Homework 1

Numerical Analysis (CMPS/MATH 305)

Hani Mehrpouyan

Homework 1 Solution

QUESTION 1 (5 POINTS)

Produce the linear and quadratic Taylor polynomials for the following cases. Graph the function and these Taylor

polynomials.

a) f(x) =√x, a = 1

b) f(x) = sin(x), a = π/4

c) f(x) = ecos(x), a = 0

d) f(x) = log(1 + ex) , a = 0

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QUESTION 2 (5 POINTS)

Produce a general formula for the degree n Taylor polynomials for the following functions, all using a = 0 as the

point of approximation.

a) 1/(1− x)

b) (1 + x)1/3

c)√1 + x

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QUESTION 3 (5 POINTS)

Compare f(x) = sin(x) with its Taylor polynomials of degrees 1, 3, and 5 on the interval −π/2 ≤ x ≤ π; a = 0.

Produce a Matlab plot or a table in the manner of Table 1.1.

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QUESTION 4 (5 POINTS)

The quotient g(x) = ex−1x is undefined for x = 0. Approximate ex by using Taylor polynomials of degrees 1,2,

and 3, in turn, to determine a natural definition of g(0).

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QUESTION 5 (5 POINTS)

a) As an alternative to the linear Taylor polynomial, construct a linear polynomial q(x), satisfying q(a) = f(a),

q(b) = f(b) for given points a and b.

b) Apply this to f(x) = ex with a = 0 and b = 1. For 0 ≤ x ≤ 1, numerically compare q(x) with the linear Taylor

polynomial of this section.

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QUESTION 6 (5 POINTS)

Find the degree 2 Taylor polynomial for f(x) = ex sin(x), about the point a = 0. Bound the error in this

approximation when −π/4 ≤ x ≤ π/4.

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QUESTION 7(5 POINTS)

Let Pn(x) be the Taylor polynomial of degree n of the function f(x) = log(1−x) about a = 0. How large should

n be chosen to have |f(x)− Pn(x)| ≤ 10−4 for −1/2 < x < 1/2? For −1 < x < 1/2?

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QUESTION 8 (5 POINTS)

Use Taylor polynomials with remainder term to evaluate the following limits:

a) limx→0

1− cos(x)

x2,

b) x→0log(1 + x2)

2x,

c) limx→0

log(1− x) + xex/2

x3.

(Hint: Use Taylor polynomials for the standard functions [e.g., cos(t), log(1 + t), and et] to obtain polynomial

approximations to the numerators of these fractions; and then simplify the results.)

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QUESTION 9 (5 POINTS)

Evaluate I =

1∫0

ex − 1

x∂x within an accuracy of 10−6.

(Hint: Replace ex by a general Taylor polynomial approximation p]us its remainder)

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QUESTION 10 (5 POINTS)

Define f(x) =

x∫0

log(1 + t)

t∂x

a) Give a Taylor polynomial approximation to f(x) about x = 0,

b) Bound the error in the degree n approximation for |x| ≤ 1/2,

c) Find n so as to have a Taylor approximation with an error of at most 10−7 on [−1/2, 1/2].

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QUESTION 11 (5 POINTS)

Evaluate

p(x) = 1− x3

3!+x6

6!− x9

9!+x12

12!− x15

15!,

as efficiently as possible. How many multiplications are necessary? Assume all coefficients have been computed

and stored for later use.