I need some help with some algebra problems please.
I don't under stand how to do that following.
Find and simplify the difference quotient f(x+h) -f(x)/h, h cannot equal 0 for the given function.
1. f(x)=5x^2
2. f(x)x^2+3x+9
The first:
f(x+h)=5(x+h)^2=5x^2+10xh+5h^2
f(x+h)-f(x)= 5x^2+10hx + 5h^2-5x^2=
10hx +5h^2
divided by h
10x + 5h
as the limit of h>>0
10x
Sure, I can help you with these algebra problems. Let's start with the first one.
1. f(x) = 5x^2
To find the difference quotient for this function, we need to substitute f(x + h) and f(x) into the formula and simplify it.
The formula for the difference quotient is:
f(x + h) - f(x) / h
For our given function f(x) = 5x^2, let's substitute f(x + h) and f(x) into the formula:
f(x + h) = 5(x + h)^2
f(x) = 5x^2
Substituting these into the difference quotient formula, we get:
[5(x + h)^2 - 5x^2] / h
Now, let's simplify this expression:
[5(x^2 + 2xh + h^2) - 5x^2] / h
Expanding and simplifying further:
[5x^2 + 10xh + 5h^2 - 5x^2] / h
Canceling out the common terms, we have:
[10xh + 5h^2] / h
Factoring out an h from the numerator, we get:
h(10x + 5h) / h
Canceling out the h terms, we obtain:
10x + 5h
So, the simplified difference quotient for the function f(x) = 5x^2 is 10x + 5h.
Now, let's move on to the second problem.
2. f(x) = x^2 + 3x + 9
Using the same process as before, let's substitute f(x + h) and f(x) into the difference quotient formula:
f(x + h) = (x + h)^2 + 3(x + h) + 9
f(x) = x^2 + 3x + 9
Substituting these into the formula, we get:
[(x + h)^2 + 3(x + h) + 9 - (x^2 + 3x + 9)] / h
Simplifying further:
[x^2 + 2xh + h^2 + 3x + 3h + 9 - x^2 - 3x - 9] / h
Canceling out the common terms, we have:
[2xh + h^2 + 3h] / h
Factoring out an h from the numerator, we get:
h(2x + h + 3) / h
Canceling out the h terms:
2x + h + 3
So, the simplified difference quotient for the function f(x) = x^2 + 3x + 9 is 2x + h + 3.
I hope this helps you understand how to find and simplify the difference quotient for these two given functions. Let me know if you have any further questions!