10th Edition

ISBN: 9781305253667

Author: Richard L. Burden, J. Douglas Faires, Annette M. Burden

Publisher: Cengage Learning

Concept explainers

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

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Textbook Question

Chapter 1.1, Problem 8ES

Suppose f ∈ C[a, b] and f ′(x) exists on (a, b). Show that if f ′(x) ≠ 0 for all x in (a, b), then there can exist at most one number p in [a, b] with f(p) = 0.

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Chapter 1.1, Problem 7ES

Chapter 1.1, Problem 9ES

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Chapter 1 Solutions

Numerical Analysis

Ch. 1.1 - Show that the following equations have at least...Ch. 1.1 - Show that the following equations have at least...Ch. 1.1 - Find intervals containing solutions to the...Ch. 1.1 - Find intervals containing solutions to the...Ch. 1.1 - Find maxaxb |f(x)| for the following functions and...Ch. 1.1 - Find maxaxb | f(x)| for the following functions...Ch. 1.1 - Show that f(x) is 0 at least once in the given...Ch. 1.1 - Suppose f C[a, b] and f (x) exists on (a, b)....Ch. 1.1 - Let f(x) = x3. a. Find the second Taylor...Ch. 1.1 - Find the third Taylor polynomial P3(x) for the...

Ch. 1.1 - Find the second Taylor polynomial P2(x) for the...Ch. 1.1 - Repeat Exercise 11 using x0 = /6. 11. Find the...Ch. 1.1 - Prob. 13ES Ch. 1.1 - Prob. 14ES Ch. 1.1 - Prob. 15ES Ch. 1.1 - Use the error term of a Taylor polynomial to...Ch. 1.1 - Use a Taylor polynomial about /4 to approximate...Ch. 1.1 - Let f(x) = (1 x)1 and x0 = 0. Find the nth Taylor...Ch. 1.1 - Let f(x) = ex and x0 = 0. Find the nth Taylor...Ch. 1.1 - Prob. 20ES Ch. 1.1 - The polynomial P2(x)=112x2 is to be used to...Ch. 1.1 - Use the Intermediate Value Theorem 1.11 and Rolles...Ch. 1.1 - Prob. 23ES Ch. 1.1 - In your own words, describe the Lipschitz...Ch. 1.2 - Compute the absolute error and relative error in...Ch. 1.2 - Compute the absolute error and relative error in...Ch. 1.2 - Prob. 3ES Ch. 1.2 - Find the largest interval in which p must lie to...Ch. 1.2 - Perform the following computations (i) exactly,...Ch. 1.2 - Use three-digit rounding arithmetic to perform the...Ch. 1.2 - Use three-digit rounding arithmetic to perform the...Ch. 1.2 - Repeat Exercise 7 using four-digit rounding...Ch. 1.2 - Repeat Exercise 7 using three-digit chopping...Ch. 1.2 - Prob. 10ES Ch. 1.2 - Prob. 11ES Ch. 1.2 - Prob. 12ES Ch. 1.2 - Let f(x)=xcosxsinxxsinx. a. Find limx0 f(x). b....Ch. 1.2 - Let f(x)=exexx. a. Find limx0(ex ex )/x. b. Use...Ch. 1.2 - Use four-digit rounding arithmetic and the...Ch. 1.2 - Prob. 16ES Ch. 1.2 - Prob. 17ES Ch. 1.2 - Repeat Exercise 16 using four-digit chopping...Ch. 1.2 - Use the 64-bit-long real format to find the...Ch. 1.2 - Prob. 23ES Ch. 1.2 - Discuss the difference between the arithmetic...Ch. 1.2 - Prob. 2DQ Ch. 1.2 - Discuss the various different ways to round...Ch. 1.2 - Discuss the difference between a number written in...Ch. 1.3 - The Maclaurin series for the arctangent function...Ch. 1.3 - Prob. 4ES Ch. 1.3 - Prob. 5ES Ch. 1.3 - Find the rates of convergence of the following...Ch. 1.3 - Find the rates of convergence of the following...Ch. 1.3 - Prob. 8ES Ch. 1.3 - Prob. 9ES Ch. 1.3 - Suppose that as x approaches zero,...Ch. 1.3 - Prob. 11ES Ch. 1.3 - Prob. 12ES Ch. 1.3 - Prob. 13ES Ch. 1.3 - Prob. 14ES Ch. 1.3 - a. How many multiplications and additions are...Ch. 1.3 - Write an algorithm to sum the finite series i=1nxi...Ch. 1.3 - Construct an algorithm that has as input an...Ch. 1.3 - Let P(x) = anxn + an1xn1 + + a1x + a0 be a...Ch. 1.3 - Prob. 4DQ Ch. 1.3 - Prob. 5DQ Ch. 1.3 - Prob. 6DQ

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Suppose f ∈ C [ a , b ] and f ′( x ) exists on ( a , b ). Show that if f ′( x ) ≠ 0 for all x in ( a , b ), then there can exist at most one number p in [ a , b ] with f ( p ) = 0.

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Chapter 1 Solutions