Questions: Find the difference quotient (f(x+h)-f(x))/h for f(x)=x^2-8x.

Transcript text: Find the difference quotient $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}-8 x$.

Solution

Solution Steps

Step 1: Substitute $ f(x+h) $ and $ f(x) $ into the difference quotient

Given $ f(x) = x^{2} - 8x $, substitute $ x+h $ into the function to find $ f(x+h) $: \[ f(x+h) = (x+h)^{2} - 8(x+h) \]

Step 2: Expand $ f(x+h) $

Expand $ (x+h)^{2} $ and simplify $ f(x+h) $: \[ f(x+h) = x^{2} + 2xh + h^{2} - 8x - 8h \]

Step 3: Compute $ f(x+h) - f(x) $

Subtract $ f(x) = x^{2} - 8x $ from $ f(x+h) $: \[ f(x+h) - f(x) = (x^{2} + 2xh + h^{2} - 8x - 8h) - (x^{2} - 8x) \] \[ f(x+h) - f(x) = 2xh + h^{2} - 8h \]

Step 4: Divide by $ h $ to find the difference quotient

Divide $ f(x+h) - f(x) $ by $ h $: \[ \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^{2} - 8h}{h} \] \[ \frac{f(x+h) - f(x)}{h} = 2x + h - 8 \]

This completes the calculation of the difference quotient.

Final Answer

$\boxed{2x + h - 8}$

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