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Chapter 1: Problem 83

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=-2 x^{2}+5 x+7$$

Short Answer

Expert verified
The simplified difference quotient for the function \( f(x) = -2 x^{2}+5 x+7 \) is \( -4x -2h +5 \).

Step by step solution

01

Substitute the function into the difference quotient

First, substitute \( f(x) = -2 x^{2}+5 x+7 \) into the difference quotient \(\frac{f(x+h)-f(x)}{h}\), to obtain \( \frac{[-2 (x+h)^{2}+5 (x+h)+7] - [-2 x^{2}+5 x+7]}{h} \). This expresses the difference quotient in terms of the given function and its shifted version.
02

Expand and simplify the numerator

Next, expand the brackets in the numerator. This yields \( \frac{-2 (x^{2}+2 xh+h^{2})+5 (x+h)+7 +2 x^{2}-5 x-7}{h} \). Now, combine like terms to simplify the numerator further, to get \( \frac{-2x^{2}-4xh-2h^{2}+5x+5h+7+2x^{2}-5x-7}{h} \). After simplification, the numerator becomes \( -4xh-2h^2+5h \).
03

Cancel common factors

In the final step, cancel the common factor of \( h \) from the numerator and denominator. This gives the simplified difference quotient, which is \( -4x -2h +5 \).

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