Review problems-Exam#3
MATH 1220, Spring 2005
(1) (§10.6) Find the convergence set of the following power series:
(2) (§10.7) Find the power series representation for f (x) =
Z
∞
X
(−1)n (x − 1)n
.
n
n=1
x
ln(1 + t) dt.
0
1
.
(1 + x)3
1
.
(4) (§10.8) Find the first three nonzero terms of the Maclaurin series for the function f (x) =
1 + sin x
π
(5) (§10.8) Find the Taylor series in x − a through (x − a)2 of the function f (x) = sin x with a = .
3
(6) (§10.6-10.8) Is it true or false?
(a) If f (x) = 2 − x + x2 − x3 + x4 − · · · , then f ′′ (0) > 0.
∞
X
(n + 1)! n
1
(b) If f (x) =
x , then f ′′′ (0) = .
(2n)!
5
n=1
(3) (§10.7 or 10.8) Find a formula for the the n-th term (an ) of the Maclaurin Series for f (x) =
(c) If f (x) =
∞
X
n=1
an (x − a)n , then an = f (n) (a).
(d) The radius of convergence is the series
∞ n
X
x
2
n=1
is
1
.
2
(7) (§11.1) Find the Maclaurin polynomial of order 4 of f (x) = sin(2x) and use it to approximate sin 2.
√
(8) (§11.1) Find the Taylor polynomial P3 (x) for f (x) = x based at a = 1.
Z 3
2
(9) (§11.2) Using the Trapezoidal Rule with n = 3, approximate the following integral:
e−x dx.
Z 20
√
sin x dx with an error
(10) (§11.2) Determine n so that the Trapezoidal Rule will approximate the integral
1
En satisfying |En | ≤ 0.01.
(11) (§11.3) Approximate the real root of the equation f (x) = x3 + 2x − 6 on the interval [1, 2] using the Bisection
Method by filling in the following table
n
hn
mn
f (mn )
1
2
3
(12) (§11.3) Using the Newton’s Method approximate
√
3
6 by filling in the following table:
n
xn
1
2
3
3
Hint: use f (x) = x − 6.
1
2
1
(13) (§11.4) Consider the equation x = 1 + . (a) Apply the Fixed-point Algorithm with x1 = 1 to find x2 , x3 and
x
1
x4 . (b) Solve the equation x = 1 + algebraically.
x
(14) (§12.6) Find the Cartesian equation of the graphs of the polar
√ equation r − 2 cos θ = 0 and sketch the graph.
(15) (§12.6) Find all possible polar coordinates of the point: (−2 3, 2).
(16) (§12.8) The graph of the limaçon r = 3 − 4 cos θ is:
Find the area of the region inside its small loop.
(17) (§12.7) Match each polar equation with its graph.
2
(a) r = 1 − 2 sin θ, (b) r = 2 sin θ, (c) r =
, (d) r = 1 − 2 cos θ, (e) r = 2 sin(3θ), (f) r = 2 − cos θ.
sin θ
3