Chapter 9

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine whether \(x-3\) is a factor of \(x^{4}+2 x^{3}-21 x^{2}+19 x-3\)

According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-15 e^{-0.9 t / 30} \cos \left(\sqrt{\left(\frac{\pi}{3}\right)^{2}-\frac{0.81}{900}} t\right) $$

The function \(d\) models the distance (in meters) of the bob of a pendulum of mass \(m\) (in kilograms) from its rest position at time \(t\) (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at \(t=0 ?\) (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? $$ d(t)=-10 e^{-0.8 t / 50} \cos \left(\sqrt{\left(\frac{2 \pi}{3}\right)^{2}-\frac{0.64}{2500}} t\right) $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=\sqrt{x},\) find \(\frac{f(x)-f(4)}{x-4},\) for \(x=5,4.5,\) and 4.1 Round results to three decimal places.

Loudspeaker A loudspeaker diaphragm is oscillating in simple harmonic motion described by the function \(d(t)=a \cos (\omega t)\) with a frequency of 520 hertz (cycles per second) and a maximum displacement of 0.80 millimeter. Find \(\omega\) and then find a function that describes the movement of the diaphragm.

A cow is tethered to one corner of a square barn, 10 feet by 10 feet, with a rope 100 feet long. What is the maximum grazing area for the cow?

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve \(2 \sin ^{2} \theta-\sin \theta+5=6\) for \(0 \leq \theta<2 \pi\)

Colossus Added to Six Flags St. Louis in \(1986,\) the Colossus is a giant Ferris wheel. Its diameter is 165 feet; it rotates at a rate of about 1.6 revolutions per minute; and the bottom of the wheel is 15 feet above the ground. Find a function that relates a rider's height \(h\) above the ground at time \(t\). Assume the passenger begins the ride at the bottom of the wheel.

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the function \(d(t)=a \sin (\omega t)\) If a tuning fork for the note A above middle \(\mathrm{C}\) on an even-tempered scale \(\left(A_{4},\right.\) the tone by which an orchestra tunes itself) has a frequency of 440 hertz (cycles per second), find \(\omega\). If the maximum displacement of the end of the tuning fork is 0.01 millimeter, find a function that describes the movement of the tuning fork.

A perfect triangle is one having integers for sides for which the area is numerically equal to the perimeter. Show that the triangles with the given side lengths are perfect. (a) 9,10,17 (b) 6,25,29

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the function \(d(t)=a \sin (\omega t)\) If a tuning fork for the note \(\mathrm{E}\) above middle \(\mathrm{C}\) on an even-tempered scale \(\left(\mathrm{E}_{4}\right)\) has a frequency of approximately 329.63 hertz (cycles per second), find \(\omega\). If the maximum displacement of the end of the tuning fork is 0.025 millimeter, Find a function that describes the movement of the tuning fork.

A forest ranger is walking on a path inclined at \(5^{\circ}\) to the horizontal directly toward a 100 -foot-tall fire observation tower. The angle of elevation from the path to the top of the tower is \(40^{\circ} .\) How far is the ranger from the tower at this time?

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. What is the remainder when \(P(x)=2 x^{4}-3 x^{3}-x+7\) is divided by \(x+2 ?\)

The Sawtooth Curve An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes. (a) Use a graphing utility to graph the following function, which can be used to approximate the sawtooth curve. $$ f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x) \quad 0 \leq x \leq 4 $$ (b) A better approximation to the sawtooth curve is given by $$ f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x)+\frac{1}{8} \sin (8 \pi x) $$ Use a graphing utility to graph this function for \(0 \leq x \leq 4\) and compare the result to the graph obtained in part (a). (c) A third and even better approximation to the sawtooth curve is given by \(f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x)+\frac{1}{8} \sin (8 \pi x)+\frac{1}{16} \sin (16 \pi x)\) Use a graphing utility to graph this function for \(0 \leq x \leq 4\) and compare the result to the graphs obtained in parts (a) and (b). (d) What do you think the next approximation to the sawtooth curve is?

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the equation of a circle with radius \(r=\sqrt{5}\) and center (-4,0) in standard from.

Two sensors are spaced 700 feet apart along the approach to a small airport. When an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is \(20^{\circ},\) and from the second sensor to the aircraft it is \(15^{\circ} .\) Determine how high the aircraft is at this time.

A Clock Signal A clock signal is a non-sinusoidal signal used to coordinate actions of a digital circuit. Such signals oscillate between two levels, high and low, "instantaneously" at regular intervals. The most common clock signal has the form of a square wave and can be approximated by the sum of simple harmonic sinusoidal waves, such as $$ f(x)=2.35+\sin x+\frac{\sin (3 x)}{3}+\frac{\sin (5 x)}{5}+\frac{\sin (7 x)}{7}+\frac{\sin (9 x)}{9} $$ Graph this function for \(-4 \pi \leq x \leq 4 \pi\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the domain of \(g(x)=3\left|x^{2}-1\right|-5\)

Clint is building a wooden swing set for his children. Each supporting end of the swing set is to be an A-frame constructed with two 10 -foot-long 4 by 4 's joined at a \(45^{\circ}\) angle. To prevent the swing set from tipping over, Clint wants to secure the base of each A-frame to concrete footings. How far apart should the footings for each A-frame be?

Non-Sinusoidal Waves Both the sawtooth and square waves (see Problems 58 and 59 ) are examples of non-sinusoidal waves. Another type of non-sinusoidal wave is illustrated by the function $$ f(x)=1.6+\cos x+\frac{1}{9} \cos (3 x)+\frac{1}{25} \cos (5 x)+\frac{1}{49} \cos (7 x) $$ Graph the function for \(-5 \pi \leq x \leq 5 \pi\).

For any triangle, show that $$ \sin \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{a b}} $$ \text { where } s=\frac{1}{2}(a+b+c)

Challenge Problem Beats When two sinusoidal waves travel through the same medium, a third wave is formed that is the sum of the two original waves. If the two waves have slightly different frequencies, the sum of the waves results in an interference pattern known as a beat. Musicians use this idea when tuning an instrument with the aid of a tuning fork. If the instrument and the tuning fork play the same frequency, no beat is heard. Suppose two waves given by the functions, \(y_{1}=3 \cos \left(\omega_{1} t\right)\) and \(y_{2}=3 \cos \left(\omega_{2} t\right)\) where \(\omega_{1}>\omega_{2}\) pass through the same medium, and each has a maximum at \(t=0\) sec. (a) How long does it take the sum function \(y_{3}=y_{1}+y_{2}\) to equal 0 for the first time? (b) If the periods of the two functions \(y_{1}\) and \(y_{2}\) are \(T_{1}=19 \mathrm{sec}\) and \(T_{2}=20 \mathrm{sec},\) respectively, find the first time the \(\operatorname{sum} y_{3}=y_{1}+y_{2}=0\) (c) Use the values from part (b) to graph \(y_{3}\) over the interval \(0 \leq x \leq 600 .\) Do the waves appear to be in tune?

Use the Law of Cosines to prove the identity $$ \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{a^{2}+b^{2}+c^{2}}{2 a b c} $$

Graph the function \(f(x)=\frac{\sin x}{x}, x>0 .\) Based on the graph what do you conjecture about the value of \(\frac{\sin x}{x}\) for \(x\) close to \(0 ?\)

State the formula for finding the area of an SAS triangle in words.

What do you do first if you are asked to solve a triangle and are given two sides and the included angle?

Without graphing, determine whether the quadratic function \(f(x)=-3 x^{2}+12 x+5\) has a maximum value or a minimum value, and then find the value.

Make up three problems involving oblique triangles. One should result in one triangle, the second in two triangles, and the third in no triangle.

How would you explain simple harmonic motion to a friend? How would you explain damped motion?

\(P=\left(-\frac{\sqrt{7}}{3}, \frac{\sqrt{2}}{3}\right)\) is the point on the unit circle that corresponds to a real number \(t\). Find the exact values of the six trigonometric functions of \(t\).

Establish the identity: \(\csc \theta-\sin \theta=\cos \theta \cot \theta\)

State the Law of Cosines in words.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write as a single logarithm: \(\log _{7} x+3 \log _{7} y-\log _{7}(x+y)\)

Find the domain of \(f(x)=\ln \left(x^{2}-25\right)+3\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: } \log (x+1)+\log (x-2)=1 $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(3 x^{3}+4 x^{2}-27 x-36=0\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(\cos \alpha=\frac{4}{5}, 0<\alpha<\frac{\pi}{2},\) find the exact value of: (a) \(\cos \frac{\alpha}{2}\) (b) \(\sin \frac{\alpha}{2}\) (c) \(\tan \frac{\alpha}{2}\)

List all potential rational zeros of \(P(x)=2 x^{3}-5 x^{2}+13 x+6\)

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(\tan \theta=-\frac{2 \sqrt{6}}{5}\) and \(\cos \theta=-\frac{5}{7},\) find the exact value of each of the four remaining trigonometric functions.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(f(x)=\sqrt{3-5 x}\) and \(g(x)=x^{2}+7,\) find \(g(f(x))\) and its domain.

Solve: \(|(5 x-7)-5| \leq 0.05\)

If \(\cos \theta=\frac{5}{7}\) and \(\tan \theta<0,\) what is the value of \(\csc \theta ?\)

Solve: \(x(x-7)=18\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The normal line to a graph at a point is the line perpendicular to the tangent line of the graph at the point. If the tangent line is \(y=\frac{2}{3} x-1\) when \(f(3)=1,\) find an equation of the normal line.

The slope \(m\) of the tangent line to the graph of \(f(x)=3 x^{4}-7 x^{2}+2\) at any number \(x\) is given by \(m=f^{\prime}(x)=12 x^{3}-14 x\). Find an equation of the tangent line at \(x=1\).

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the equation \(100=a^{0.2 x}\) in logarithmic form.

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. $$ \text { Solve: } \frac{x^{2} \cdot \frac{1}{x}-\ln x \cdot 2 x}{\left(x^{2}\right)^{2}}=0 $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\frac{4 \cdot 3^{x} \cdot \ln 3 \cdot x^{1 / 2}-4 \cdot 3^{x} \cdot \frac{1}{2} \cdot x^{-1 / 2}}{(\sqrt{x})^{2}}\)

If \(h(x)\) is a function with range \([-5,8],\) what is the range of \(h(2 x+3) ?\)