Question

Consider the function f(x) = 9x - x^2Compute the derivative values of the following:f’ (0) =f'(1) =f'(2) =f'(3)Conjecture a formula for f(a) that depends only on the value a. That is, in the same way that we have a formula for f(x) (recall f(x) = 9x - x^2), see if you can use your work above to guess a formula for f'(a) in terms

Answer 1

`\begin{gathered} \text{Given} \\ f(x)=9x-x^2 \end{gathered}`
`\begin{gathered} \text{Using the power rule} \\ f^(\prime)(x)=9x^(1-1)-(2)x^(2-1) \\ f^(\prime)(x)=9-2x \end{gathered}`

Solve for f'(0), f'(1), f'(2), f'(3)

`\begin{gathered} f^(\prime)(x)=9-2x \\ f^(\prime)(0)=9-2(0) \\ f^(\prime)(0)=9-0 \\ f^(\prime)(0)=9 \\ \\ f^(\prime)(1)=9-2(1) \\ f^(\prime)(1)=9-2 \\ f^(\prime)(1)=7 \\ \\ f^(\prime)(2)=9-2(2) \\ f^(\prime)(2)=9-4 \\ f^(\prime)(2)=5 \\ \\ f^(\prime)(3)=9-2(3) \\ f^(\prime)(3)=9-6 \\ f^(\prime)(3)=3 \end{gathered}`

the conjecture for f'(a)

`\begin{gathered} \text{Since }f^(\prime)(x)=9-2x,\text{ then} \\ f^(\prime)(a)=9-2a \end{gathered}`