approximating a derivative in exercises 67 and 68, evaluate f(2) and f(2.1) and use the results to…

Explanation:

Step1: Evaluate $f(2)$ for $f(x)=x(4 - x)$

Substitute $x = 2$ into $f(x)$: [ \begin{align*} f(2)&=2\times(4 - 2)\ &=2\times2\ &=4 \end{align*} ]

Step2: Evaluate $f(2.1)$ for $f(x)=x(4 - x)$

Substitute $x = 2.1$ into $f(x)$: [ \begin{align*} f(2.1)&=2.1\times(4 - 2.1)\ &=2.1\times1.9\ &=3.99 \end{align*} ]

Step3: Approximate $f^{\prime}(2)$

Use the formula for the approximate derivative $f^{\prime}(a)\approx\frac{f(a + h)-f(a)}{h}$, where $a = 2$, $h=0.1$. [ \begin{align*} f^{\prime}(2)&\approx\frac{f(2.1)-f(2)}{2.1 - 2}\ &=\frac{3.99 - 4}{0.1}\ &=\frac{- 0.01}{0.1}\ &=- 0.1 \end{align*} ]

Answer:

$f(2)=4$, $f(2.1)=3.99$, $f^{\prime}(2)\approx - 0.1$