Let f(x) be the function 1/[x+12] . Then the quotient [f(4+h)-f(4)]/h
can be simplified to for: -1/[ah+b]
a=
b=
[1/(4+h+12) - 1/(4+12) ] / h
[ 1/(16+h) - 1/16 ] / h
[ 16 - 16-h ]/[(16+h)(16)h]
-h / [(256+16h)h]
-1 / [16 h + 256]
Plug and chug :)
a: 1
b: 12
To simplify the given quotient and express it as -1/[ah + b], let's substitute the expressions for f(4+h) and f(4) using the given function f(x).
f(4+h) = 1 / [4+h+12]
= 1 / (h+16)
f(4) = 1 / [4+12]
= 1 / 16
Now let's substitute these values back into the quotient:
[f(4+h) - f(4)] / h = [1 / (h+16) - 1 / 16] / h
To simplify this expression, let's find a common denominator:
[1 / (h+16) - 1 / 16] / h = [(16 - (h+16)) / ((h+16) * 16)] / h
= [(-h) / ((h+16) * 16)] / h
= -h / (h * (h+16) * 16)
Now, we want this to be in the form -1/[ah + b]. Let's rewrite the expression as:
- (1 / [(h+16) * 16]) * (h / (h * (h+16)))
= - (1 / 16(h+16)) * (h / (h * (h+16)))
Comparing this with -1/[ah + b], we can conclude that:
a = 1 / 16
b = h+16
Therefore, the simplified expression is -1/[ah + b], where a = 1 / 16 and b = h+16.