You can use the function given and the inputs to evaluate the difference quotient needed.
The difference quotient for the given function is given by
[tex]\dfrac{f(5+h) - f(5)}{h} = h + 5[/tex]
How to evaluate a given mathematical expression with variables if values of the variables are known?
You can simply replace those variables with the value you know of them and then operate on those values to get a final value. This is the result of that expression at those values of the considered variables.
How to evaluate the difference quotient for the given function?
The given function is [tex]f(x) = 1 + 5x - x^2[/tex]
The difference quotient is [tex]\dfrac{f(5+h) - f(5)}{h}[/tex]
Evaluating the function at 5+h and 5, we get
[tex]f(x) = 1 + 5x - x^2\\f(5+h) = 1 + 5(5+h) - (5+h)^2 = 1 + (5+h)(5 - (5+h)) = 1 + (5+h)(h) \\f(5+h) = h^2 + 5h + 1\\\\f(5) = 1 + 5(5) - (5)^2 = 1 + 25 - 25 = 1\\[/tex]
Thus, the difference quotient becomes
[tex]\dfrac{f(5+h) - f(5)}{h} = \dfrac{h^2 + 5h + 1 - 1}{h} = \dfrac{h^2 + 5h}{h} = h + 5[/tex]
Thus,
The difference quotient for the given function is given by
[tex]\dfrac{f(5+h) - f(5)}{h} = h + 5[/tex]
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