Solve the problems in the assignmentMATH2004C Assignment 1 Last Name: First Name: Student ID: • You...

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Solve the problems in the assignmentMATH2004C Assignment 1 Last Name: First Name: Student ID: • You may either write your answers on a copy of this assignment, or on your own paper or on your electronic devices (you do not need to copy the questions). • Due date: Sunday, February 6th, at 10pm. • The assignment is out of 40 points • No email submission will be accepted. • No excuses regarding technical issues will be accepted. It is your responsibility to double check that you submitted the right file. Don’t wait until too close to the deadline to start working. • Submission Requirements: Submit your work as only one .pdf document. Files in format different then .pdf or that are not in one document will not be marked and you will obtain 0. Submit your file at the appropriate link on Brightspace. • If you have changes after submission, you can resubmit before the deadline. Only the last submitted file before the deadline will be marked. • In the next page, you must certified that this is your own work by signing at the different places. • You .pdf must be legible. The questions must be in the right order and the files should have the correct orientation • Your file must be the following format: LastName,FirstName/Name of file. For example, if my name is Matt Lemire, I would name my file as: Lemire, Matt Assignment 1.pdf • Show your work: Means that you must show all your steps with justification. For example, solving an integral by submission means that you must show your substitution. Basically, only Question 4a and Question 7a do not need any justification and/or work to be shown. • No decimal answers will be accepted. We only want exact and simplified answers in the form of fractions. For example, 0.125 is not accepted but 1 8 would be. An expression of the form 3π − 1 2 would an example of an exact answer. • You can use the Discord forum to write privately to other people in the class regarding answers and work but please do not post any kind of solutions or major hints on the forum. The goal is for you to learn as much as you can your own. It is okay to get help from others as long you understand it yourself eventually. Question 0a. This assignment is open book. I would kindly ask you to do this assignment without just copying down other people answers. I would kindly ask you to promise (code of honour) that you accept the following: I promise not to have someone else doing my assignment. I am allowed to consult textbooks, notes, the internet, some classmates, but I will only do so to help my understanding and not for others to do my work. Signature: (For students who do not write on a printed version of the exam, simply write 0a: and then put your signature.) Question 0b: By signing here, I hereby certify that I have read all the instruc- tions and conditions on the first page and that I will follow them. Signature: (For students who do not write on a printed version of the exam, simply write 0b: and then put your signature.) Question 0c: By signing here, I hereby certify that I understand that I must submit all my work no later than Friday February 4th, no later than 22:00 at the appropriate link on Brightspace. I also know that my work will not be accepted passed that day and time. Signature: (For students who do not write on a printed version of the exam, simply write 0c: and then put your signature.) Important Trigonometric Identities: sin2(x) + cos2(x) = 1 cos2(x) = 1 + cos(2x) 2 sin2(x) = 1− cos(2x) 2 sin(x) sin(y) = 1 2 cos(x−y)− 1 2 cos(x+y) = 1 2 cos(y−x)− 1 2 cos(x+y) = sin(y) sin(x) cos(x) cos(y) = 1 2 cos(x−y)+ 1 2 cos(x+y) = 1 2 cos(y−x)+ 1 2 cos(x+y) = cos(y) sin(x) sin(2x) = 2 sin(x)cos(x) cos(2x) = cos2(x)− sin2(x) Mathieu Lemire 1. a) Find the equation of the plane that passes through the points (2, 3,−1), (3, 4, 2) and (1,−1, 0). Show all your work. (2 points) b) Find the equation of the plane that passes through the point (1, 2, 3) and that is parallel to the plane 4x− 3y + 2z = 1. Show your work. (1 point) 2. Find a parametrization (parametric equation or parametric curve) of the line in space that intersect the planes 2x + y − 3z = 0 and x + y = 1. Show all your work. (2 points) 3. A curve C in space is given as the triangle that begins at (-1,0,7) until (5,4,-2), from (5,4,-2) to (-3,1,4) and then, from (−3, 1, 4) to (−1, 0, 7). Hence, C can be seen as the union of the curves C1, C2 and C3, where each of C1, C2, C3 is one side of the triangle. Give a parametrization of each of C1, C2 and C3. For each of them, do it in such a way that each depends of a parameter t with 0 ≤ t ≤ 1. Show all your work and make sure that the orientation of your parametrization follow the right direction given above. (3 points) 4. A curve C in R2 begins at (−1, 1) and ends at (6,−4). It consists of 3 curves C1, C2 and C3. • Curve C1 goes along the parabola y = x2 line segment from (−1, 1) and ends at (2, 4). It involves a parameter t with 0 ≤ t ≤ 3. • Curve C2 goes along the lower part of the circle of radius 2 center at (4, 4) from (2, 4) to (6, 4). It involves a parameter t with π ≤ t ≤ 2π. • Curve C3 goes the left half of the ellipse with equation (x− 6)2 4 + y2 16 = 1 from (6, 4) to (6,−4). It involves a parameter t from π 2 to 3π 2 . a) Give a sketch of the curve C. (1 point) b) Give the parametrization of each of the curves C1, C2 and C3. Don’t forget that your parametrization must satisfy the given intervals for the param- eter t. Show your work. (3 points) 5. Consider the parametric curve C : r(t) = (2 cot t, 2 sin2 t), where 0

Sudarshan K · Accepted Answer

Math 2004C
1) Equation of planes the passes through points A(2,3,-1), B(3,4,2) and C(1,-1,0)
a. Equation of line
Equation of three dimensional plane can be given by ax+by+cz+d=0
By plugging the points in the plane equation,
Point A:
2a+3b-c+d=0    ----(1)
Point B:
3a+4b+3c+d=0     -----(2)
Point C:
a-b+d=0    -------(3)
Solving this equation.
a=b-d
Subs. it in (1)
5b-c-d=0  c=5b-d
Subs a and c in (2)
3(b-d)+4(b)+3(5b-d)+d=0  22b-5d=0  b=5d/22, a = -17d/22, c=3d/22
Subs. a, b, c in plane equation.
-17/22 (dx)+5/22 (dy)+3/22(dz)+d=0
Cancelling d and multiplying by 22
-17x+5y+3z+22=0 is the equation of plane passing though point A,B and C
b. Equation of plane through point (1,2,3) and parallel to the plane (4x-3y+2z=1).
Equation of plane parallel to plane 4x-3y+2z=1 is given by 
4x-3y+2z=a
Subs point A(1,2,3) in above equation
4-6+6=a  a=4
So final equation is 4x-3y+2z=4
2) Find a parameterization of the line in space that intersect the planes 2x+y-3z=0 and x+y=1.
Plane A=2x+y-3z =0,

MATH2004C Assignment 1 Last Name: First Name: Student ID: • You may either write your answers on a copy of this assignment, or on your own paper or on your electronic devices (you do not need to copy...

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