Chapter 11

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=4 t-3 \\ y=6 t-2 \end{array} \text { for } 0 \leq t \leq 1\right. $$

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\langle 12,-5\rangle, \vec{w}=\langle 3,4\rangle $$

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=9+9 i $$

Graph the following equations. $$ x^{2}+2 x y+y^{2}-x \sqrt{2}+y \sqrt{2}-6=0 $$

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Circle: \(r=6 \sin (\theta)\)

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(2, \frac{\pi}{3}\right) $$

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ a=7, b=12, \gamma=59.3^{\circ} $$

The sounds we hear are made up of mechanical waves. The note 'A' above the note 'middle \(\mathrm{C}^{\prime}\) is a sound wave with ordinary frequency \(f=440\) Hertz \(=440 \frac{\text { cycles }}{\text { second }}\). Find a sinusoid which models this note, assuming that the amplitude is 1 and the phase shift is \(0 .\)

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=4 t-1 \\ y=3-4 t \end{array} \text { for } 0 \leq t \leq 1\right. $$

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\langle-7,24\rangle, \vec{w}=\langle-5,-12\rangle $$

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=5+5 i \sqrt{3} $$

Graph the following equations. $$ 7 x^{2}-4 x y \sqrt{3}+3 y^{2}-2 x-2 y \sqrt{3}-5=0 $$

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Circle: \(r=2 \cos (\theta)\)

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(5, \frac{7 \pi}{4}\right) $$

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ \alpha=104^{\circ}, b=25, c=37 $$

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=73.2^{\circ}, \beta=54.1^{\circ}, a=117 $$

The voltage \(V\) in an alternating current source has amplitude \(220 \sqrt{2}\) and ordinary frequency \(f=60\) Hertz. Find a sinusoid which models this voltage. Assume that the phase is \(0 .\)

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=2 t \\ y=t^{2} \end{array} \text { for }-1 \leq t \leq 2\right. $$

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\langle 2,-1\rangle, \vec{w}=\langle-2,4\rangle $$

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Rose: \(r=2 \sin (2 \theta)\)

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=6 i $$

Graph the following equations. $$ 5 x^{2}+6 x y+5 y^{2}-4 \sqrt{2} x+4 \sqrt{2} y=0 $$

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(\frac{1}{3}, \frac{3 \pi}{2}\right) $$

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=95^{\circ}, \beta=85^{\circ}, a=33.33 $$

The London Eye is a popular tourist attraction in London, England and is one of the largest Ferris Wheels in the world. It has a diameter of 135 meters and makes one revolution (counterclockwise) every 30 minutes. It is constructed so that the lowest part of the Eye reaches ground level, enabling passengers to simply walk on to, and off of, the ride. Find a sinsuoid which models the height \(h\) of the passenger above the ground in meters \(t\) minutes after they board the Eye at ground level.

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=t-1 \\ y=3+2 t-t^{2} \end{array} \text { for } 0 \leq t \leq 3\right. $$

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\langle 10,4\rangle, \vec{w}=\langle-2,5\rangle $$

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Rose: \(r=4 \cos (2 \theta)\)

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-3 \sqrt{2}+3 i \sqrt{2} $$

Graph the following equations. $$ x^{2}+2 \sqrt{3} x y+3 y^{2}+2 \sqrt{3} x-2 y-16=0 $$

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(\frac{5}{2}, \frac{5 \pi}{6}\right) $$

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ a=3, b=4, \gamma=90^{\circ} $$

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=95^{\circ}, \beta=62^{\circ}, a=33.33 $$

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\langle-\sqrt{3}, 1\rangle, \vec{w}=\langle 2 \sqrt{3}, 2\rangle $$

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Rose: \(r=5 \sin (3 \theta)\)

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-6 \sqrt{3}+6 i $$

Graph the following equations. $$ 13 x^{2}-34 x y \sqrt{3}+47 y^{2}-64=0 $$

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(12,-\frac{7 \pi}{6}\right) $$

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ \alpha=120^{\circ}, b=3, c=4 $$

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=117^{\circ}, a=35, b=42 $$

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=\frac{1}{9}\left(18-t^{2}\right) \\ y=\frac{1}{3} t \end{array} \text { for } t \geq-3\right. $$

In Exercises \(1-20\), use the pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. $$ \vec{v}=\langle-3 \sqrt{3}, 3\rangle \text { and } \vec{w}=\langle-\sqrt{3},-1\rangle $$

Use the given pair of vectors \(\vec{v}\) and \(\vec{w}\) to find the following quantities. State whether the result is a vector or a scalar. $$ \begin{array}{lllll} \bullet \vec{v}+\vec{w} & \bullet \vec{w}-2 \vec{v} & \bullet\|\vec{v}+\vec{w}\| & \bullet\|\vec{v}\|+\|\vec{w}\| & \bullet\|\vec{v}\| \vec{w}-\|\vec{w}\| \vec{v} & \bullet\|\vec{w}\| \hat{v} \end{array} $$ Finally, verify that the vectors satisfy the Parallelogram Law $$\|\vec{v}\|^{2}+\|\vec{w}\|^{2}=\frac{1}{2}\left[\|\vec{v}+\vec{w}\|^{2}+\|\vec{v}-\vec{w}\|^{2}\right]$$ $$ \vec{v}=\left\langle\frac{3}{5}, \frac{4}{5}\right\rangle, \vec{w}=\left\langle-\frac{4}{5}, \frac{3}{5}\right\rangle $$

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Rose: \(r=\cos (5 \theta)\)

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-2 $$

Graph the following equations. $$ x^{2}-2 \sqrt{3} x y-y^{2}+8=0 $$

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(3,-\frac{5 \pi}{4}\right) $$

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ a=7, b=10, c=13 $$

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=117^{\circ}, a=45, b=42 $$

Suppose an object weighing 10 pounds is suspended from the ceiling by a spring which stretches 2 feet to its equilibrium position when the object is attached. (a) Find the spring constant \(k\) in \(\frac{\mathrm{lbs} .}{\mathrm{ft} .}\) and the mass of the object in slugs. (b) Find the equation of motion of the object if it is released from 1 foot below the equilibrium position from rest. When is the first time the object passes through the equilibrium position? In which direction is it heading? (c) Find the equation of motion of the object if it is released from 6 inches above the equilibrium position with a downward velocity of 2 feet per second. Find when the object passes through the equilibrium position heading downwards for the third time.