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(A) (B) ƒ(5) - ƒ(3) 5-3 f(3 +h)-f(3) h (C) lim h→0 || f(3+h)-f(3) h

(A) (B) ƒ(5) - ƒ(3) 5-3 f(3 +h)-f(3) h (C) lim h→0 || f(3+h)-f(3) h

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Let f(x)=−2−x2f(x)=−2−x2. Find each of the following:

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(A) \(\frac{f(5) - f(3)}{5 - 3} =\) [blank] (B) \(\frac{f(3 + h) - f(3)}{h} =\) [blank] (C) \(\lim_{{h \to 0}} \frac{f(3 + h) - f(3)}{h} =\) [blank] ### Explanation - **Equation (A)** represents the average rate of change of the function \(f(x)\) between \(x = 3\) and \(x = 5\). This is a basic slope formula for a secant line. - **Equation (B)** formulates the difference quotient which is used to approximate the derivative. It measures the rate of change around a point \(x = 3\). - **Equation (C)** represents the derivative of the function at \(x = 3\). This is calculated by taking the limit as \(h\) approaches zero, which gives the slope of the tangent line at that point.
Transcribed Image Text:(A) \(\frac{f(5) - f(3)}{5 - 3} =\) [blank] (B) \(\frac{f(3 + h) - f(3)}{h} =\) [blank] (C) \(\lim_{{h \to 0}} \frac{f(3 + h) - f(3)}{h} =\) [blank] ### Explanation - **Equation (A)** represents the average rate of change of the function \(f(x)\) between \(x = 3\) and \(x = 5\). This is a basic slope formula for a secant line. - **Equation (B)** formulates the difference quotient which is used to approximate the derivative. It measures the rate of change around a point \(x = 3\). - **Equation (C)** represents the derivative of the function at \(x = 3\). This is calculated by taking the limit as \(h\) approaches zero, which gives the slope of the tangent line at that point.
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