Sketch the graph of the function in previous question. Show that the slope of the tangent can be found using the properties of circles.

(a). Sketch the graph of $f(x)=-2(x-3)^2+4$.

(b). Draw a tangent line at the point $(5, f(5))$, and estimate its slope.

($c$). Estimate the instantaneous rate of change in $f(x)$ when $x=5$.

Show, using the properties of limits, that if $\lim _{x \rightarrow 5} f(x)=3$, then $\lim _{x \rightarrow 5} \frac{x^2-4}{f(x)}=7$.

Use a graphing calculator to graph each function and estimate the limits, if they exist.

(a). $f(x)=\frac{1}{x^2}, \lim _{x \rightarrow 0} f(x)$

(b). $g(x)=x(x-5), \lim _{x \rightarrow 0} g(x)$

($c$). $h(x)=\frac{x^3-27}{x^2-9}, \lim _{x \rightarrow 4} h(x)$ and $\lim _{x \rightarrow-3} h(x)$

Find the equation of the tangent to the graph of the function at the given value of $x$.

(a). $f(x)=\frac{x}{x+3}, x=-5$ (b). $f(x)=\frac{2 x+5}{5 x-1}, x=-1$

Determine the slope of the tangent to each curve at the given point.

(a). $y=x^2-3 x,(2,-2)$ $\quad$(d). $y=\sqrt{x-7},(16,3)$

(b). $f(x)=\frac{4}{x},(-2,-2)$ $\quad$(e). $y=\sqrt{25-x^2},(3,4)$

($c$). $y=3 x^3,(1,3)$ $\quad\quad\quad$ (f). $y=\frac{4+x}{x-2},(8,2)$

For each function, sketch the graph of the function. Determine the indicated limit if it exists.

(a).

$$f(x)=\left\{\begin{array}{r}x+2, \text { if } x<-1 \\ -x+2, \text { if } x \geq-1\end{array} ; \lim _{x \rightarrow-1} f(x)\right.$$

(b).

$$f(x)=\left\{\begin{array}{c}-x+4, \text { if } x \leq 2 \\ -2 x+6, \text { if } x>2\end{array} ; \lim _{x \rightarrow 2} f(x)\right.$$

($c$).

$$f(x)=\left\{\begin{array}{l}4 x, \text { if } x \geq \frac{1}{2} \\ \frac{1}{x}, \text { if } x<\frac{1}{2}\end{array} ; \lim _{x \rightarrow \frac{1}{2}} f(x)\right.$$

(d).

$$f(x)=\left\{\begin{array}{r}1, \text { if } x<-0.5 \\ x^2-0.25, \text { if } x \geq-0.5\end{array} ; \lim _{x \rightarrow-0.5} f(x)\right.$$

Sacha drains the water from a hot tub. The hot tub holds 1600 L of water. It takes 2 h for the water to drain completely. The volume of water in the hot tub is modelled by $V(t)=1600-\frac{t^2}{9}$, where $V$ is the volume in litres at $t$ minutes and $0 \leq t \leq 120$.

(a). Determine the average rate of change in volume during the second hour.

(b). Estimate the instantaneous rate of change in volume after exactly $60 \mathrm{~min}$.

($c$). Explain why all estimates of the instantaneous rate of change in volume where $0 \leq t \leq 120$ result in a negative value.

The time t, in seconds, taken by an object dropped from a height of s metres to reach the ground is given by the formula $t=\sqrt{\frac{s}{5}}$. Determine the rate of change in time with respect to height when the object is 125 m above the ground.

Jacques Charles (1746-1823) discovered that the volume of a gas at a constant pressure increases linearly with the temperature of the gas. To obtain the data in the following table, one mole of hydrogen was held at a constant pressure of one atmosphere. The volume $V$ was measured in litres, and the temperature $T$ was measured in degrees Celsius.

$$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \boldsymbol{T}\left({ }^{\circ} \mathrm{C}\right) & -40 & -20 & 0 & 20 & 40 & 60 & 80 \\ \hline \boldsymbol{V}(\mathbf{L}) & 19.1482 & 20.7908 & 22.4334 & 24.0760 & 25.7186 & 27.3612 & 29.0038 \\ \hline \end{array}$$

(a). Calculate first differences, and show that $T$ and $V$ are related by a linear relation.

(b). Find the linear equation for $V$ in terms of $T$.

($c$). Solve for $T$ in terms of $V$ for the equation in part (b).

(d). Show that $\lim _{V \rightarrow 0^{+}} T$ is approximately $-273.15$. Note: This represents the approximate number of degrees on the Celsius scale for absolute zero on the Kelvin scale ( $0 \mathrm{~K})$.

(e). Using the information you found in parts b and d, draw a graph of $V$ versus $T$.

Find the equation of the tangent at the given value of x.

(a). $y=x^2+x-3, x=4$ (b). $y=2 x^2-7, x=-2$ ($c$). $f(x)=3 x^2+2 x-5, x=-1$ (d). $f(x)=5 x^2-8 x+3, x=1$

Examine the continuity of the following function:

$$f(x)=\left\{\begin{array}{l} x, \text { if } x \leq 1 \\ 1, \text { if } 1<x \leq 2 \\ 3, \text { if } x>2 \end{array}\right.$$

Consider the function $f(x)=\frac{2 x-2}{x^2+x-2}$.

(a). For what values of $x$ is $f$ discontinuous?

(b). At each point where $f$ is discontinuous, determine the limit of $f(x)$, if it exists.

Determine whether $f(x)=\frac{x^2+2 x-8}{x+4}$ is continuous at $x=-4$.