Question

Given f (x) = x² + 4x + 5, what is

Answer 1

8+h

Explanation:

Simply substitute x = 2+h and x = 2 in.

`\displaystyle \large{(f(2+h)-f(2))/(h)=((2+h)^(2) +4(2+h)+5-[(2)^2+4(2)+5])/(h)}\\\displaystyle \large{(f(2+h)-f(2))/(h)=(4+4h+h^2 +8+4h+5-17)/(h)}\\\displaystyle \large{(f(2+h)-f(2))/(h)=(h^2 +8h)/(h)=h+8}`

Therefore, the answer is h+8 or 8+h.

Another way is to differentiate the function with respect to x. This equation or expression is rate of changes from 2 to 2+h with h being any numbers.

The definition of derivative is:

`\displaystyle \large{f\prime(x)= \lim_(h \to 0)(f(x+h)-f(x))/(h)}`

Notice how both f(2+h)-f(2) over h and the definition of derivative look same except the derivative has limit of h approaching to 0, making h not having any values by default.

Since f(2+h)-f(2) over h does not have limit, we have to add +h when differentiating the function.

`\displaystyle \large{f\prime(x)=2x+4}\\\displaystyle \large{f\prime(2)=2(2)+4}\\\displaystyle \large{f\prime(2)=8}`

Since f’(2) is 8 but because f(2+h)-f(2)/h does not have limit of h, therefore we add +h.

`\displaystyle \large{(f(2+h)-f(2))/(h)=8+h}}`