Questions: Evaluate the difference quotient for the given function. Simplify your answer. f(x)=1+5x-x^2, (f(5+h)-f(5))/h

Evaluate the difference quotient for the given function. Simplify your answer. f(x)=1+5x-x^2, (f(5+h)-f(5))/h
Transcript text: Evaluate the difference quotient for the given function. Simplify your answer. \[ f(x)=1+5 x-x^{2}, \quad \frac{f(5+h)-f(5)}{h} \]
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Solution

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Solution Steps

Step 1: Define the Function

We start with the function \( f(x) = 1 + 5x - x^2 \).

Step 2: Calculate \( f(5+h) \)

Next, we substitute \( x = 5 + h \) into the function: \[ f(5+h) = 1 + 5(5+h) - (5+h)^2 = 1 + 25 + 5h - (25 + 10h + h^2) = 5h - h^2 + 1 \]

Step 3: Calculate \( f(5) \)

Now, we find \( f(5) \) by substituting \( x = 5 \): \[ f(5) = 1 + 5(5) - 5^2 = 1 + 25 - 25 = 1 \]

Step 4: Compute the Difference Quotient

We compute the difference quotient: \[ \frac{f(5+h) - f(5)}{h} = \frac{(5h - h^2 + 1) - 1}{h} = \frac{5h - h^2}{h} \]

Step 5: Simplify the Expression

Finally, we simplify the expression: \[ \frac{5h - h^2}{h} = 5 - h \]

Thus, the simplified difference quotient is \( -h - 5 \).

Final Answer

\(\boxed{5 - h}\)

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