Q. 9 - Calculus

Question

In Example 3 we estimated the slope of the tangent line to $f\left(x\right)=-\frac{1}{2}{x}^2+3x$ at $x=2$. Get a better estimate by calculating the slopes of secant lines with values of $z$ even closer to $x=2$, for example, $z=2.01,z=2.001$, and $z=2.0001$.

Step-by-Step Solution

Answer

Slope of secant line at $x=2$, 

for point $z=2.01$ is $0.995$.

for point $z=2.001$ is $0.9995$.

for point  $z=2.0001$ is $0.99995$.

Step 1. Given information

Function $f\left(x\right)=-\frac{1}{2}{x}^2+3x$

Step 2. Slope of secant line for z = 2 . 01

The slope of the secant line at $x=2$ for the point $z=2.01$ will be :-

$$\begin{gathered} f\text{'}\left(2\right)=\frac{f(2.01)-f\left(2\right)}{2.01-2} \\ =\frac{-{\displaystyle \frac{1}{2}}{(2.01)}^2+3(2.01)+{\displaystyle \frac{1}{2}}{\left(2\right)}^2-3\left(2\right)}{0.01} \\ =\frac{0.00995}{0.01} \\ =0.995 \end{gathered}$$

Step 3. Slope of the secant line for z = 2 . 001

The slope of the secant line at $x=2$ for the point $z=2.001$will be :

$$\begin{gathered} f\text{'}\left(2\right)=\frac{f(2.001)-f\left(2\right)}{2.001-2} \\ =\frac{-{\displaystyle \frac{1}{2}}{(2.001)}^2+3(2.001)+{\displaystyle \frac{1}{2}}{\left(2\right)}^2-3\left(2\right)}{0.001} \\ =\frac{0.0009995}{0.001} \\ =0.9995 \end{gathered}$$

Step 4. Slope of secant line for z = 2 . 0001

The slope of the secant line at $x=2$ for the point $z=2.0001$ will be :

$$\begin{gathered} f\text{'}\left(2\right)=\frac{f(2.0001)-f\left(2\right)}{2.0001-2} \\ =\frac{-{\displaystyle \frac{1}{2}}{(2.0001)}^2+3(2.0001)+{\displaystyle \frac{1}{2}}{\left(2\right)}^2-3\left(2\right)}{0.0001} \\ =\frac{0.000099995}{0.001} \\ =0.99995 \end{gathered}$$