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Consider the function f(x) = 8x - 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) = 8x- x^2), see if you can use your work above to guess a formula for f'(a) in termsof a.f'(a)=

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Question:

Solution:

Consider the following function:

[tex]f(x)=8x-x^2[/tex]

The derivative of this function is:

[tex]f^{\prime}(x)=8-2x^{}[/tex]

Now, if we evaluate this function at x = 0, x=1, x= 2 and x= 3, we get respectively:

[tex]f^{\prime}(0)=8-2(0)=8-0=8^{}[/tex]

[tex]f^{\prime}(1)=8-2(1)=8-2=6^{}[/tex]

[tex]f^{\prime}(2)=8-2(2)=8-4=4^{}[/tex]

and

[tex]f^{\prime}(3)=8-2(3)=8-6=2^{}[/tex]

According to this:

[tex]f^{\prime}(a)=8-2(a)^{}[/tex]

So that, we can conclude that the correct answers are:

[tex]f^{\prime}(0)=8^{}[/tex]

[tex]f^{\prime}(1)=6^{}[/tex]

[tex]f^{\prime}(2)=4^{}[/tex]

[tex]f^{\prime}(3)=2^{}[/tex]

and

[tex]f^{\prime}(a)=8-2a[/tex]

Note that the above formula is a function of a, so it depends only of a.

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