Question

Evaluate f(2) and f(2.1) and use the results to approximate f '(2). (Round your answer to one decimal place.)f(x) = x(9 − x)f '(2) ≈

Answer 1

Given a function f(x) = x(9 - x).

We need to find the value of f(2) and f(2.1) and use them to approximate the value of f'(2).

The value of f(2) is calculated below:

`\begin{gathered} f(2)=2(9-2) \\ =2(7) \\ =14 \end{gathered}`

The value of the f(2.1) is calculated as follows:

`\begin{gathered} f(2.1)=2.1(9-2.1) \\ =2.1(6.9) \\ =14.49 \end{gathered}`

Now, by the definition of f'(x), we know that

`f^(\prime)(x)=(f(x+\Delta x)-f(x))/((x+\Delta x)-x)=(f(x+\Delta x)-f(x))/(\Delta x)`

For the given condition, x = 2, and delta x = 0.1. So, the value of f'(2) is

`\begin{gathered} f^(\prime)(2)=(f(2+0.1)-f(2))/(0.1) \\ =(f(2.1)-f(2))/(0.1) \\ =(14.49-14)/(0.1) \\ =(0.49)/(0.1) \\ =4.9 \end{gathered}`

Thus, the approximate value of f'(2) is 4.9.