Question

In Exercises 47 and \(48,\) evaluate \(f(2)\) and \(f(2.1)\) and use the results to approximate \(f^{\prime}(2)\) \(f(x)=x(4-x)\)

Short answer

The approximate value of the derivative \(f'(2)\) is -0.1

Step-by-step solution

Function Evaluation at x = 2

Firstly, substitute \(x = 2\) into the function to get \(f(2)\). Thus, \(f(2) = 2(4-2) = 4\).

Function Evaluation at x = 2.1

Next, substitute \(x = 2.1\) into the function to get \(f(2.1)\). Therefore, \(f(2.1) = 2.1(4-2.1) = 3.99\).

Approximation of the derivative

Finally, use the definition of the derivative to approximate \(f'(2)\) as \((f(2.1)-f(2))/(2.1-2)\), which is equal to \((3.99-4) / (2.1-2) = -0.1\).