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Calculus
Found in: Page 183
Calculus

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Q. Problem Zero: Read the section and make your own sum-

mary of the material.

The Derivative of a Function f at x=c is defined as f'(c)=limh→0f(c+h)-f(c)h or f'(c)=limz→cf(z)-f(c)z-c.

If a function f is differentiable at x=c then f'(c)=limh→0f(c+h)-f(c)hmust exist.

The left-hand derivative of a function f is defined as f'-(c)=limh→0-f(c+h)-f(c)h.

The right-hand derivative of a function f is defined as f'+(c)=limh→0+f(c+h)-f(c)h.

If a function is differentiable at any point then the function will also be continuous at that point.

The tangent line to the graph of a function f at x=c is defined as y=f(c)+f'(c)(x-c) where f'(c) is the slope.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1. Given information.

The topic of the given section is the Formal Definition of the Derivative.

Step 2. Summary of section.

The Derivative of a Function f at x=c is defined as role="math" localid="1649815739115" f'(c)=limh→0f(c+h)-f(c)h or f'(c)=limz→cf(z)-f(c)z-c

If a function f is differentiable at x=c then role="math" localid="1649815763221" f'(c)=limh→0f(c+h)-f(c)hmust exist.

The left-hand derivative of a function f is defined as role="math" localid="1649815781651" f'-(c)=limh→0-f(c+h)-f(c)h.

The right-hand derivative of a function f is defined as f'+(c)=limh→0+f(c+h)-f(c)h.

If a function is differentiable at any point then the function will also be continuous at that point.

The tangent line to the graph of a function f at x=c is defined as role="math" localid="1649815817628" y=f(c)+f'(c)(x-c) where f'(c) is the slope.

Most popular questions for Math Textbooks

For each function graphed in Exercises 65-68, determine the values of x at which f fails to be continuous and/or differentiable. At such points, determine any left or right continuity or differentiability. Sketch secant lines supporting your answers.

Use the definition of the derivative to find the equations of the lines described in Exercises 59-64.The line tangent to the graph of y=1-x-x2 at the point (1,-1)

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Stuart left his house at noon and walked north on Pine Street for 20 minutes. At that point he realized he was late for an appointment at the dentist, whose office was located south of Stuart’s house on Pine Street; fearing he would be late, Stuart sprinted south on Pine Street, past his house, and on to the dentist’s office. When he got there, he found the office closed for lunch; he was 10 minutes early for his 12:40 appointment. Stuart waited at the office for 10 minutes and then found out that his appointment was actually for the next day, so he walked back to his house. Sketch a graph that describes Stuart’s position over time. Then sketch a graph that describes Stuart’s velocity over time.

Use the definition of the derivative to find f for each function f in Exercises 39-54

f(x)=1x2-1
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