Chapter 7: Q. 103 (page 487)

Show that the difference quotient for
$f (x) = \sin x$ is given by
role="math" localid="1646430733990" $\frac{f (x + h) - f (x)}{h} = \frac{\sin (x + h) - \sin (x)}{h} = \cos x \cdot \frac{\sin h}{h} - \sin x \cdot \frac{1 - \cos h}{h}$

Short Answer

Expert verified

The difference quotient for $f (x) = \sin x$ is determined by using the Sum formula for the sine function as $\frac{f (x + h) - f (x)}{h} = \frac{\sin (x + h) - \sin (x)}{h} = \frac{(\sin x) (\cos h) + (\sin h) (\cos x) - \sin (x)}{h} = \cos x \cdot \frac{\sin h}{h} - \sin x \cdot \frac{1 - \cos h}{h}$

Step by step solution

Step 1. Given data

The given function is

$f (x) = \sin x$

the equation that needs to prove is

role="math" localid="1646430725441" $\frac{f (x + h) - f (x)}{h} = \frac{\sin (x + h) - \sin (x)}{h} = \cos x \cdot \frac{\sin h}{h} - \sin x \cdot \frac{1 - \cos h}{h}$

Step 2. Derivation

Use Sum formula for the sine function

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

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Show that the difference quotient for
$f (x) = \sin x$ is given by
role="math" localid="1646430733990" $\frac{f (x + h) - f (x)}{h} = \frac{\sin (x + h) - \sin (x)}{h} = \cos x \cdot \frac{\sin h}{h} - \sin x \cdot \frac{1 - \cos h}{h}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Derivation

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Show that the difference quotient for f(x)=sinx is given byrole="math" localid="1646430733990" f(x+h)-f(x)h=sin(x+h)-sin(x)h=cosx·sinhh-sinx·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

Discrete Mathematics

Pure Maths

Geometry

Logic and Functions

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Show that the difference quotient for
$f (x) = \sin x$ is given by
role="math" localid="1646430733990" $\frac{f (x + h) - f (x)}{h} = \frac{\sin (x + h) - \sin (x)}{h} = \cos x \cdot \frac{\sin h}{h} - \sin x \cdot \frac{1 - \cos h}{h}$