I need help with this calculus problem please!?

Question

I've spend all night on this long multi-step problem. I can't figure out the last step and if I guess wrong one more time I will have to start all over again. Can someone please help me out with this?

f(x)=7x^2

the instantaneous rate of change of y=f(x) with respect to x at x = 4

so far I have :

( 7(4 + h)^2 - 7(2)^2 ) / ( h )

Thanks a lot.

Answer 1

ok. to find the instantaneous rate of change at x = 4, you want to find the derivative of f(x) at x=4. if you guys don't know that, i'll re-answer.
the derivative of any function
g(x) = x^n = g'(x)= n(x)^(n-1)
if there is a coefficient, c, like in this case it is 7
g'(x) = cn(x)^(n-1)
so,
f'(x) = (7)(2)(x)^(2-1)
f'(4) = 14(4)^1 = 14 * 4 = 56

going off of what you're doing.....

we want
lim h->0 (f(x + h) - f(x)) / (x+h - x)
which is the instantaneous rate of change at x
in this specific case, x = 4
so, you should replace that 2 with a 4
now,
(7(4 +h)^2 -7(4)^2) / (h)

(4 + h)^2 = 16 + 8h + h^2
now, multiply it by 7
7(16 + 8h +h^2) = 112 +56h +7h^2

now 7(4)^2 = 7(16) = 112 so when we subtract it 112 - 112 cancels out, so we are left with

56h + 7h^2 on the top / h on the bottom
NOW, this is so sweet, pull out the h on the top
so we have
(h (56 + 7h)) / (h)

the h on top and h on bottom cancel

so now we are taking
lim h -->0 56 + 7h
since it is the limit as h goes to 0, the term 7h goes to 0 and we are left with just 56

so, the instantaneous rate of change is 56.
=)

Answer 2

The first derivative of 7x^2 would be 14x as I recall.
At x=4, the rate of change (slope or first derivative) would be 14*4 = 56.

How do you know that your guess is wrong?

Answer 3

I take it you are learning about derivatives. And right now you are learning about them as a limiting process.

y = 7x²

Find the limit as h → 0 of {[7(x + h)² - 7x²]/h}
= limit as h → 0 of {[7x² + 14xh + 7h² - 7x²]/h}
= limit as h → 0 of {[14xh + 7h²]/h}
= limit as h → 0 of {14x + 7h} = 14x = 14*4 = 56

Answer 4

This is a derrivative problem.
f(x)=7x^2
f'(x)=14x
f'(4)=14(4)
f'(4)=56