Math Problem

Weekly Problem 13

Due: Friday, December 2 by 11:59pm

Directions. You are welcome to collaborate with your classmates on this assignment, but all submitted work must be written by you. Show all of your work for the problems below. An answer without proper work will receive little to no credit. If there are multiple parts to the problem, then make sure you have the parts in order on your paper. When submitting your work to Gradescope, you must submit it properly, selecting each problem when submitting. See Blackboard for a quick video on submitting properly. If your work is not submitted properly, it will receive a 0.

1. (10 points) Sketch both parametric curves below. Then determine if they intersect. If they intersect, find the point (x, y) of intersection.

C1 : x1(t) = 3 cos t, y1(t) = 3 sin t

C2 : x2(t) = 4 sin t, y2(t) = 4 sin t

2. (10 points) Plot and label the following polar points (r, θ) in the xy-plane.

(a) A = (−1, 5π/4)

(b) B = (2, 3π)

(c) C = (−2, 4π/3)

(d) D = (3,−4) [remember this point is in polar coordinates]

3. (10 points) Sketch the set of points (r, θ) that satisfy the following conditions:

{(r, θ) : −2 ≤ r < 2, 0 ≤ θ ≤ π/4}

4. (10 points) Consider the polar curve r = sin(3θ). Sketch this polar curve on the interval 2π/3 ≤ θ ≤ π.

5. (10 points) Use Gauss-Jordan elimination to solve the system

5x+ 7y = −11

2x+ y = 1

Make sure you create the augmented matrix and write down all of your elementary row operations.

6. (10 points) Consider the matrix

A =

[ 4 1 −1 7

] (a) Find the determinant of A.

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Weekly Problem 13

(b) Give a one sentence reason why A−1 exists and then find A−1.

(c) Give an example of a 2×2 matrix B where all it’s elements are distinct (i.e. different) numbers and B−1 does not exist. Give a one sentence reason why your B−1 does not exist.

7. (5 points) Multiply

[ 8 3 −1 10

] · [

0 2 −7 0

] . Show your work!

8. (15 points) Find the eigenvalues of

[ 1 3 4 2

] and find an associated eigenvector for each

eigenvalue.

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