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Math
CalculusCalculus (MindTap Course List)11th Edition

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Chapter 5.3, Problem 63E
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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
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Problem 10E
Problem 11E
Problem 12E
Problem 13E
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Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
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Problem 26E
Problem 27E
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Problem 30E
Problem 31E
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Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
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Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
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Problem 53E
Problem 54E
Problem 55E
Problem 56E
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Problem 60E
Problem 61E
Problem 62E
Problem 63E
Problem 64E
Problem 65E
Problem 66E
Problem 67E
Problem 68E
Problem 69E
Problem 70E
Problem 71E
Problem 72E
Problem 73E
Problem 74E
Problem 75E
Problem 76E
Problem 77E
Problem 78E
Problem 79E
Problem 80E
Problem 81E
Problem 82E
Problem 83E
Problem 84E
Problem 85E
Problem 86E
Problem 87E
Problem 88E
Problem 89E
Problem 90E
Problem 91E
Problem 92E
Problem 93E
Problem 94E
Problem 95E
Problem 96E
Problem 97E
Problem 98E
Problem 99E
Problem 100E
Problem 101E
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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Evaluating the Derivative of an Inverse Function In Exercises 63-70, verify that f has an inverse function. Then use the function f and the given real number a to find ( f − 1 ) ' ( a ) (Hint: See Example 5.)

f ( x ) = 5 − 2 x 3 ,       a = 7

To determine

To prove: The function f(x)=5−2x3 has an inverse function and calculate (f−1)′(a).

Explanation

Given:

The value of f(x)=5−2x3 and a=7.

Formula used:

The formula used to calculate derivative of inverse of function is g′(x)=1f′(g(x)).

Proof:

Let, f(x)=5−2x3

Differentiate both sides,

f′(x)=−6x2

Since, f′(x) is always negative.

then, f(x) is always decreasing and f(x) is strictly monotonic on its entire domain.

Hence, f(x) is one-one and f(x) has an inverse.

Calculation:

Let, f(x)=5−2x3

When x=−1

f(x)=5−2x3=5−2(−1)3=5+2=7

Hence, f−1(7)=−1

Since, f(x) is differentiable and has an inverse function

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Chapter 5.3,
Problem 62E
Chapter 5.3,
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P-28ESect-5.7 P-29ESect-5.7 P-30ESect-5.7 P-31ESect-5.7 P-32ESect-5.7 P-33ESect-5.7 P-34ESect-5.7 P-35ESect-5.7 P-36ESect-5.7 P-37ESect-5.7 P-38ESect-5.7 P-39ESect-5.7 P-40ESect-5.7 P-41ESect-5.7 P-42ESect-5.7 P-43ESect-5.7 P-44ESect-5.7 P-45ESect-5.7 P-46ESect-5.7 P-47ESect-5.7 P-48ESect-5.7 P-49ESect-5.7 P-50ESect-5.7 P-51ESect-5.7 P-52ESect-5.7 P-53ESect-5.7 P-54ESect-5.7 P-55ESect-5.7 P-56ESect-5.7 P-57ESect-5.7 P-58ESect-5.7 P-59ESect-5.7 P-60ESect-5.7 P-61ESect-5.7 P-62ESect-5.7 P-63ESect-5.7 P-64ESect-5.7 P-65ESect-5.7 P-66ESect-5.7 P-67ESect-5.7 P-68ESect-5.7 P-69ESect-5.7 P-70ESect-5.7 P-71ESect-5.7 P-72ESect-5.7 P-73ESect-5.7 P-74ESect-5.7 P-75ESect-5.7 P-76ESect-5.7 P-77ESect-5.7 P-78ESect-5.7 P-79ESect-5.7 P-80ESect-5.7 P-81ESect-5.7 P-82ESect-5.7 P-83ESect-5.7 P-84ESect-5.7 P-85ESect-5.7 P-86ESect-5.7 P-87ESect-5.7 P-88ESect-5.7 P-89ESect-5.7 P-90ESect-5.7 P-91ESect-5.7 P-92ESect-5.7 P-93ESect-5.7 P-94ESect-5.7 P-95ESect-5.7 P-96ESect-5.7 P-97ESect-5.7 P-98ESect-5.7 P-99ESect-5.7 P-100ESect-5.8 P-1ESect-5.8 P-2ESect-5.8 P-3ESect-5.8 P-4ESect-5.8 P-5ESect-5.8 P-6ESect-5.8 P-7ESect-5.8 P-8ESect-5.8 P-9ESect-5.8 P-10ESect-5.8 P-11ESect-5.8 P-12ESect-5.8 P-13ESect-5.8 P-14ESect-5.8 P-15ESect-5.8 P-16ESect-5.8 P-17ESect-5.8 P-18ESect-5.8 P-19ESect-5.8 P-20ESect-5.8 P-21ESect-5.8 P-22ESect-5.8 P-23ESect-5.8 P-24ESect-5.8 P-25ESect-5.8 P-26ESect-5.8 P-27ESect-5.8 P-28ESect-5.8 P-29ESect-5.8 P-30ESect-5.8 P-31ESect-5.8 P-32ESect-5.8 P-33ESect-5.8 P-34ESect-5.8 P-35ESect-5.8 P-36ESect-5.8 P-37ESect-5.8 P-38ESect-5.8 P-39ESect-5.8 P-40ESect-5.8 P-41ESect-5.8 P-42ESect-5.8 P-43ESect-5.8 P-44ESect-5.8 P-45ESect-5.8 P-46ESect-5.8 P-47ESect-5.8 P-48ESect-5.8 P-49ESect-5.8 P-50ESect-5.8 P-51ESect-5.8 P-52ESect-5.8 P-53ESect-5.8 P-54ESect-5.8 P-55ESect-5.8 P-56ESect-5.8 P-57ESect-5.8 P-58ESect-5.8 P-59ESect-5.8 P-60ESect-5.8 P-61ESect-5.8 P-62ESect-5.8 P-63ESect-5.8 P-64ESect-5.8 P-65ESect-5.8 P-66ESect-5.8 P-67ESect-5.8 P-68ESect-5.8 P-69ESect-5.8 P-70ESect-5.8 P-71ESect-5.8 P-72ESect-5.8 P-73ESect-5.8 P-74ESect-5.8 P-75ESect-5.8 P-76ESect-5.8 P-77ESect-5.8 P-78ESect-5.9 P-1ESect-5.9 P-2ESect-5.9 P-3ESect-5.9 P-4ESect-5.9 P-5ESect-5.9 P-6ESect-5.9 P-7ESect-5.9 P-8ESect-5.9 P-9ESect-5.9 P-10ESect-5.9 P-11ESect-5.9 P-12ESect-5.9 P-13ESect-5.9 P-14ESect-5.9 P-15ESect-5.9 P-16ESect-5.9 P-17ESect-5.9 P-18ESect-5.9 P-19ESect-5.9 P-20ESect-5.9 P-21ESect-5.9 P-22ESect-5.9 P-23ESect-5.9 P-24ESect-5.9 P-25ESect-5.9 P-26ESect-5.9 P-27ESect-5.9 P-28ESect-5.9 P-29ESect-5.9 P-30ESect-5.9 P-31ESect-5.9 P-32ESect-5.9 P-33ESect-5.9 P-34ESect-5.9 P-35ESect-5.9 P-36ESect-5.9 P-37ESect-5.9 P-38ESect-5.9 P-39ESect-5.9 P-40ESect-5.9 P-41ESect-5.9 P-42ESect-5.9 P-43ESect-5.9 P-44ESect-5.9 P-45ESect-5.9 P-46ESect-5.9 P-47ESect-5.9 P-48ESect-5.9 P-49ESect-5.9 P-50ESect-5.9 P-51ESect-5.9 P-52ESect-5.9 P-53ESect-5.9 P-54ESect-5.9 P-55ESect-5.9 P-56ESect-5.9 P-57ESect-5.9 P-58ESect-5.9 P-59ESect-5.9 P-60ESect-5.9 P-61ESect-5.9 P-62ESect-5.9 P-63ESect-5.9 P-64ESect-5.9 P-65ESect-5.9 P-66ESect-5.9 P-67ESect-5.9 P-68ESect-5.9 P-69ESect-5.9 P-70ESect-5.9 P-71ESect-5.9 P-72ESect-5.9 P-73ESect-5.9 P-74ESect-5.9 P-75ESect-5.9 P-76ESect-5.9 P-77ESect-5.9 P-78ESect-5.9 P-79ESect-5.9 P-80ESect-5.9 P-81ESect-5.9 P-82ESect-5.9 P-83ESect-5.9 P-84ESect-5.9 P-85ESect-5.9 P-86ESect-5.9 P-87ESect-5.9 P-88ESect-5.9 P-89ESect-5.9 P-90ESect-5.9 P-91ESect-5.9 P-92ESect-5.9 P-93ESect-5.9 P-94ESect-5.9 P-95ESect-5.9 P-96ESect-5.9 P-97ESect-5.9 P-98ESect-5.9 P-99ESect-5.9 P-100ESect-5.9 P-101ESect-5.9 P-102ESect-5.9 P-103ESect-5.9 P-104ESect-5.9 P-105ESect-5.9 P-106ECh-5 P-1RECh-5 P-2RECh-5 P-3RECh-5 P-4RECh-5 P-5RECh-5 P-6RECh-5 P-7RECh-5 P-8RECh-5 P-9RECh-5 P-10RECh-5 P-11RECh-5 P-12RECh-5 P-13RECh-5 P-14RECh-5 P-15RECh-5 P-16RECh-5 P-17RECh-5 P-18RECh-5 P-19RECh-5 P-20RECh-5 P-21RECh-5 P-22RECh-5 P-23RECh-5 P-24RECh-5 P-25RECh-5 P-26RECh-5 P-27RECh-5 P-28RECh-5 P-29RECh-5 P-30RECh-5 P-31RECh-5 P-32RECh-5 P-33RECh-5 P-34RECh-5 P-35RECh-5 P-36RECh-5 P-37RECh-5 P-38RECh-5 P-39RECh-5 P-40RECh-5 P-41RECh-5 P-42RECh-5 P-43RECh-5 P-44RECh-5 P-45RECh-5 P-46RECh-5 P-47RECh-5 P-48RECh-5 P-49RECh-5 P-50RECh-5 P-51RECh-5 P-52RECh-5 P-53RECh-5 P-54RECh-5 P-55RECh-5 P-56RECh-5 P-57RECh-5 P-58RECh-5 P-59RECh-5 P-60RECh-5 P-61RECh-5 P-62RECh-5 P-63RECh-5 P-64RECh-5 P-65RECh-5 P-66RECh-5 P-67RECh-5 P-68RECh-5 P-69RECh-5 P-70RECh-5 P-71RECh-5 P-72RECh-5 P-73RECh-5 P-74RECh-5 P-75RECh-5 P-76RECh-5 P-77RECh-5 P-78RECh-5 P-79RECh-5 P-80RECh-5 P-81RECh-5 P-82RECh-5 P-83RECh-5 P-84RECh-5 P-85RECh-5 P-86RECh-5 P-87RECh-5 P-88RECh-5 P-89RECh-5 P-90RECh-5 P-91RECh-5 P-92RECh-5 P-93RECh-5 P-94RECh-5 P-95RECh-5 P-96RECh-5 P-97RECh-5 P-98RECh-5 P-99RECh-5 P-100RECh-5 P-101RECh-5 P-102RECh-5 P-103RECh-5 P-104RECh-5 P-105RECh-5 P-106RECh-5 P-107RECh-5 P-108RECh-5 P-109RECh-5 P-110RECh-5 P-111RECh-5 P-112RECh-5 P-113RECh-5 P-114RECh-5 P-115RECh-5 P-116RECh-5 P-117RECh-5 P-118RECh-5 P-119RECh-5 P-120RECh-5 P-121RECh-5 P-122RECh-5 P-123RECh-5 P-124RECh-5 P-125RECh-5 P-126RECh-5 P-127RECh-5 P-128RECh-5 P-129RECh-5 P-130RECh-5 P-131RECh-5 P-132RECh-5 P-133RECh-5 P-134RECh-5 P-1PSCh-5 P-2PSCh-5 P-3PSCh-5 P-4PSCh-5 P-5PSCh-5 P-6PSCh-5 P-7PSCh-5 P-8PSCh-5 P-9PSCh-5 P-10PSCh-5 P-11PSCh-5 P-12PSCh-5 P-13PSCh-5 P-14PSCh-5 P-15PSCh-5 P-16PSCh-5 P-17PS

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