Find f(a), f(a + h), and the difference quotient f(a + h) − f(a) /h , where h ≠ 0.

Question

f(x)= 4x / x-5

a. f(a)=

b. f(h)=

c. f(a+h)=

d. f(a+h)-f(a)

------------- =

h

Answer 1

1. Just substitute a for x in f(x)=4x/(x-5) to get f(a)=4a/(a-5)
2. just substitute h for x, to get f(h)=4h/(x-5)
3. just substitute a+h for x to get f(a+h)=4(a+h)/(a+h-5)
4. Take 3 above and subtract 1 above, to get f(a+h)-f(a) = 4(a+h)/(a+h-5) - 4a/(a-5) Try putting that expression all over a common denominator which is the product of the two denominators

That gives numerator = 4(a+h)(a-5) -4a(a+h-5) all over the new denominator: (a+h-5)(a-5)

Factor out the 4: 4[(a+h)(a-5)-a(a+h-5)] expand the factors inside the brackets:

4[a^2+ha-5a-5h-a^2-ha+5a] = 4(-5h)=-20h Divide by h to get -20. That leaves

[f(a+h)-f(a)]/h = -20/(a+h+5)(a-5) That seems to be the answer to 4, as given

But the whole point of this exercise was really to lead up to finding the limit of that expression as h goes to 0 as a limit.

so last step, substitute h=0 into the

denominator (a+h-5)(a-5) to get (a-5)^2

That leaves f'(a)=-20/(a-5)^2

That is the derivative of the original function f(a)=4x/(x-5) the derivative is f'(a)=-20/(x-5)^2

You later learn a short cut method to get that answer, without going through all this, to get the derivative of f(a) = -20/(x-5)^2 or for taking the derivative of a fraction involving the variable.

Answer 2

It MUST be f(x) = 4x / (x-5); otherwise the x cancels and the function becomes -1

Moreover, we do not need f(h) to find the difference quotient

If you substitute dummy var x=a, then f(a) = 4a / (a-5)

For some very small number, h, that is approximately zero

f(a+h) = 4(a+h)/ (a+h-5)

Subtracting them....

4(a+h)/(a+h-5) - 4a / (a-5)

[4(a+h)(a-5)-4a (a+h-5) ]/ [ (a-5)(a+h-5)]

[ 4a^2 -20a + 4ah - 20h -4a^2 - 4ah + 20a ] / [ (a-5)(a+h-5)]

[-20h] / [ (a-5)(a+h-5)]

Dividing by h:

-20 / [ (a-5)(a+h-5)]