2020 Summer Math280_New_Midterm_Take_Home
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Coastline Community College *
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Course
280
Subject
Mathematics
Date
Feb 20, 2024
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docx
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Math 280 Midterm Take Home Name
:
1
2
You must show some work to get full credit. Page 3: Page 4: Page 5: Page 6: Page 7: Page 8: Page 9: Please leave this page blank. Excel with Integrity Pledge
The Excel with Integrity pledge affirms Coastline commitment to excel with integrity both on and off campus, in academic, professional, and research endeavors. According to the International Center for Academic Integrity, academic integrity means having
the courage to act in ways that are honest, fair, responsible, respectful & trustworthy even
when it is difficult. Creating work with integrity is important because otherwise we are
misrepresenting our knowledge and abilities and the College is falsely certifying our
accomplishments. And when this happens, the degree loses its value and we’ve all wasted
our time and talents! Student Name: Excel with Integrity Pledge I understand I may not collaborate or communicate with
any other humans while working on this exam. I am fair to my classmates and instructors by
not using any unauthorized aids. I respect myself and my college by upholding educational
and evaluative goals. I am honest in my representation of myself and of my work. I accept
responsibility for ensuring my actions are in accord with academic integrity. I show that I am
trustworthy even when no one is watching. Affirm your adherence to this pledge by writing the following statement in the space below: I Excel with Integrity. Signature: Date: 4
Part A1: Quick Response Questions 1. True or False: The cross product of two vectors in R
3 may not be
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a vector in R
3
. 2. True or False: If a region in the plane is not open, then it must be close
. 3. True or False: The entire plane (our usual x - y plane), is an example of a set in the plane that is both open and close. 4. Fill in the blank: The directional derivative of a scalar valued
function of several variables in the direction of a unit vector is a . 5. Fill in the blank: The gradient of a scalar valued function of several variables is a valued function. 6. True or False: Let u and v be vectors. If u • v = 0 (dot product), then either u is the zero vector or v is the zero vector. 7. True or False: Let u and v be vectors. If u × v = 0 (cross product), then either u is the zero vector or v is the zero vector. 8. True or False: Let α be a scalar and v be a vector. If αv = 0 (scalar
product), then either α is the zero number or v is the zero vector. 5 Part A2: Multiple Choice Questions. You must show some work to get full credit. 1. Given A = <-3,2,-4> and B = <-1,4,1>. Find the unit vector in the direction of 6A - 2B. a) <-16,4,-26> b) 1 /
16
2 + 4
2 + 26
2 <-16,4,-26> c) <-20,4,-26> d) <-20,20,-22> e) None of the above. 2. Given A = <-3,2,-4> and B = <-1,4,1>. Find the area of the parallelogram formed by A and B.
a) <18,7,-10> b) <-18,-7,10> c) /
18
2 + 7
2 + 10
2 d) <14,7,-14> e) None of the above. 3. Given A = <-3,2,-4> and B = <-1,4,1>. Find the vector proj
A
B .
a) 1
/29
<3,8,-4> • <-3,2,-4> b) 7
29
<-3,2,-4> c) 3
/
2 cosθ d) 7
29 e) None of the above. 6 4. Given A = <-3,2,-4> and B = <-1,4,1>. Find the vector B - proj
A
B. a) <-1,4,1> - 〈
-6, 8
3
,-
2
9
〉 b) <-1,4,1> - 〈
-2,3,-
3
2
〉 c) <-1,4,1> - 29
7<-3,2,-4> d) B - A |B|
• 2 B B e) None of the above. 5. Let u and v be vectors in R
3 (in the x-y-z space). perpendicular to proj
v
u? Is u-proj
v
u a) Only if either u - proj
v
u or proj
v
u is the zero vector . b) Only if either u or v is the zero vector . c) No. u - proj
v
u is never perpendicular to proj
v
u d) Yes, always. e) None of the above 7 6. Find the vector equation r(t) of the line through the points (3,-4,-1) and (6,-2,-2). a) 3(x - 3) + 2(y + 4) - (z +1)=0 b) r(t) = <3,2,-1> + t<3,-4,-1> c) r(t) = <3,-4,-1> + t<3,2,-1> d) There is not enough information. e) None of the above 7. Given r(t) = 〈
t
2
, 2
3
t
3
,t
〉
. Find the vector T at the point 〈
1, 2
3
,1
〉
. a) T(-1) b) r
/
(t) |r
/
(t)| c) r
/
(1, 2
3
,1) ∣
∣r
/
(1, 2
3
,1)
∣
∣ d) T(1, 2
3
,1) e) None of the above 8
Part B: Free Response Questions. (You must show your work to get full credit.)
1. Find r(t) if r (t) = e
2t , (2t - t
2
) sin
(
t
2 - t
3
3 and r(0) = 0. and r(0) = 0. 2. Find the equation of the plane that contains the point (3,-4,-1) and the line r(t) = <2 - 4t,1+3t,-1 - t>. 9
3. Given z = sin (x + 4y), x = sln
/
t, y = tln
/
s. Compute ∂z
∂s and ∂z
∂t
. ∂z
∂s = ∂z
∂t = 1
0
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4. Given f(x, y) = sin (3x + 4y). Compute the gradient of f, Vf(x, y). Note that the gradient of a function of two variables (which is what
we have in this case) gives us a 2D vector valued function of two
variables. For three variables, we will get a 3D vector valued
function of three variables. This will be the main object of the last
chapter. 5. Give an example of a closed set in the plane (our usual x-y plane) that is not bounded. You can neatly draw the picture. 1
1 6. Give an example of a open set in the plane (our usual x-y plane) that is bounded. You can draw the picture.
7. Give an example of a set in the plane (our usual x-y plane) that is
not closed, not open, and not bounded. You can draw the picture but
note that it has to satisfy all three conditions. 1
2 Part C: Proof Questions. Let A = <a
1
,a
2
,a
3
> be a vector and α be a scalar. Show that if αA = 0, then either α = 0 or A
= 0. 13 Choose one of the following statements and prove it. Do not prove both. (a) Let A and B be two non-zero vectors. Show that if proj
B
A = 0, then proj
A
B = 0. (b) Let A and B be two non-zero vectors. If proj
A
B = 0, then proj
B
A = 0. 1
4 Challenge: Let r(t) be a differentiable vector valued function of 1
variable over the real line(so we can take the derivative of each
component). Show that if r(t) • r (t) = 0 for every t, then |r(t)| = |r(0)| for every t That is, |r(t)| must be a constant function. Note that this means r(t) must be a curve that lies on a sphere center at the origin.