91Ó°ÊÓ

Refer to the following: The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula \(h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]\) where \(x\) is the number of hours since midnight and \(h\) is the height of the tide in feet. What is the height of the tide at 3.00 P.M.? (IMAGE CANNOT COPY)

In Exercises \(61-66,\) sketch the graph of the function over the indicated interval. $$y=-3+4 \sin [\pi(x-2)],[0,4]$$

Determine whether each statement is true or false. \(\cos (2 n \pi+\theta)=\cos \theta, n\) an integer.

How many solutions are there to the equation \(\tan x=x ?\) Explain.

In calculus, the definite integral \(\int_{a}^{b} f(x) d x\) is used to find the area below the graph of a continuous function \(f\), above the \(x\) -axis, and between \(x=a\) and \(x=b\). The Fundamental Theorem of Calculus cstablishes that the definite integral \(f_{a}^{b} f(x) d x\) equals \(F(b)-F(a),\) where \(F\) is any antiderivative of a continuous function \(f.\) In Exercises \(87-90\), first shade the area corresponding to the definite integral and then use the information below to find the exact value of the area. $$\begin{array}{|l|c|c|c|c|} \hline \text { Function } & \tan x & \cot x & \sec x & \csc x \\ \hline \text { Antiderivative } & -\ln |\cos x| & \ln |\sin x| & \ln |\sec x+\tan x| & -\ln |\csc x+\cot x| \\ \hline \end{array}$$ $$\int_{\pi / 4}^{\pi / 2} \cot x d x$$

In calculus, the definite integral \(\int_{a}^{b} f(x) d x\) is used to find the area below the graph of a continuous function \(f\), above the \(x\) -axis, and between \(x=a\) and \(x=b\). The Fundamental Theorem of Calculus cstablishes that the definite integral \(f_{a}^{b} f(x) d x\) equals \(F(b)-F(a),\) where \(F\) is any antiderivative of a continuous function \(f.\) In Exercises \(87-90\), first shade the area corresponding to the definite integral and then use the information below to find the exact value of the area. $$\begin{array}{|l|c|c|c|c|} \hline \text { Function } & \tan x & \cot x & \sec x & \csc x \\ \hline \text { Antiderivative } & -\ln |\cos x| & \ln |\sin x| & \ln |\sec x+\tan x| & -\ln |\csc x+\cot x| \\ \hline \end{array}$$ $$\int_{0}^{\pi / 4} \sec x d x$$

For Exercises \(87-90\), refer to the following: Set the calculator in parametric and radian modes and let $$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$ (TABLE CANNOT COPY) Set the window so that \(0 \leq \mathrm{T} \leq 2 \pi,\) step \(=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2\) and \(-2 \leq Y \leq 2 .\) To approximate the sine or cosine of a T value, use the \([\text { TRACE }]\) key, type in the T value, and read the corresponding coordinates from the screen. Approximate \(\cos \left(\frac{5 \pi}{4}\right)\) to four decimal places.

For Exercises \(95-98,\) refer to the following: A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. (IMAGE CAN'T COPY). This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function \(y=A \cos (t \sqrt{\frac{k}{m}}),\) where \(|A|\) is the amplitude, \(t\) is the time in seconds, \(m\) is the mass of the weight, and \(k\) is a constant particular to the spring. If the height of the spring is measured in centimeters and the mass in grams, then what are the amplitude and mass if \(y=4 \cos \left(\frac{t \sqrt{k}}{2}\right) ?\)

For Exercises \(95-98,\) refer to the following: A weight hanging on a spring will oscillate up and down about its equilibrium position after it is pulled down and released. (IMAGE CAN'T COPY). This is an example of simple harmonic motion. This motion would continue forever if there were not any friction or air resistance. Simple harmonic motion can be described with the function \(y=A \cos (t \sqrt{\frac{k}{m}}),\) where \(|A|\) is the amplitude, \(t\) is the time in seconds, \(m\) is the mass of the weight, and \(k\) is a constant particular to the spring. The frequency of the oscillations in cycles per second is determined by \(f=\frac{1}{p},\) where \(p\) is the period. What is the frequency for the oscillation modeled by \(y=3 \cos \left(\frac{t}{2}\right) ?\)

If a sound wave is represented by \(y=0.008 \sin (750 \pi t) \mathrm{cm},\) what are its amplitude and frequency? See Exercise 99.