MATH 152, Fall 2014
Week In Review
Week 12
1. if f (x) =
∞
X
bn (x − 5)n , for all x . Write a formula for b8 .
n=0
2. Find the Maclaurin series for f (x). Also nd the associated radius of convergence.
(a) f (x) =
1
.
(1 + x)2
(b) f (x) =
x
.
1−x
3. Find the Taylor series for f (x) at the given value a.
(a) f (x) = ln(x) a = 2 .
√
(b) f (x) = x a = 4 .
(c) f (x) = cos(x) a = −π/4 .
4. Use a Maclaurin series developed previously to obtain a Maclaurin Series for the given function.
(a) f (x) = cos(x3 ) .
(b) f (x) = xe−x .
(c) f (x) = sin2 (x) .
5. Find the Maclaurin series for f and its radius of convergence.
(a) f (x) = √
1
.
1 + 2x
(b) f (x) = (1 + x)−3 .
(c) f (x) = 2x .
6. Use the series to approximate the denite integral to within the indicated approximation
(a)
Z
0.5
cos(x2 )dx
( Three decimal places ).
0
(b)
Z
0.1
√
0
(c)
Z
0
0.5
1
dx
1 + x3
2
x2 e−x dx
( error < 10−8 ).
( error < 0.001 ).
7. Find the limit lim
x→0
1 − cosx
.
1 + x − ex
8. Find the third nonzero terms of the Maclaurin seris for each function.
(a) f (x) = e−x cos(x) .
2
(b) f (x) = sec(x) .
9. Find the sum of the series.
(a)
∞
X
(−1)n
n=0
π 2n+1
.
42n+1 (2n + 1)!
(b) f (x) =
∞
X
x3n+1
.
n!
n=2
(c) f (x) =
∞
X
xn+1 n
.
(n + 1)!
n=0
10. (a)Approximate f by a Taylor polynomial with degree n at the number a.(b) Use Taylor's Inequality to estimate
the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval.
(a) f (x) = sin(x) a = π/4, n = 5, 0 ≤ x ≤ π/2 .
(b) f (x) = ex
2
a = 0,
n = 3,
0 ≤ x ≤ 0.1 .
(c) f (x) = ln(x) a = 4, n = 3, 3 ≤ x ≤ 5 .
11. Estimate sin(35o ) correct to ve decimal places.
12. Use Taylor's Inequality to determine the number of terms of the Maclaurin Series for ex that should be used to
estimate e0.1 to within 0.0001.
13. Use the Alternating Series Estimation or Taylor's Inequality to estimate the range of values of x for which the
given approximation is accurate to within the stated error.
sinx ≈ x −
.
x3
,
6
error < 0.01
14. Find a power series representation for the following functions, and determine the interval of convergence.
(a) f (x) =
1
.
(1 + x)3
(b) f (x) = xln(1 + x) .
(c) f (x) =
x2
.
(1 − 2x)2
15. Evaluate the indenite integral as a power series
(a)
Z
x
dx .
1 + x5
(b)
Z
tan−1 (x)
dx .
x
(c)
Z
tan−1 (x2 )dx .
16. Use a power series to approximate the denite integral to six decimal places.
(a)
Z
1/2
tan−1 (x2 )dx .
0
(b)
Z
0.5
0
(c)
Z
1/3
1
dx .
1 + x6
x2 tan−1 (x4 )dx .
0
17. Dened the function f by the power series
f (x) =
∞
X
xn
.
n!
n=0
(a) Show that f is solution of the dierential equation
f 0 (x) − f (x) = 0
.
(b) Prove that f (x) = ex .
f or
all
x