Explanation We know that lim x → a f ( x ) = A means | f ( x ) − A | becomes arbitrarily small when x is sufficiently close to a , then the limit of f ( x ) as x → a is A . Given: The following function is given: f ( t ) = 3 t + 5 , a = − 1 Formula Used: The derivative of the function f ( x ) is given by the limit f ' ( x ) = lim h → 0 f ( x + h ) − f ( x ) h … ( 1 ) Equation of the tangent line is: y = f ( a ) + f ' ( a ) ( x − a ) … ( 2 ) Calculation: In order to get the value of f ' ( a ) , use the equation (1) as below: f ' ( a ) = lim h → 0 f ( a + h ) − f ( a ) h Now for a = − 1 , we get f ' ( − 1 ) = lim h → 0 f ( − 1 + h ) − f ( − 1 ) h = lim h → 0 ( 3 ( − 1 + h ) + 5 ) − ( 3 ( − 1 ) + 5 ) h = lim h → 0 3 h + 2 − 2 h = lim h → 0 3 h + 2 − 2 h ⋅ 3 h

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Concept explainers

Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0…

Question

Chapter 3.1, Problem 40E

To determine

The value of f'(a) using the limit definition and also find the equation of the tangent line

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Chapter 3.1, Problem 39E

Chapter 3.1, Problem 41E

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Chapter 3 Solutions

Calculus - Standalone book

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The value of f ' ( a ) using the limit definition and also find the equation of the tangent line

Solution Summary: The author explains how to determine the value of f'(a) using the limit definition and the equation of the tangent line.

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The value of f'(a) using the limit definition and also find the equation of the tangent line

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