Q. 104 Show that the difference quotien... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 104 (page 487)

Show that the difference quotient for $f (x) = \cos x$ is given by

role="math" localid="1646431168099" $\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x 路 \frac{\sin h}{h} - \cos x 路 \frac{1 - \cos h}{h}$

Short Answer

Expert verified

The difference quotient for $f (x) = \cos (x)$ is determined by using the Sum formula for the cosine function as

$\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = \frac{(\cos x) (\cos h) - (\sin x) (\sin h) - \cos (x)}{h} = - \sin x 路 \frac{\sin h}{h} - \cos x 路 \frac{1 - \cos h}{h}$

Step by step solution

Step 1. Given data

The given function is

$f (x) = \cos x$

the equation that needs to prove is

$\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x 路 \frac{\sin h}{h} - \cos x 路 \frac{1 - \cos h}{h}$

Step 2. Derivation

Use Sum formula for the cosine function

$\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = \frac{(\cos x) (\cos h) - (\sin x) (\sin h) - \cos (x)}{h} = \frac{- (\sin x) (\sin h)}{h} - \frac{\cos (x) - (\cos x) (\cos h)}{h} = - (\sin x) \frac{(\sin h)}{h} - \frac{(\cos x) (1 - (\cos h))}{h} = - \sin x 路 \frac{\sin h}{h} - \cos x 路 \frac{1 - \cos h}{h}$

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91影视

Show that the difference quotient for $f (x) = \cos x$ is given by

role="math" localid="1646431168099" $\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x 路 \frac{\sin h}{h} - \cos x 路 \frac{1 - \cos h}{h}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Derivation

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91影视

Show that the difference quotient for f(x)=cosxis given byrole="math" localid="1646431168099" f(x+h)-f(x)h=cos(x+h)-cos(x)h=-sinx路sinhh-cosx路1-coshh

Short Answer

Step by step solution

Step 1. Given data

Step 2. Derivation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Show that the difference quotient for $f (x) = \cos x$ is given by

role="math" localid="1646431168099" $\frac{f (x + h) - f (x)}{h} = \frac{\cos (x + h) - \cos (x)}{h} = - \sin x 路 \frac{\sin h}{h} - \cos x 路 \frac{1 - \cos h}{h}$