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 the questions).

• Due date: Sunday, Fe

uary 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

e 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 co

ect orientation

• Your file must be the following format: LastName,FirstName/Name of file. Fo

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 you

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

egarding answers and work but please do not post any kind of solutions o

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 fo

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 Fe

uary 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)

) 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

ight 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)

) 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 < t < π.

Determine the equations of the tangent lines of that curve at the points

(

− 2√

3

,

3

2

)

and

(

2

√

3,

1

2

)

. Show all your work for each line. (4 points)

6. Find the exact value of the length of the parametric curve

(t) = (2 cos(t) − cos(2t), 2 sin(t) − sin(2t)), 0 ≤ t ≤ π

2

. Show all your work. We

expect you to show your steps when solving integrals. (4 points)

Hint:

1− cos(2x)

2

= sin2(x) for all x.

7. a) Describe each shaded section of the figures below using inequalities for r and θ.

(4 points)

) Convert the Cartesian equation 3x− y+ 2 = 0 to a Polar equation. Show all you

work. (2 points)

c) Convert the Polar equation r = 2 csc(θ) to a Polar equation. Show all your work.

(2 points)

Mathieu Lemire

(In orange)

Mathieu Lemire

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Mathieu Lemire

(Or in blue)

Mathieu Lemire

Mathieu Lemire

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Cartesian

8. Find the exact area of the interior of the polar curve r = 3 + 2 sin(θ) that satisfies

x ≤ 0, y ≥ 0. Sketching the curve might be useful (but is not necessary) to fully

understand the exact area that we want. Show all your work. (4 points)

Note: The hint given for Problem 6 may be useful here too.

9. Consider the polar curves r = 2 cos(2θ) and r = 1.

a) Sketch (by hand and only by hand) the two curves on the same graph. (1 point)

) Determine two angles θ such that −π

4

≤ θ ≤ π

4

, where the two curves inter-

sect. You must find the angles alge

aically and not from your sketch. (1 point)

c) Find the exact area of the region giving all the points that are inside the first

curve and inside the second curve. Show all your work. (3 points)

10. Find the arc length of the polar curve r =

√

1 + sin(2θ), 0 ≤ θ ≤ π

4

. Show all

your work. (3 points)

Answered 1 days AfterFeb 05, 2022

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

. 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, plane B =x+y=1

Normal vector of plane A <2,1,-3> and normal vector of plane B<1,1,0

Cross product of the above two vectors give

AxB = | i j k |

| 2 1 -3 |

| 1 1 0 |

AxB=3i-3j+k

So normal vector of the cross product is <3, -3, 1

By taking the value of y as zero for finding the plane

Plane B gives x=1.

Subs. it in plane A, gives z=2/3

So the vector becomes <1, 0, 2/3

So the parameterization equation can be given by

(1+3t)i+(0-3t)j+(2/3+t)k

So the x, y, and z values are given by

X=1+3t, y = -3t, z =2/3+t

3) A curve space is given at triangle that begins at A(-1,0,7) to B(5,4,-2), from B(5,4,-2) to C(-3,1,4) and from point C to point A. Find parameterization of three curves given by these 3 lines depending upon ‘t’ and within 0Answer:

For line 1 from A to B so it is starting from point A and moving towards point B. So the starting point can be taken as point A and then using the difference between two points

V=

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

. 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, plane B =x+y=1

Normal vector of plane A <2,1,-3> and normal vector of plane B<1,1,0

Cross product of the above two vectors give

AxB = | i j k |

| 2 1 -3 |

| 1 1 0 |

AxB=3i-3j+k

So normal vector of the cross product is <3, -3, 1

By taking the value of y as zero for finding the plane

Plane B gives x=1.

Subs. it in plane A, gives z=2/3

So the vector becomes <1, 0, 2/3

So the parameterization equation can be given by

(1+3t)i+(0-3t)j+(2/3+t)k

So the x, y, and z values are given by

X=1+3t, y = -3t, z =2/3+t

3) A curve space is given at triangle that begins at A(-1,0,7) to B(5,4,-2), from B(5,4,-2) to C(-3,1,4) and from point C to point A. Find parameterization of three curves given by these 3 lines depending upon ‘t’ and within 0

For line 1 from A to B so it is starting from point A and moving towards point B. So the starting point can be taken as point A and then using the difference between two points

V=

SOLUTION.PDF## Answer To This Question Is Available To Download

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