Question

1-(Partial Fraction Decomposition Revisited) Consider the rational function 1/(1-x)(1-2x)

(a) Find power series expansions separately for 1/(1 − x) and 1/(1 − 2x).

(b) Multiply these two power series expansions together to get a power series ex-pansion for

1 (1−x)(1−2x)

(This involves doing an infinite amount of distributing and combining coeffi-cients, but you should be able to figure out the pattern here.)

c) Separate the power series in terms of power series for A/(1 − x) and B/(1 − 2x) for some constants A and B. (This was definitely not easier, but it is interesting

to know there is another way.

2- Time to remember some things from Calculus I)

(a) If the function f has a local maximum at a, must the second
Taylor polynomial for f at a also have a local maximum at a?

(b) If the second Taylor polynomial for f at a has a local maximum
at a, must the function f have a local maximum at a?

(c) If the function f has an inflection point at a, what does the second Taylor polynomial for f at a look like?

Answer #1

1. Consider the function f(x) = 2x^2 - 7x + 9
a) Find the second-degree Taylor series for f(x) centered at x =
0. Show all work.
b) Find the second-degree Taylor series for f(x) centered at x =
1. Write it as a power series centered around x = 1, and then
distribute all terms. What do you notice?

Find the power series representation for the function: f(x) =
x2 / (1+2x)2 Determine the radius of
convergence.

let
f(x)=ln(1+2x)
a. find the taylor series expansion of f(x) with center at
x=0
b. determine the radius of convergence of this power
series
c. discuss if it is appropriate to use power series
representation of f(x) to predict the valuesof f(x) at x= 0.1, 0.9,
1.5. justify your answe

Use differentiation to find a power series representation for
the following function: f(x) = x/ (1 + 2x)^2

Calculus, Taylor series Consider the function f(x) = sin(x) x .
1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s
remainder theorem to get the same result: (a) Write down P1(x), the
first-order Taylor polynomial for sin(x) centered at a = 0. (b)
Write down an upper bound on the absolute value of the remainder
R1(x) = sin(x) − P1(x), using your knowledge about the derivatives
of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x...

Find a power series representation for the function; find the
interval of convergence. (Give your power series representation
centered at x = 0.) f(x) = (x2 + 1)/ (2x − 1)

The function f(x)=lnx has a Taylor series at a=4 . Find the
first 4 nonzero terms in the series, that is write down the Taylor
polynomial with 4 nonzero terms.

consider the function f(x) = x/1-x^2
(a) Find the open intervals on which f is increasing or
decreasing. Determine any local minimum and maximum values of the
function. Hint: f'(x) = x^2+1/(x^2-1)^2.
(b) Find the open intervals on which the graph of f is concave
upward or concave downward. Determine any inflection points. Hint
f''(x) = -(2x(x^2+3))/(x^2-1)^3.

Find the Taylor series for f(x) = e^3+2x centered at x = −1.

Consider the following function.
f(x) = ln(1 + 2x), a =
1, n = 3, 0.8 ≤ x ≤
1.2
(a) Approximate f by a Taylor polynomial with degree
n at the number a.
T3(x) =
(b) Use Taylor's Inequality to estimate the accuracy of the
approximation
f(x) ≈ Tn(x) when x lies in the given
interval. (Round your answer to six decimal places.)
|R3(x)| ≤
(c) Check your result in part (b) by graphing
|Rn(x)|.

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