Find f(a), f(a + h),
and the difference quotient
f(a + h) − f(a)
------------------
h
,
where h ≠ 0.
f(x) = x^2+7
f(a)=
f(a + h) =
f(a + h) − f(a)
----------------- =
h
2,5,6
To find f(a), substitute a into the function f(x) = x^2 + 7:
f(a) = a^2 + 7
To find f(a + h), substitute a + h into the function:
f(a + h) = (a + h)^2 + 7
To find the difference quotient, subtract f(a) from f(a + h) and divide it by h:
[f(a + h) - f(a)] / h = [(a + h)^2 + 7 - (a^2 + 7)] / h
Simplifying the expression above:
[f(a + h) - f(a)] / h = [(a^2 + 2ah + h^2 + 7) - (a^2 + 7)] / h
The a^2 term and the 7 term cancels out, leaving:
[f(a + h) - f(a)] / h = [2ah + h^2] / h
Now, simplify the expression further:
[f(a + h) - f(a)] / h = 2a + h
Therefore, f(a) = a^2 + 7, f(a + h) = (a + h)^2 + 7, and the difference quotient is 2a + h.