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Chapter 1: Problem 71

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=4 x$$

Short Answer

Expert verified
The simplified difference quotient for the function \(f(x)=4x\) is \(4\).

Step by step solution

01

Substitute the Function into the Formula

Start by substituting \(f(x)=4 x\) into the difference quotient formula, \(\frac{f(x+h)-f(x)}{h}\). So you get \(\frac{4(x+h)-4x}{h}\)
02

Simplify the Equation

Simplify the equation inside the numerator. Distribute \(4\) through the expression \((x+h)\) to get \(4x+4h\). Then you will have \(\frac{4x+4h-4x}{h}\)
03

Cancel out Like Terms

Within the numerator of the difference quotient, \(4x\) and \(-4x\) are like terms and cancel out, leaving us with \(\frac{4h}{h}\)
04

Simplify the Fraction

The fraction \(\frac{4h}{h}\) simplifies to \(4\), because the \(h\) in the numerator and denominator cancel out. Therefore, the difference quotient is \(4\)

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Most popular questions from this chapter

You invested \(\$ 80,000\) in two accounts paying \(5 \%\) and \(7 \%\) annual interest. If the total interest earned for the year was \(\$ 5200,\) how much was invested at each rate? (Section \(\mathrm{P.8}\) Example 5 )

$$\text { Solve for } y: \quad x=y^{2}-1, y \geq 0$$

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.

$$\text { Solve and check: } \frac{x-1}{5}-\frac{x+3}{2}=1-\frac{x}{4}$$
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