The problem is
Find the difference quotient and simplify your answer
F(x)=5x - x^2, F(5+h) - F(5) over h
^2 this means power of two
F(5+h) = [5(5+h) - (25 + 10h +h^2)]
= -5h -h^2
F(5) = 25 - 25 = 0
[F(5+h) -F(5)]/h = -5 -h
To find the difference quotient, we need to evaluate the expression (F(5+h) - F(5)) / h for the given function F(x) = 5x - x^2.
Step 1: Evaluate F(5+h)
To do this, substitute (5+h) into the function F(x) = 5x - x^2:
F(5+h) = 5(5+h) - (5+h)^2
Simplifying:
F(5+h) = 25 + 5h - (25 + 10h + h^2)
F(5+h) = 25 + 5h - 25 - 10h - h^2
F(5+h) = -h^2 - 5h
Step 2: Evaluate F(5)
Again, substitute 5 into the function:
F(5) = 5(5) - (5)^2
F(5) = 25 - 25
F(5) = 0
Step 3: Calculate the difference quotient
Plug the values of F(5+h) and F(5) into the formula (F(5+h) - F(5)) / h:
(F(5+h) - F(5)) / h = (-h^2 - 5h - 0) / h
(F(5+h) - F(5)) / h = (-h^2 - 5h) / h
Step 4: Simplify the expression
To simplify the expression further, we can factor out a common factor of h from the numerator:
(F(5+h) - F(5)) / h = h(-h - 5) / h
Cancel out the common factor of h:
(F(5+h) - F(5)) / h = -h - 5
Thus, the simplified difference quotient is -h - 5.