Solved: (a) Experiment with a calculator to find an interval 0 u u1, where u is measured

Chapter 3, Problem 22

(choose chapter or problem)

Discussion Problems

(a) Experiment with a calculator to find an interval \(0 \leq \theta<\theta_{1}\), where \(\theta\) is measured in radians. for which you think \(\sin \theta \approx \theta\) is a fairly good estimate. Then use a graphing utility to plot the graphs of y=x and y=sin x on the same coordinate axes for \(0 \leq x \leq \pi / 2\). Do the graphs confirm your observations with the calculator?

(b) Use a numerical solver to plot the solutions curves of the initial-value problems

\(\frac{d^{2} \theta}{d t^{2}}+\sin \theta=0\), \(\theta(0)=\theta_{0}\), \(\theta^{\prime}(0)=0\)

and \(\frac{d^{2} \theta}{d t^{2}}+\theta=0\), \(\theta(0)=\theta_{0}\), \(\theta^{\prime}(0)=0\)

for several values of \(\theta_{0}\) in the interval \(0 \leq \theta<\theta_{1}\) found in part (a). Then plot solution curves of the initial-value problems for several values of \(\theta_{0}\) for which \(\theta_{0}>\theta_{1}\).