Ch 5.3 - 22E

Chapter 5, Problem 22E

(choose chapter or problem)

(a) Experiment with a calculator to find an interval \(0 \leq \theta<\theta_{1}\), where \(\boldsymbol{\theta}\) 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 solution curves of the initial-value problems

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

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

for several values of \(\theta_{0}\) 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 \(\theta_{0}>\theta_{1}\).

Text Transcription:

0 leq theta < theta_1

theta

theta approx theta

0 leq x leq pi/2

d^2 theta/dt^2+sin theta=0,theta(0)=theta_0,theta^prime(0)=0

d^2 theta/dt^2+theta=0,theta(0)=theta_0,theta^prime(0)=0

theta_0

0 leq theta<theta_1

theta_0>theta_1