Find the quotient and simplify your answer.
f(x)=5x-x^2, f(5+h)-f(5) over h
Please help I am so lost! All I know is that you have to put (5+h) wherever there is an x.
so let's do that
f(5+h) = 5(5+h) - (5+h)^2
= 25 + 5h - 25 - 10h - h^2
= -5h - h^2
and f(5) = 25 - 25 = 0
so f(5+h)-f(5) over h
= (-5h - h^2 - 0)/h
= -5 - h , if h is not equal to zero
(looks like you are starting to find the derivative of a function from basic principles in an introductory Calculus course, right?)
To find the quotient and simplify the expression, you need to evaluate the given function for f(5+h) and f(5), and then find the difference of the two values and divide it by h.
Step 1: Evaluate f(5+h)
Replace x with (5+h) in the function f(x)=5x-x^2:
f(5+h) = 5(5+h) - (5+h)^2
Step 2: Simplify f(5+h)
f(5+h) = 25 + 5h - (25 + 10h + h^2)
= 25 + 5h - 25 - 10h - h^2
= -h^2 - 5h
Step 3: Evaluate f(5)
Replace x with 5 in the function f(x)=5x-x^2:
f(5) = 5(5) - (5)^2
= 25 - 25
= 0
Step 4: Find the difference between f(5+h) and f(5)
f(5+h) - f(5) = (-h^2 - 5h) - 0
= -h^2 - 5h
Step 5: Divide the difference by h
(f(5+h) - f(5)) / h = (-h^2 - 5h) / h
Step 6: Simplify the expression
Dividing each term by h:
= (-h^2/h) - (5h/h)
= -h - 5
Therefore, the quotient is -h - 5.
To find the quotient and simplify the expression (f(5+h) - f(5)) / h, we first need to evaluate f(5+h) and f(5). Let's start by finding f(5+h):
Given f(x) = 5x - x^2, substitute (5+h) in place of x:
f(5+h) = 5(5+h) - (5+h)^2
Now, expand and simplify:
f(5+h) = 25 + 5h - (25 + 10h + h^2)
= 25 + 5h - 25 - 10h - h^2
= -h^2 - 5h
Next, let's find f(5):
Given f(x) = 5x - x^2, substitute 5 in place of x:
f(5) = 5(5) - 5^2
= 25 - 25
= 0
Now, substitute these values back into the expression (f(5+h) - f(5)) / h:
[(f(5+h) - f(5)) / h] = [(-h^2 - 5h - 0) / h]
To simplify, factor out h from the numerator:
[(-h^2 - 5h) / h]
Finally, cancel out h from the numerator and denominator:
= -h - 5
So, the quotient of (f(5+h) - f(5)) / h is -h - 5.