Exercise 5. $f\left(x\right)=x^2+3x;x_0=2$

Solution:

$$\begin{array}{cc}\frac{\Delta y}{\Delta x}& =\frac{f(2+\Delta x)-f\left(2\right)}{\Delta x}\\ & =\frac{(10+7\Delta x+(\Delta x{)}^2)-10}{\Delta x}\\ & =\frac{7\Delta x+(\Delta x{)}^2}{\Delta x}\\ & =7+\Delta x\end{array}$$ Let $\Delta x$ become very small. The required instantaneous velocity at $x_0=2$ is 7 meters per second.

Exercise 2. $y=3x^2+x$

Solution:

• $\Delta y=f(2+0.5)-f\left(2\right)=7.25$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{7.25}{0.5}=14.5$

• $\Delta y=f(2+0.01)-f\left(2\right)=0.1303$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{0.1303}{0.01}=13.01$

• $\Delta y=f(4+0.1)-f\left(4\right)=2.53$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{2.53}{0.1}=25.3$

• $\Delta y=f(4+0.01)-f\left(4\right)=0.2503$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{0.2503}{0.01}=25.03$

Exercise 3. $y=x^2+10x$

Solution:

• $\Delta y=f(2+0.5)-f\left(2\right)=7.25$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{7.25}{0.5}=14.5$

• $\Delta y=f(2+0.01)-f\left(2\right)=0.1401$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{0.1401}{0.01}=14.01$

• $\Delta y=f(4+0.1)-f\left(4\right)=1.81$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{1.81}{0.1}=18.1$

• $\Delta y=f(4+0.01)-f\left(4\right)=0.1801$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{0.2503}{0.01}=18.01$

Exercise 4. $y=2x$

Solution:

• $\Delta y=f(2+0.5)-f\left(2\right)=1$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{1}{0.5}=0.5$

• $\Delta y=f(2+0.01)-f\left(2\right)=0.02$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{0.02}{0.01}=2$

• $\Delta y=f(4+0.1)-f\left(4\right)=.2$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{.2}{0.1}=2$/p>

• $\Delta y=f(4+0.01)-f\left(4\right)=0.02$. Avg. Velocity = $\frac{\Delta y}{\Delta x}=\frac{0.02}{0.01}=2$