.772454
5 1.772454
√
(a) The approximation of π to four decimal places is
√
π≈
(b) Explain how x4 was obtained from x3 . ( c You should be able, using your calculator, to get
x4 plugging x3 in an appropriate expression.)
8
c Problem 7. Consider the equation x =
to find x2 , x3 and x4 .
√
1 + x. Apply the Fixed-point Algorithm with x1 = 1
9
√
√
Problem 8. (a) Find all possible (give a formula) polar coordinates of the point: (− 2, − 2).
(b) Find the Cartesian equation of the graph of r =
2
.
sin x
10
Problem 9. Match each polar equation with its graph. Explain your choices but plugging some
values of θ in the appropriate equations and by plotting the points obtained.
(a) r = 2 − 3 sin θ,
(b) r = 3 − 2 sin θ,
3
,
(c) r = 3 cos θ,
(d) r =
cos θ
(e) r = 3 sin(2θ),
(f) r = 3 cos(2θ).
11
Problem 10. Find the area of the region bounded by the cardioid r = 2 + 2 cos θ.
12
Useful formulas
x2 x3 x4 x5
+
−
+
− · · · , for −1 < x ≤ 1.
2
3
4
5
x3 x5 x7 x9
−1
+
−
+
+ · · · , for −1 ≤ x ≤ 1.
tan x = x −
3
5
7
9
x2 x3 x4 x5
+
+
+
+ ···.
ex = 1 + x +
2!
3!
4!
5!
x3 x5 x7 x9
sin x = x −
+
−
+
− ···.
3!
5!
7!
9!
x2 x4 x6 x8
+
−
+
− ···.
cos x = 1 −
2! 4! 6! 8! p
p
p
p
p
(1 + x)p = 1 +
x+
x2 +
x3 +
x4 +
x5 + · · · , for −1 < x < 1.
1
2
3
4
5
The remainder in the Taylor’s Formula:
1. ln(1 + x) = x −
2.
3.
4.
5.
6.
7.
f (n+1) (c)
(x − a)n+1 ,
(n + 1)!
where c is some point in the interval (a − r, a + r).
8. The error for the Trapezoidal Rule is given by the formula:
(b − a)3 ′′
f (c)
En = −
12n2
where c is some point between a and b.
Rn (x) =