Explanation Given: Δ f ( x ) = lim h → 0 f ( x + h ) + f ( x − h ) − 2 f ( x ) h 2 Calculation: (a) f ( x ) = x . Therefore, we get, Δ f ( x ) = lim h → 0 x + h + x − h − 2 x h 2 = lim h → 0 0 h 2 = 0 Thus, we get Δ f ( x ) = 0 , where f ( x ) = x (b) f ( x ) = x 2 Therefore, we get Δ f ( x ) = lim h → 0 ( x + h ) 2 + ( x − h ) 2 − 2 x 2 h 2 = lim h → 0 x 2 + h 2 + 2 x h + x 2 + h 2 − 2 x h + 2 x 2 h 2 = lim

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 3.5, Problem 56E

To determine

The value of Δf(x)=limh→0f(x+h)+f(x−h)−2f(x)h2 for the given functions

(a) f(x)=x

(b) f(x)=x2

(c) f(x)=x3

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Chapter 3.5, Problem 55E

Chapter 3.6, Problem 1PQ

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Chapter 3 Solutions

Calculus - Standalone book

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The value of Δ f ( x ) = lim h → 0 f ( x + h ) + f ( x − h ) − 2 f ( x ) h 2 for the given functions (a) f ( x ) = x (b) f ( x ) = x 2 (c) f ( x ) = x 3

Solution Summary: The author explains that Delta f(x) represents the second derivative of a function.

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The value of Δf(x)=limh→0f(x+h)+f(x−h)−2f(x)h2 for the given functions

(a) f(x)=x

(b) f(x)=x2

(c) f(x)=x3

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Chapter 3 Solutions

The value of Δ f ( x ) = lim h → 0 f ( x + h ) + f ( x − h ) − 2 f ( x ) h 2 for the given functions (a) f ( x ) = x (b) f ( x ) = x 2 (c) f ( x ) = x 3

Solution Summary: The author explains that Delta f(x) represents the second derivative of a function.

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The value of Δf(x)=limh→0f(x+h)+f(x−h)−2f(x)h2 for the given functions (a) f(x)=x (b) f(x)=x2 (c) f(x)=x3

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Chapter 3 Solutions

The value of Δf(x)=limh→0f(x+h)+f(x−h)−2f(x)h2 for the given functions

(a) f(x)=x

(b) f(x)=x2

(c) f(x)=x3