PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

8th Edition

ISBN: 9781337904285

Author: Larson

Publisher: CENGAGE L

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Question

Chapter 8.4, Problem 84E

To determine

To simplify: The difference quotient $f(x+h)−f(x)h$ , where $f(x)=x4$ by using the Binomial Theorem.

Expert Solution & Answer

Answer to Problem 84E

The simplified form is $4x3+6x2h+4xh2+h3$ .

Explanation of Solution

Given information: The functionis $f(x)=x4$ .

Theorem used:The Binomial Theorem: $(x+y)n=xn+nxn−1y+⋯+nCrxn−ryr+⋯+nxyn−1+yn$ , the coefficient of $xn−ryr$ is $nCr=n!(n−r)!r!$ .

Calculation:

Here, $n=4$ .

First write out the Pascal’s triangle such that the row that begins with 1, 4.

$111121133114641$

From the Pascal’s triangle, it is observed that the binomial coefficients of the expression are 1, 4, 6, 4 and 1.

Simplify $f(x+h)−f(x)h$ , where $f(x)=x4$ by using formula in Binomial theorem.

$f(x+h)−f(x)h=(x+h)4−x4h=x4+4x3h+6x2h2+4xh3+h4−x4h=4x3h+6x2h2+4xh3+h4h=4x3+6x2h+4xh2+h3$

Thus, the simplified form is $4x3+6x2h+4xh2+h3$ .

Chapter 8.4, Problem 83E

Chapter 8.4, Problem 85E

Chapter 8 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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The difference quotient f ( x + h ) − f ( x ) h , where f ( x ) = x 4 by using the Binomial Theorem.

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