Find f(a), f(a + h),
and the difference quotient
f(a + h) − f(a)
------------------
h
,
where h ≠ 0.
f(x) = 6x + 7
f(a)=
f(a + h) =
f(a + h) − f(a)
----------------- =
h
To find f(a), substitute the value of a into the function f(x) = 6x + 7:
f(a) = 6a + 7
To find f(a + h), substitute the value of (a + h) into the function:
f(a + h) = 6(a + h) + 7
To find the difference quotient, subtract f(a) from f(a + h) and then divide by h:
f(a + h) − f(a)
----------------- =
h
Now, let's substitute the expressions we found earlier into the difference quotient formula:
(f(a + h) − f(a)) / h = (6(a + h) + 7 - (6a + 7)) / h
Simplifying the numerator:
= (6a + 6h + 7 - 6a - 7) / h
= (6h) / h
Finally, cancel out the h terms in the numerator and denominator:
= 6
Therefore, the difference quotient is simply 6.