9th Edition

ISBN: 9781337613927

Author: Stewart

Publisher: CENGAGE L

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 9.3, Problem 38E

Find a function f such that f(3) = 2 and

(t² + 1)f′(t) + [f(t)]² + 1 = 0 t ≠ 1

[Hint: Use the addition formula for tan(x + y) on Reference Page 2.]

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Chapter 9.3, Problem 37E

Chapter 9.3, Problem 39E

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Consider the function f(x) = x²-1. (a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative. Show all your steps clearly. (b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the graph where x 1 and x-> 1+h (for a small positive value of h, illustrate conceptually). Then, draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the value you found in part (a). (c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in the context of the graph of f(x). How does the rate of change of this function vary at different points?

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Does the series converge or diverge

Chapter 9 Solutions

Calculus, Early Transcendentals

Ch. 9.1 - Write a differential equation that models the...Ch. 9.1 - Write a differential equation that models the...Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Determine whether the given function is a solution...Ch. 9.1 - Determine whether the given function is a solution...Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 12E Ch. 9.1 - Show that the given function is a solution of the...

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Find a function f such that f (3) = 2 and ( t 2 + 1) f ′( t ) + [ f ( t )] 2 + 1 = 0 t ≠ 1 [ Hint: Use the addition formula for tan( x + y) on Reference Page 2.]

Solution Summary: The author explains how the differential equation can be written as follows: l(t

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Chapter 9 Solutions