find and simplify each of the following for f(x)=5x^2−6x+8.
A) f(x+h)
B) f(x+h)-f(x)
C) )(f(x+h)-f(x))/h
f(x+h) = 5(x+h)^2 - 6(x+h) + 8 = 5x^2 + 10xh + 5h^2 - 6h - 6x + 8
and I think you can manage the rest, right? It's just Algebra I.
Post your work if you get stuck.
Looks like you are learning "derivatives from First Principles"
f(x)=5x^2−6x+8
f(x+h) = 5(x+h)^2 - 6(x+h) + 8
= 5x^2 + 10hx + 5h^2 - 6x - 6h + 8
f(x+h)-f(x) = 5x^2 + 10hx + 5h^2 - 6x - 6h + 8 - (5x^2 - 6x + 8)
= 5x^2 + 10hx + 5h^2 - 6x - 6h + 8 - 5x^2 + 6x - 8
= 10hx + 5h^2 - 6h
(f(x+h)-f(x))/h
= (10hx + 5h^2 - 6h)/h
= 10x + 5h - 6
I bet your next step in this section will be to take
lim 10x + 5h - 6 , as h ----> 0
Didn't see oobleck aready did this
To find and simplify each of the given expressions, we need to substitute the expression f(x) = 5x^2 - 6x + 8 into the respective expressions.
A) f(x + h):
To find f(x + h), we need to replace every occurrence of x in the function f(x) with (x + h). So we have:
f(x + h) = 5(x + h)^2 - 6(x + h) + 8
Now, we can expand and simplify the expression:
f(x + h) = 5(x^2 + 2xh + h^2) - 6(x + h) + 8
= 5x^2 + 10xh + 5h^2 - 6x - 6h + 8
= 5x^2 - 6x + 10xh - 6h + 5h^2 + 8
Simplified form: f(x + h) = 5x^2 + (10h - 6)x + (5h^2 - 6h + 8)
B) f(x + h) - f(x):
To find f(x + h) - f(x), we need to subtract f(x) from f(x + h). So we have:
f(x + h) - f(x) = (5x^2 + (10h - 6)x + (5h^2 - 6h + 8)) - (5x^2 - 6x + 8)
Now, we can collect like terms and simplify the expression:
f(x + h) - f(x) = 5x^2 + 10hx - 6x + 5h^2 - 6h + 8 - 5x^2 + 6x - 8
= (10hx - 6x + 6x) + (5h^2 - 6h) + (8 - 8)
= 10hx + 5h^2 - 6h
Simplified form: f(x + h) - f(x) = 10hx + 5h^2 - 6h
C) (f(x + h) - f(x))/h:
To find (f(x + h) - f(x))/h, we need to divide f(x + h) - f(x) by h. Using the simplified form we obtained in part B:
(f(x + h) - f(x))/h = (10hx + 5h^2 - 6h)/h
Now, we can simplify the expression by canceling out the h terms:
(f(x + h) - f(x))/h = 10x + 5h - 6
Simplified form: (f(x + h) - f(x))/h = 10x + 5h - 6