Tangent Lines, Velocities, and Growth

Practice 1: $\quad y=x^{2}$

If $x=1.994$, then $y=3.976036$ so the slope between $(2,4)$ and $(x, y)$ is

$\frac{4-y}{2-x}=\frac{4-3.976036}{2-1.994}=\frac{0.023964}{0.006} \approx 3.994$

If $x=2.0003$, then $y \approx 4.0012$ so the slope between $(2,4)$ and $(x, y)$ is

$\frac{4-y}{2-x}=\frac{4-4.0012}{2-2.0003}=\frac{-0.0012}{0.0003} \approx 4.0003$

Practice 2: $\quad\mathrm{m}_{\mathrm{sec}}=\frac{\mathrm{f}(-1+\mathrm{h})-(1)}{(-1+\mathrm{h})-(-1)}=\frac{(-1+\mathrm{h})^{2}-1}{\mathrm{~h}}=\frac{1-2 \mathrm{~h}+\mathrm{h}^{2}-1}{\mathrm{~h}}=\frac{\mathrm{h}(-2+\mathrm{h})}{\mathrm{h}}=\mathbf{- 2}+\mathbf{h}$

As $\mathrm{h} \rightarrow 0, \mathrm{~m}_{\mathrm{sec}}=-2+\mathrm{h} \rightarrow \mathbf{- 2}$.

Practice 3: The average velocity between $\mathrm{t}=1.5$ and $\mathrm{t}=2.0$ is $\frac{36-64 \mathrm{feet}}{2.0-1.5 \mathrm{sec}}=-56$ feet per second.

The average velocity between $t=2.0$ and $t=2.5$ is $\frac{0-36 \text { feet }}{2.5-2.0 \mathrm{sec}}=-72$ feet per second.

The velocity at $\mathrm{t}=2.0$ is somewhere between $-56 \mathrm{ft} / \mathrm{sec}$ and $-72 \mathrm{ft} / \mathrm{sec}$, probably about the middle of this interval: $\frac{(-56)+(-72)}{2}=-\mathbf{6 4} \mathbf{~ f t} / \mathrm{sec}$.

Practice 4: (a) When $\mathrm{t}=9$ days, the population is approximately $\mathrm{P}=4,200$ bacteria. When $\mathrm{t}=13$, $\mathrm{P} \approx 5,000$. The average change in population is approximately

$\frac{5000-4200 \text { bacteria }}{13-9 \text { days }}=\frac{800 \text { bacteria }}{4 \text { days }}=200 \text { bacteria per day. }$

(b) To find the rate of population growth at $\mathrm{t}=9$ days, sketch the line tangent to the population curve at the point $(9,4200)$ and then use $(9,4200)$ and another point on the tangent line to calculate the slope of the line. Using the approximate values $(5,2800)$ and $(9,4200)$, the slope of the tangent line at the point $(9,4200)$ is approximately $\frac{4200-2800 \text { bacteria }}{9-5 \text { days }}=\frac{1400 \text { bacteria }}{4 \text { days }} \approx \text{350 bacteria/day}$.