if f(x) = 4x-x^2, find (f(x+h)-f(x-h))/2h

Question

if f(x) = 4x-x^2, find (f(x+h)-f(x-h))/2h

Answer 1

f(x + h) = 4(x + h) - (x + h)²

f(x + h) = 4x + 4h - x² - 2hx - h²

f(x - h) = 4(x - h) - (x - h)²

f(x - h) = 4x - 4h - x² + 2hx - h²

f(x + h) - f(x + h) = 4x + 4h - x² - 2hx - h² - (4x - 4h - x² + 2hx - h²)

Can you simplify and divide by 2h?

Answer 2

The work for the problem would be the following:

f(x+h)=4(x+h)-(x+h)²--I replace x+h wherever I see an x in the original function

f(x)=4x-x²

So we put it together as the following:

4(x+h)-(x+h)²-(4x-x²)/2h--In step 1 make sure to utilize parentheses on the top when simplifying the numerator

1. We will foil and distribute the numerator to start the problem:

4x+4h-(x+h)(x+h)-4x+x²/2h

4x+4h-(x²+2xh+h²)-4x+x²/2h

4x+4h-x²-2xh-h²-4x+x²/2h--Next we will combine like terms and cancel things in the numerator

such as 4x and x² are canceled out:

4h-2xh-h²/2h-This is what we have left and the final goal is to eliminate h from the bottom by factoring h from the numerator to have:

h(4-2x-h)/2h--h is canceled out to remain:

4-2x-h/2---Finally we do the limit h approaching 0 which we plug in:

4-2x-0/2---which will leave us with

4-2x/2--finally lets simplify by breaking up the terms:

4/2-2x/2---simlify to the derivative as:

f'(x)=2-x as the final answer.

Any other questions please let me know.