Evaluate the difference quotient for the given function. Simplify your answer.
f(x) = 3 + 5x − x2,
f(4 + h) − f(4)
h
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To evaluate the difference quotient for the given function, we need to find f(4 + h) and f(4), and then simplify the expression.
First, we substitute (4 + h) into the function f(x) to find f(4 + h):
f(4 + h) = 3 + 5(4 + h) − (4 + h)^2
Simplifying the expression inside the parentheses:
f(4 + h) = 3 + 20 + 5h − (16 + 8h + h^2)
Combining like terms:
f(4 + h) = 23 + 5h - 16 - 8h - h^2
Simplifying further:
f(4 + h) = -h^2 - 3h + 7
Next, we substitute 4 into the function f(x) to find f(4):
f(4) = 3 + 5(4) − 4^2
Simplifying:
f(4) = 3 + 20 - 16
f(4) = 7
Finally, we plug these values into the difference quotient formula:
(f(4 + h) - f(4)) / h = (-h^2 - 3h + 7 - 7) / h
Simplifying the numerator:
(-h^2 - 3h + 7 - 7) = -h^2 - 3h
The simplified difference quotient is:
(-h^2 - 3h) / h
Now, we can cancel out the common factor "h" in the numerator and denominator:
(-h^2 - 3h) / h = -h - 3
Therefore, the simplified difference quotient for the given function is -h - 3.