Chapter 9: Problem 1
The area \(K\) of a triangle whose base is \(b\) and whose altitude is \(h\) is ________
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(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ h(x)=\cos (2 x) \cos (x) $$
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 \(-5 \pi \leq x \leq 5 \pi\).
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?
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? $$ d(t)=-30 e^{-0.5 t / 70} \cos \left(\sqrt{\left(\frac{\pi}{2}\right)^{2}-\frac{0.25}{4900}} t\right) $$
State the Law of Cosines in words.
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