Please clearly show the working if willing to help.

Here is the process outlined in detail:

Pick your problem. r = 3cosθ

Start by picking an easy-to-work-with angle θ (or pick a difficult angle if you like a challenge)Plug that angle in for θ in your equation, and compute the corresponding value of r.

Write your answer in the polar coordinates form (r, θ)

Repeat the steps above for another chosen value of θ.

Use the conversion formulas for converting polar points to Cartesian coordinates.

x = rcosθ and y = rsinθ

Write your equivalent Cartesian coordinates points in form (x,y)

CHECK: Do your polar points and their corresponding Cartesian coordinates land on the same point when graphed?

r = 3cosθ

2. Start by drawing a diagram of the situation in your selected word problem. Clearly label which direction is north, south, east, and west in your diagram, and label the known sides and angles in your triangle. Then use the Law of Cosines or Law of Sines to solve the problem.

A vehicle travels due west for 30 miles. Then it turns and goes 30 miles in the direction of S68ᵒW. How far is it from the starting place?

3. Pick a set of vectors u and v from the list below, and do the following 5 items:

A. Draw each vector in standard position.

B. Find the magnitude of vector u: ||u||

C. Find the difference of vectors < u - v >

D. Find the magnitude of vector < u - v > : || u - v ||

E. Find the angle θ, in degrees, between the vectors u and v. θ should be between 0° and 180°

Here is the pairs of vectors:

u = < 4, 5 > and v = < -1, -15 >

4. Determine if the given equation is a parabola, ellipse, circle, or hyperbola.

(y+2)2 = 4(x+6)

If it is a parabola find the vertex point, focus, directrix, and determine which way it will open (up, down, right, or left).

If it is an ellipse, find the center point and vertices, and determine if it is short and wide or tall and thin.

If it is a circle, find the center and radius.

If it is a hyperbola, find the center point and vertices, and determine which way it will open (up and down, or right and left).

5. GRAPH the conic section. You can graph by hand, or use a graphing calculator such as Desmos. Also find two points (in the form of ordered pairs (x,y)) that fall on the graph of your conic section.

12x2 + 3y2 − 30y + 39 = 0

5x2 + 18y2 − 30x + 72y + 27 = 0

25x2 + 20x + 5y − 1 = 0

6. Below you will find systems of equations, along with the augmented matrix formed for the system. Pick any problem, and use Gauss-Jordan elimination on the augmented matrix to find the solution of the system. That is, you need to use row-equivalent operations on the matrix until it is in row-reduced echelon form, and looks like the identity matrix in the first two columns. And once you have that, you can read off the solution to the system, in the third column.

The purpose of this exercise is to practice using Gauss-Jordan elimination on an easy matrix. There are LOTS of faster easier ways to solve these little systems with 2 equations and 2 variables, but we are doing this to practice the method. And the best way to start practicing a new technique is to start small.

The augmented matrix is given below, for the system of equations x + y = 1 and 3x + y = 7

Use Gauss-Jordan elimination to solve this system.

7. Pick a system of equations. Solve it using elimination, substitution, or Gauss-Jordan elimination on a matrix. Show all your work.

5x + 3y + z = 18

x - 3y + 2z = 0

14x - 2y + 3z = 40

8. pick a sequence from the list below.

A. Determine if the sequence is arithmetic, geometric, or neither.

B. Find the explicit formula, also know as the "general rule", an for the sequence.

C. Use your explicit formula to find the 10th term a10.

11, 7, 3, -1, …..

9. pick a sequence from the list below and answer the 3 questions.

A. Find the term that is requested in the problem. To do this, find the explicit formula for the sequence, and then use that to find the specified term.

B. Use summation notation to write the sum of some number of terms. This is an exercise in using the summation notation properly. No calculations are needed here; it is just practice in using mathematical notation.

C. Find the requested sum. Use one of the formulas for either the sun of a geometric sequence or sum of an arithmetic sequence, or the sum of an infinite number of terms of a geometric sequence.

4, 16, 64, 256, …..

A. Find the tenth term, a10

B. Use summation notation to write the sum of the first ten terms.

C. Use the Sum of Arithmetic or Geometric Sequences formulas (page 666) to find S10, the sum of the first 10 terms.