91Ó°ÊÓ

For \(y=x^{3},\) describe how the graph to the left of the \(y\) -axis compares to the graph to the right of the \(y\) -axis. Show that for \(f(x)=x^{3},\) we have \(f(-x)=-f(x),\) In general, if you have the graph of \(y=f(x)\) to the right of the \(y\) -axis and \(f(-x)=-f(x)\) for all \(x,\) describe how to graph \(y=f(x)\) to the left of the \(y\) -axis.

Determine the number of (real) solutions. Solve for the intersection points exactly if possible and estimate the points if necessary. $$\sin x=x^{2}+1$$

Discuss whether you think \(y\) would be a function of \(x\). \(y=\) probability of getting lung cancer, \(x=\) number of cigarettes smoked per day.

Use the range for \(\theta\) to determine the indicated function value. $$\sin \theta=\frac{1}{2}, \frac{\pi}{2} \leq \theta \leq \pi ; \quad \text { find } \tan \theta$$

In exercise \(55,\) if you had 20 tickets with a 1 -in-20 chance of winning, would you expect your probability of winning at least once to increase or decrease? Compute the probability \(1-\left(\frac{19}{20}\right)^{20}\) to find out.

Use a triangle to simplify each expression. Where applicable, state the range of \(x\) 's for which the simplification holds. $$\cos \left(\sin ^{-1} x\right)$$

Use a graphing calculator or computer graphing utility to estimate all zeros. $$f(x)=x^{3}-3 x+1$$

Iterations of functions are important in a variety of applications. To iterate \(f(x)\), start with an initial value \(x_{0}\) and compute \(x_{1}=f\left(x_{0}\right), x_{2}=f\left(x_{1}\right), x_{3}=f\left(x_{2}\right)\) and so on. For example, with \(f(x)=\cos x\) and \(x_{0}=1\), the iterates are \(x_{1}=\cos 1 \approx 0.54, x_{2}=\cos x_{1} \approx \cos 0.54 \approx 0.86\) \(x_{3} \approx \cos 0.86 \approx 0.65\) and so on. Keep computing iterates and show that they get closer and closer to \(0.739085 .\) Then pick your own \(x_{0}\) (any number you like) and show that the iterates with this new \(x_{0}\) also converge to 0.739085

Discuss whether you think \(y\) would be a function of \(x\). \(y=\) a person's weight, \(x=\) number of minutes exercising per day.

In general, if you have \(n\) chances of winning with a 1 -in- \(n\) chance on each try, the probability of winning at least once is \(1-\left(1-\frac{1}{n}\right)^{n} .\) As \(n\) gets larger, what number does this probability approach? (Hint: There is a very good reason that this question is in this section!)