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)

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