Victor Camocho
math2250fall2011-2
WeBWorK assignment number Homework 8 is due : 10/20/2011 at 11:00pm MDT.
The
(* replace with url for the course home page *)
for the course contains the syllabus, grading policy and other information.
This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.
The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making
some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are
having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor for
help. Don’t spend a lot of time guessing – it’s not very efficient or effective.
Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,
you can if you wish enter elementary expressions such as 2 ∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e ∧ (ln(2)) instead of 2,
(2 + tan(3)) ∗ (4 − sin(5)) ∧ 6 − 7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.
You can use the Feedback button on each problem page to send e-mail to the professors.
0
7
• D.
,
1. (1 pt) Library/Rochester/setLinearAlgebra10Bases/ur la 10 32.pg
0 -6 4
Find a basis
-16
-8
 of R consisiting of all vectors of
 of the subspace
• E.
,
x1
-10
-5  −8x1 + x2 
-3
-7
8

the form
• F.
,
,
 −9x1 + 3x2  .
-9
6
5
−2x1 + 9x2
 


5.
(1 pt) Library/Rochester/setLinearAlgebra9Dependencela
9
10.pg
/ur
 


 ,
.

-28
 


Express the vector v =
as a linear combination of
25
2
5
.
x=
and y =
1
-4
2. (1 pt) Library/Rochester/setLinearAlgebra10Bases/ur la 10 25.pg
v=
x+
y.
Find a basis of the subspace of R3 defined by the equation
−9x
6.
(1 pt) Library/TCNJ/TCNJ BasesLinearlyIndependentSet 1 + 5x2 −8x3 = 0.
/problem8.pg
,


 
6
-2
Let W be the set:  3  , -1  .
5
0
Determine if W is a basis for R3 and check the correct answer(s)
below.

.
3. (1 pt) Library/Rochester/setLinearAlgebra10Bases/ur la 10 1.pg
The vectors






2
-5
-4
v1 = -4  , v2 = 2  , and v3 = -8 
0
6
k
.
form a basis for R3 if and only if k 6=
4.
(1
pt)
• A. W is not a basis because it is linearly dependent.
• B. W is not a basis because it does not span R3 .
• C. W is a basis.
Library/Rochester/setLinearAlgebra9Dependence-
7. (1 pt) hw8/p7.pg
Determine which of the following pairs of functions are linearly
independent.
/ur la 9 9.pg
Which of the following sets of vectors are linearly independent?

• A. 
• B.

• C. 
-5
-2
-4
-1
3
9
-6
0
 
 

0
-3
 , -7  , 6 
1
-8
9
,
-1
   

3
2
 , 4  , -5 
0
0
f (θ) = cos(3θ) , g(θ) = 5 cos3 (θ) − 10 cos(θ)
f (x) = e5x , g(x) = e5(x−3)
f (x) = x2 , g(x) = 4|x|2
f (t) = 17t 3 , g(t) = ex
2x − 4y =
12
? 5.
−3x + 6y = −18
?
?
?
?
1
1.
2.
3.
4.
?
?
?
?
8.
6.
7.
8.
9.
f (t) = eλt cos(µt) , g(t) = eλt sin(µt) , µ 6= 0
f (t) = 5t 2 + 35t , g(t) = 5t 2 − 35t
f (θ) = cos(3θ) , g(θ) = 5 cos3 (θ) − 5 cos(θ)
f (t) = 3t , g(t) = |t|
Answer: y(x) = C1
+C2
.
NOTE: The order of your answers is important in this problem. For example, webwork may expect the answer ”A+B” but
the answer you give is ”B+A”. Both answers are correct but
webwork will only accept the former.
(1 pt) Library/TCNJ/TCNJ BasesLinearlyIndependentSet-
/problem10.pg
11.
Determine whether each set {p1 , p2 } is a linearly independent
set in P3 . Type ”yes” or ”no” for each answer.
(1 pt) Library/Utah/Calculus II/set13 Differential Equations-
/set13 pr9.pg
Solve the following differential equation:
The polynomials p1 (t) = 1 + t 2 and p2 (t) = 1 − t 2 .
y00 − 3y0 − 10y = 0;
y = 1, y0 = 10 at x = 0
Answer: y(x) =
The polynomials p1 (t) = 2t + t 2 and p2 (t) = 1 + t.
.
12. (1 pt) hw8/p13.pg
Match the third order linear equations with their fundamental
solution sets.
The polynomials p1 (t) = 2t − 4t 2 and p2 (t) = 6t 2 − 3t.
9. (1 pt) Library/TCNJ/TCNJ LinearIndependence/problem9.pg
Let u = (−5, 0, −1), v = (0, −5, −2), w = (−10, −15, h − 3).
Determine the value for h so that w is in the span of the vectors
u and v.
h=
.
Determine the value for h so that u is in the span of the vectors v and w.
h=
.
1.
2.
3.
4.
y000 − y00 − y0 + y = 0
y000 − 6y00 + 8y0 = 0
ty000 − y00 = 0
y000 + 3y00 + 3y0 + y = 0
A.
B.
C.
D.
1, e4t , e2t
et , tet , e−t
1, t, t 3
e−t , te−t , t 2 e−t
13. (1 pt) hw8/p14.pg
Find y as a function of x if
? 1. If h equals the value in the first question above, determine whether or not the set {u, v, w} is linearly independent.
y000 − 13y00 + 40y0 = 0,
y(0) = 5, y0 (0) = 7, y00 (0) = 8.
y(x) =
10. (1 pt) Library/Utah/Calculus II/set13 Differential Equations/set13 pr10.pg
Solve the following differential equation:
NOTE: You have all the tools you need to solve this problem.
Generalize what you have learned about second order linear
ODE’s to help you solve this 3rd order problem.
y00 + 10y0 + 25y = 0
c
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Team, Department of Mathematics, University of Rochester
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