Given the function f(x)=8+2x2, calculate the following values

Question

f(a)=  

f(a+h)=  

f(a+h)−f(a)/h=

Answer 1

Answer is 4a+2h

f(a)=8+2a²

f(a+h)=8+2(a+h)² = 8 +2a²+4ah+2h²

[f(a+h)-f(a)}/h = (8+2a²+4ah+2h²-8-2a²)/h

the 8's cancel the 2a² terms cancel leaving

the right side as (4ah+2h²)/h

or 4a+2h as h goes to zero the value goes to 4a

which is the same as the first derivative of the original function f'(x)=(2x²)=4x

Take the power and multiply it by the coefficient, then reduce the power by 1

2 times 2 = 4 is the new coefficient and 2-1=1 is the new power for the 1st derivative

Taking f(a+h)-f(a) all divided by h gives an expression that reduces to the derivative if you let h approach zero.

This question may be college algebra, but it's also the beginning of calculus