Can someone please help me with this question?
If f(x) = 7x2 – x + 3 and h ≠ 0, find the following and simplify.
a.) f(x+h)
b.)( f(x+h)-f(x))/h
LOL, Calculus really
f(x+h) = 7(x+h)^2 -x-h + 3
= 7 x^2 + 14 xh + 7 h^2 -x -h + 3
now subtract f(x)
= 14 xh +7h^2 -h
now divide by h
= 14x + 7 h - 1
That is the end of the problem. To do the derivative you now look at it when h = 0
df(x)dx = 14 x - 1
f(x) = 7x^2 – x + 3
f(x+h) = 7(x+h)x^2 – (x+h) + 3
= 7x^2 + 14xh + 7h^2 - x - h + 3
( f(x+h)-f(x))/h
= (7x^2 + 14xh + 7h^2 - x - h + 3 - (7x^2 – x + 3) )/h
= (7x^2 + 14xh + 7h^2 - x - h + 3 - 7x^2 + x - 3)/h
= 14xh + 7h^2 - h)/h
= 14x + 7h - 1 , h ≠ 0
Now all you have to do is take the limit of that as h ---> 0
and you have the derivative of f(x)
LOL oh well :)
Thank you guys so much! Turns out I was missing a key part.
Sure! Let's break down the problem step by step.
a.) To find f(x+h), we substitute (x+h) into the expression for f(x).
So, f(x+h) = 7(x+h)^2 - (x+h) + 3.
To simplify this, we need to expand the expression (x+h)^2 using the binomial theorem. The binomial theorem states that (a+b)^2 = a^2 + 2ab + b^2. In our case, (x+h)^2 can be expanded as x^2 + 2xh + h^2.
Now, let's substitute this expansion into f(x+h):
f(x+h) = 7(x^2 + 2xh + h^2) - (x+h) + 3.
Next, distribute the 7 into the expanded expression:
f(x+h) = 7x^2 + 14xh + 7h^2 - x - h + 3.
This is the simplified expression for f(x+h).
b.) To find (f(x+h) - f(x))/h, we first need to find f(x) in terms of x.
Given that f(x) = 7x^2 - x + 3, there's no need for further simplification.
Now, substitute these expressions into (f(x+h) - f(x))/h:
(f(x+h) - f(x))/h = ((7x^2 + 14xh + 7h^2 - x - h + 3) - (7x^2 - x + 3))/h.
Simplify by subtracting the parentheses and simplifying further:
(f(x+h) - f(x))/h = (7x^2 + 14xh + 7h^2 - x - h + 3 - 7x^2 + x - 3)/h.
Combine like terms:
(f(x+h) - f(x))/h = (14xh + 7h^2 - h)/h.
Further simplify:
(f(x+h) - f(x))/h = 14x + 7h - 1.
So, the simplified expression for (f(x+h) - f(x))/h is 14x + 7h - 1.