Question

Given f ( x ) = 1/ x + 4 , find the average rate of change of f ( x ) on the interval [ 4 , 4 + h ] . Your answer will be an expression involving h .
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Answer 1

Average rate of change of f ( x ) on the interval [ 4 , 4 + h ] is $\frac{-1}{64+8h}$

Given :

$f(x)=\frac{1}{x+4}$

Find the average rate of change of f(x) over the interval [4,4+h]

Average rate of change formula is

$Average =\frac{f(b)-f(a)}{b-a}$

Where interval is [a,b]

From the given information , a=4, b=4+h

$f(x)=\frac{1}{x+4}\\a=4, \\f(a)=f(4)=\frac{1}{4+4}=\frac{1}{8} \\b=4+h\\f(b)=f(4+h)=\frac{1}{4+h+4}=\frac{1}{8+h}$

Replace the values inside the formula

$Average =\frac{f(b)-f(a)}{b-a}\\Average =\frac{f(4+h)-f(4)}{4+h-4}\\Average =\frac{\frac{1}{8+h} -\frac{1}{8} }{h}$

Simplify the numerator part

$Average =\frac{\frac{1}{8+h} -\frac{1}{8} }{h} =\frac{\frac{8-8-h}{8(8+h)} }{h}=\frac{\frac{-h}{64+8h} }{h} =\frac{-1}{64+8h}$

Average rate of change is $\frac{-1}{64+8h}$

Learn more : brainly.com/question/19961434