Find f(a), f(a + h),
and the difference quotient
f(a + h) − f(a)
------------------
h
,
where h ≠ 0.
f(x) = 2/x+5
f(a)=
f(a + h) =
f(a + h) − f(a)
----------------- =
h
To find f(a), we substitute a into the function f(x) = 2/x+5:
f(a) = 2/a + 5
To find f(a + h), we substitute (a + h) into the function f(x) = 2/x+5:
f(a + h) = 2/(a + h) + 5
Now, let's find the difference quotient:
f(a + h) − f(a) = [2/(a + h) + 5] - [2/a + 5]
Combining like terms, we get:
f(a + h) − f(a) = (2/(a + h) - 2/a)
To get the fraction over a common denominator, we multiply the first fraction by a/a, and the second fraction by (a + h)/(a + h):
f(a + h) − f(a) = (2a - 2(a + h))/(a(a + h))
Now, simplify the numerator:
f(a + h) − f(a) = (2a - 2a - 2h)/(a(a + h))
The 2a terms cancel each other out:
f(a + h) − f(a) = -2h/(a(a + h))
Finally, divide by h to complete the difference quotient:
[f(a + h) − f(a)]/h = -2h/(a(a + h) * h)
Simplifying further:
[f(a + h) − f(a)]/h = -2/(a(a + h))