91Ó°ÊÓ

sketch the graph of the function by hand. Use a graphing utility to verify your sketch. $$ y=\cos 2 \pi x $$

sketch the graph of the function by hand. Use a graphing utility to verify your sketch. $$ y=2 \tan x $$

complete the table (using a spreadsheet or a graphing utility set in radian mode) to estimate \(\lim _{x \rightarrow 0} f(x)\). $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{\sin 4 x}{2 x} $$

Meteorology The normal average daily temperature in degrees Fahrenheit for a city is given by \(T=55-21 \cos \frac{2 \pi(t-32)}{365}\) where \(t\) is the time in days, with \(t=1\) corresponding to January \(1 .\) Find the expected date of $$ \text { (a) the warmest day. (b) the coldest day. } $$

Tides Throughout the day, the depth of water \(D\) in meters at the end of a dock varies with the tides. The depth for one particular day can be modeled by \(D=3.5+1.5 \cos \frac{\pi t}{6}, \quad 0 \leq t \leq 24\) where \(t=0\) represents midnight. $$ \begin{array}{l}{\text { (a) Determine } d D / d t} \\ {\text { (b) Evaluate } d D / d t \text { for } t=4 \text { and } t=20 \text { and interpret your }} \\\ {\text { results. }} \\ {\text { (c) Find the time(s) when the water depth is the greatest }} \\ {\text { and the time(s) when the water depth is the least. }} \\ {\text { (d) What is the greatest depth? What is the least depth? }} \\ {\text { Did you have to use calculus to determine these }} \\\ {\text { depths? Explain your reasoning. }}\end{array} $$

Biology: Predator-Prey Cycle The population \(P\) of a predator at time \(t\) (in months) is modeled by \(P=5700+1200 \sin \frac{2 \pi t}{24}\) and the population \(p\) of its prey is modeled by \(p=9800+2750 \cos \frac{2 \pi t}{24}\) (a) Use a graphing utility to graph both models in the same viewing window. (b) Explain the oscillations in the size of each population.