Chapter 10

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=-3 \mathbf{j}\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\frac{1}{2}\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right)\right]^{5} $$

Identify and graph each polar equation. $$ r=2+4 \cos \theta $$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-5,-\frac{\pi}{6}\right) $$

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=3 \mathbf{i}-4 \mathbf{j}\)

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 average rate of change of \(f(x)=x^{3}-5 x^{2}+27\) from -3 to 2 .

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-6,-\frac{\pi}{4}\right) $$

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=-5 \mathbf{i}+12 \mathbf{j}\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\sqrt{3} e^{i \frac{5 \pi}{18}}\right]^{6} $$

Identify and graph each polar equation. $$ r=2 \sin (3 \theta) $$

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 exact value of \(5 \cos 60^{\circ}+2 \tan \frac{\pi}{4} .\) Do not use a calculator.

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ (-2,-\pi) $$

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=\mathbf{i}-\mathbf{j}\)

In Problems \(45-56,\) write each expression in rectangular form \(x+y i\) and in exponential form re". 53\. \((1-i)^{5}\)

Identify and graph each polar equation. $$ r=4 \sin (5 \theta) $$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-3,-\frac{\pi}{2}\right) $$

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ (\sqrt{3}-i)^{6} $$

Identify and graph each polar equation. $$ r=3 \cos (4 \theta) $$

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. Volume of a Box An open-top box is made from a sheet of metal by cutting squares from each corner and folding up the sides. The sheet has a length of 19 inches and a width of 13 inches. If \(x\) is the length of one side of each square to be cut out, write a function, \(V(x),\) for the volume of the box in terms of \(x\).

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(7.5, \frac{11 \pi}{18}\right) $$

If \(\|\mathbf{v}\|=4,\) what is the magnitude of \(\frac{1}{2} \mathbf{v}+3 \mathbf{v} ?\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ (\sqrt{2}-i)^{6} $$

Identify and graph each polar equation. $$ r^{2}=9 \cos (2 \theta) $$

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: \(7^{x-1}=3 \cdot 2^{x+4}\)

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-3.1, \frac{91 \pi}{90}\right) $$

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ (1-\sqrt{5} i)^{8} $$

Identify and graph each polar equation. $$ r^{2}=\sin (2 \theta) $$

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 function that is graphed after the graph of \(y=\sqrt[3]{x}\) is shifted left 4 units and up 9 units?

Find a vector \(\mathbf{v}\) whose magnitude is 4 and whose component in the \(\mathbf{i}\) direction is twice the component in the \(\mathbf{j}\) direction.

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 all asymptotes of the graph of \(f(x)=\frac{2 x^{2}-5}{x^{2}-2 x-15}\)

Find a vector \(\mathbf{v}\) whose magnitude is 3 and whose component in the \(\mathbf{i}\) direction is equal to the component in the \(\mathbf{j}\) direction.

Find all the complex roots. Write your answers in exponential form. The complex fourth roots of \(\sqrt{3}-i\)

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 exact value of \(\cos 80^{\circ} \cos 70^{\circ}-\sin 80^{\circ} \sin 70^{\circ}\).

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (3,0) $$

If \(\mathbf{v}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=x \mathbf{i}+3 \mathbf{j},\) find all numbers \(x\) for which \(\|\mathbf{v}+\mathbf{w}\|=5\)

Find all the complex roots. Write your answers in exponential form. The complex fourth roots of \(4-4 \sqrt{3} i\)

Identify and graph each polar equation. $$ r=1-\cos \theta $$

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 vertex and determine if the graph of \(f(x)=\frac{2}{3} x^{2}-12 x+10\) is concave up or concave down.

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (0,2) $$

If \(P=(-3,1)\) and \(Q=(x, 4),\) find all numbers \(x\) so that the vector represented by \(\overline{P Q}\) has length 5 .

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 \(f(x)=\frac{1}{\left(x^{2}+9\right)^{3 / 2}}\) and \(g(x)=3 \tan x,\) show that\((f \circ g)(x)=\frac{1}{27\left|\sec ^{3} x\right|}\)

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (-1,0) $$

Write the vector \(\mathbf{v}\) in the form \(\mathbf{ai}+ \mathbf{bj}\), given its magnitude \(\|\mathbf{v}\|\) and the angle \(\alpha\) it makes with the positive \(x\) -axis. \(|\mathbf{v}|=5, \quad \alpha=60^{\circ}\)

Identify and graph each polar equation. $$ r=1-3 \cos \theta $$

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (0,-2) $$

Write the vector \(\mathbf{v}\) in the form \(\mathbf{ai}+ \mathbf{bj}\), given its magnitude \(\|\mathbf{v}\|\) and the angle \(\alpha\) it makes with the positive \(x\) -axis. \(\mid \mathbf{v} \|=8, \quad \alpha=45^{\circ}\)

Find all the complex roots. Write your answers in exponential form. The complex cube roots of -8

Identify and graph each polar equation. $$ r=4 \cos (3 \theta) $$

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (1,-1) $$

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (-3,3) $$