Math 1502 Assignments

Mathematics 1502 Calculus II Spring, 2001

Current reading and homework assignments

Preparing for the test on 3 May at 11:30:
- Read Prof. Harrell's classnotes on eigenvectors and eigenvalues. (Note: a sign error in the last differential equation problem was corrected on Saturday morning, 27 April.)
- Review the printed reading assignments from earlier in the term and rework those problems.
- ${NEW!}$ Look at the on-line linear algebra text by Prof. Shores of the University of Nebraska, especially chapters 1-4 and sections 1-2 of chapter 5. Do exercises 1-8 in section 5.2 of Shores's text.
- ${NEW!}$ Watch the video lectures on linear algebra by Prof. Strang of MIT, especially 1-10, 14,15,18,21-23.
- Prof. Carlen's classnotes on eigenvectors and eigenvalues.
- Take some old final exams:
- Want solutions to the exercises? Remember Maple, the "mother of all solutions manuals."

Past homework assignments

Due Thursday, 11 January:

Reading:
- SHE (Salas, Hille, Etgen), sections 11.1-11.3
Exercises
- SHE, section 11.1, #3, 5, 17, 18, 23, 24, 25, 26, 27, 29, 33, 34, 36, 55, 60, 61
- SHE, section 11.2, #1, 2, 4, 5, 8, 11, 12, 16, 19, 20, 21, 33, 34, 35, 36
- SHE, section 11.3, #1,5,7,14,18,29,31,36

Due Thursday, 18 January:

Reading:
- SHE (Salas, Hille, Etgen), sections 11.4,11.5,11.6,10.5
Exercises
- SHE, section 11.4, #2,8,10,16,20,32,44
- SHE, section 11.5, #2,3,5,10,11,12,17,18,25,26,29,38,41,43
- SHE, section 11.6, #2,5,17,19
- SHE, section 10.5, #2-8,10,24-26,34,46

Due Thursday, 25 January:

Reading:
- See Maple's classnotes from Friday, with the Top Ten Taylor Table
- SHE (Salas, Hille, Etgen), sections 10.6,11.6,11.7,11.8
- One of the TA's, Jose Renom has kindly provided some Maple-based study materials.
- (optional) Watch a pi-minute long Quicktime © movie about an interesting number. (Click for free Quicktime reader if necessary, for Mac or Windows.
Exercises
- Do Problems 3-4 of a test given last semester
- Do Problems 1, 2b, and 3 of a test given in 1999
- Take Prof. Loss's Prep Test 1 and an actual test he gave
- SHE, section 10.6, #2,8.9,13,29,31,32,37.
- SHE, section 11.6, #2,5,17,19 (all from last week); and 27,28,33
- SHE, section 11.7, #1,4,5,9,19,20,33,34
- SHE, section 11.8, #1,2,6,9,10,13,16,23,31,43,44

There was a test on the 26th!

Due Thursday, 1 February:

Reading:
- Read the solutions to the recent test, according to Maple:
  - green version (as a Maple worksheet or as a web page)
  - yellow version (as a Maple worksheet or as a web page)
- SHE (Salas, Hille, Etgen), sections 11.7,11.8,8.7. We will review the material on series which are most needed, in light of the first test.
Exercises
- SHE, section 11.7, #19,20,33,34 (from last week)
- SHE, section 11.8, #1,2,6,9,10,13,16,23,31,43,44 (same as last week)
- SHE, section 11.5, # 17-32 - Do these to prepare for the quiz, which will be on Lagrange's formula.
- SHE, section 8.7, #1,3,7,9,14,18 - these have been deferred to next week!

Due Thursday, 8 February:

Reading:
- SHE (Salas, Hille, Etgen), sections 8.7-8.9
- The simple derivation of Simpson's rule
- Prof. Herod's Maple worksheet, " Area by Chance." (click here for a version which does not require Maple.)
Exercises
- SHE, section 8.7, #1,3,7,9,14,18
- Do the exercise in Prof. Herod's Maple worksheet, " Area by Chance": Use the "Monte-Carlo" method to find the area of a quarter disk with radius . Suggestion: If you do not have Maple of a similar utility, most spreadsheets and programming languages have (pseudo)random number commands. For example, you can produce a column of random numbers evenly distributed between 0 and 1 in Excel by pasting
  
  =RAND()
  
  =RAND()
  
  into the cells of the column.
- Use power series to solve the following differential equations:
  1. y'(t) = (1+t-t²)y(t), such that y(0) = 1
  2. y'(t) = (2 -t)) y(t), such that y(1) = -1.
- SHE, section 8.8, #1,5,9-12,23-26,30,32,36
- SHE, section 8.9, #7-13,26,29

=RAND()
=RAND()

There was a test on the 16th!

Due Thursday, 15 February:

Reading:
- SHE (Salas, Hille, Etgen), sections 8.8,8.9,12.3 (section 12 will not be on Fridays' test)
- H² (Hubbard and Hubbard), section 1.1
- Since the test will refer to the hyperbolic functions sinh, cosh, and tanh, please review the Maple worksheet with some information about these functions. (Click here in case you do not have maple.) please
Exercises
- SHE, section 8.8, #34,38
- SHE, section 8.9, #3-6, 14-16,25,27,28,30
- SHE, section 12.3, #1-19
- H², section 1.10, #1.1.1, 1.1.2
- Take problems 1-3 (not 4) of a test given last semester.
- Work on the ant problem:

Due Thursday, 22 February:

Reading:
- H² (Hubbard and Hubbard), section 1.2
Exercises
- H², section 1.10, #1.1.1, 1.1.2, 1.2.2, 1.2.3, 1.2.8

Due Thursday, 1 March:

Reading:
- SHE, sections 12.4, 12.5
- H² (Hubbard and Hubbard), section 1.3, 1.4 (Especially 1.4)
- Prof. Spingarn's notes, Basic Matrix Concepts.
- Demko (optional, but on reserve at the library), 1.1, 1.2
Exercises
- H², section 1.10, #1.4.2-1.4.5, 1.4.7, 1.4.8
- SHE, section 12.4, #4-12, 19-24, 27, 29, 32, 43, 44
- SHE, section 12.5, #2-6, 23-28
- Find the equations for the following lines and planes. Convert the equations to standard form. Remember, the plane passing through P = (p,q,r) and perpendicular to N = ai + bj + c k is
  a x + b y + c z = P . N (dot product) = ap + bq + cr
  
  The line in the plane through (1,2) and parallel to the vector 5 i - j
  The line in the plane through (1,2) and perpendicular to the vector 5 i - j
  The plane passing through (1,2,3) and perpendicular to the vector 5 i - j + k
  The plane passing through (5,-1,1) and perpendicular to the vector i + 2 j + 3 k
  The plane passing through the three points (1,2,3), (5,-1,1), and (-1,0,1).
- Some projection problems
  
  Find the projection of the vector 5 i - j + k onto the direction of the vector i + 2 j + 3 k
  Find the projection of the vector i + 2 j + 3 k onto the direction of the vector 5 i - j + k
  Find the vector w parallel to i + j + k which is as close as possible to 5 i - j + k. (Closest means that ||5 i - j + k - w || is as small as possible.)
  Find the vector w parallel to 5 i - j + k which is as close as possible to i + j + k.

There was a test on Friday, 16 March.
Due Thursday, 15 March:

Reading:

SHE, section 12.7
H² (Hubbard and Hubbard), section 2.1
Prof. Spingarn's notes, Matrix Operations, Elementary Row Operations, and Matrix Inversion.
Demko (optional, but on reserve at the library), 1.2, 1.3
The class notes from Monday

with Maple or
without Maple

Exercises

H², section 2.10, #2.1.1-2.1.2, 2.1.7,
SHE, section 12.7, #1,2,3-6,10,11,15,25-29,37
Find all solutions of the following systems of equations or show that there are no solutions

x - y - 2 z = 0 2x + y - z = 0 x + 2y + z = -1

2x + 5y - 2z = 5 x + y = 0 3x + 2z = 4

4x + 5y - 6z = -1 x + y + z = 2 2x + 2z = 2

(augmented matrix notation)
[9 8 7 | 14] [6 5 4 | 8 ] [3 2 1 | 2 ]

[9 8 7 | 7 ] [6 5 4 | 4 ] [3 2 1 | 7 ]

[8 1 6 | 15] [3 5 7 | 15] [4 9 2 | 15]

[8 1 6 1 | 0 ] [3 5 7 1 |-10] [4 9 2 1 |-8 ] [1 0 1 0 | 0 ]

[8 0 6 2 | 6 ] [0 5 7 1 | 7 ] [1 9 -1 1 |-1 ] [1 0 2 0 | 2 ]

Find the intersection(s) of the following planes:

The plane passing through (0,1,2) with normal vector N^t = (1 4 7)
The plane which intercepts the three positive axes 2 units from the origin, and
The plane parallel to the y-z plane and passing through the point (1,2,2)

Take the test given in 1999.
Take an old test given by Prof. Loss

Due Thursday, 22 March.

Reading:

H², sections 2.2, 2.3, 2.4, 2.5
Prof. Spingarn's notes, Matrix Inversion.
A Maple worksheet on linear operators on the plane, courtesy of José Renom.
Demko (optional, but on reserve at the library), 2.3, 4.1, 4.2, 3.4, 3.1

Exercises

(For these problems, refer to class notes from Monday, 19 March, or Demko, sections 4.1-4.2.)
Sketch the mage of the square {0 < x < 1, 0 < y < 1} when mapped by the following matrices:

[1 0] [2 1]

[3 0] [0 -1]

[1 2] [1 2]

[2 cos(1) -2 sin(1)] [2 sin(1) 2 cos(1)]

H², section 2.10, #2.3.1, 2.3.2, 2.3.4,2.3.11, 2.4.3, 2.4.5, 2.4.8
Find the 2 by 2 matrices which do the following:

First stretches by a factor of 3 in the y direction, then rotates clockwise by pi/4.
First projects onto the x-axis then rotates counterclockwise by pi/4 radians.
Projects onto the line y=x.
Projects onto the line y = -2 x.
Projects onto the x axis; then rotates counterclockwise by pi/2, then projects onto the x-axis again.

Find the 3 by 3 matrices which do the following:

Stretches by a factor of 3 in the x direction, then shrinks by a factor of 3 in the z direction, then rotates about the x axis by pi/4.
Rotates by pi/4 about the z axis then by pi/4 about the y axis.
Projects onto the vector i - 2 j + 3 k.
Projects onto the plane x - 2 y + 3 z = 0.
Maps the vector i to i - 2 j, maps the vector j to j + 3 k, and maps the vector k to i + j + k.
Maps the vector i to i - 2 j, maps the vector i + j to j + 3 k, and maps the vector i + j + k to 0.

Due Thursday, 29 March.

Reading:

H², sections 2.4, 2.5
Prof. Spingarn's notes, Describing the Solution Set of a Linear System.
A Maple worksheet on linear operators on the plane, courtesy of José Renom.
Demko (optional, but on reserve at the library), 3.4, 3.1, 4.4
Prof. Carlen's notes, Geometry of Linear Transformations and Equations.

Exercises

H², section 2.10, #2.4.8, 2.4.9, 2.4.11, 2.5.1, 2.5.2, 2.5.7, 2.5.13, 2.5.14
Do the exercises in Prof. Spingarn's notes
Calculate the determinants of the following, or explain why that is impossible:

[1 2] [3 4]

[4 2] [8 1]

[1 2] [2 4] [3 6]

[1 0 0] [2 2 3] [3 2 2]

[1 2 3 4 ] [5 6 7 8 ] [9 10 11 12]

[1 2 3 4 ] [5 6 7 8 ] [9 1 0 -1 ] [-2 -3 -4 -5]

[1 2 3 4] [5 6 11 12] [-1 1 0 2] [1 1 2 2]

[6 7 8] [1 5 9] [8 3 4]

[6 7 8 0] [1 5 9 0] [8 3 4 0] [0 0 0 2]

Find the column space (same thing as image) of the matrices in the previous exercise.
In each case, state whether

the column vectors are linearly independent
the row vectors are linearly independent

Find the inverses of the same matrices, or explain why that is impossible.

There was a test on Friday, 6 April.
Due Thursday, 5 April.

Reading:

Prof. Harrell's classnotes on overdetermined systems.
Prof. Harrell's classnotes (Maple worksheet) on column and null spaces. Note: Some graphics are missing in the text portion of this worksheet due to a Maple bug (we're working on it). For some reason the graphics which are missing from the html version of the worksheet are different from the ones missing from the Maple version. Weird.
H², sections 2.5, 2.6
Prof. Spingarn's notes, Describing the Solution Set of a Linear System and
The Gauss-Jordan Method.
Demko (optional, but on reserve at the library), 3.2, 3.3, 3.6
Prof. Carlen's notes, Subspaces, Bases, and Coordinates () and Geometry of Linear Transformations and Equations.
Look at the answer sheets available on Prof. Chen's 1502 Website.
The first five sections of Prof. Erdös's notes.
Prof. Harrell's classnotes on overdetermined systems.
Prof. Harrell's classnotes (Maple worksheet) on column and null spaces. Note: Some graphics are missing in the text portion of this worksheet due to a Maple bug (we're working on it). For some reason the graphics which are missing from the html version of the worksheet are different from the ones missing from the Maple version. Weird.

Exercises

Take an old t est given by Prof. Loss.
Work problems 3-6 of Prof. Chen's practice test 4.
Take an old test given by Prof. Harrell
Do the exercises in Prof. Spingarn's notes
Calculate the nullspaces of the following.

[1 2] [3 4]

[4 2] [8 1]

[1 2] [2 4] [3 6]

[1 0 0] [2 2 3] [3 2 2]

[1 2 3 4 ] [5 6 7 8 ] [9 10 11 12]

[1 2 3 4 ] [5 6 7 8 ] [9 1 0 -1 ] [-2 -3 -4 -5]

[1 2 3 4] [5 6 11 12] [-1 1 0 2] [1 1 2 2]

[6 7 8] [1 5 9] [8 3 4]

[6 7 8 0] [1 5 9 0] [8 3 4 0] [0 0 0 2]

Find orthogonal bases for the null spaces of the previous problem.
Find all solutions of the following equations. Relate your answers to the null spaces found above.

[1 2] [-1] [2 4] x = [-2] [3 6] [-3]

[1 2 3 4 ] [ 1] [5 6 7 8 ] x = [ 1] [9 10 11 12] [ 1]

[1 2 3 4 ] [ 1] [5 6 7 8 ] x = [-2] [9 10 11 12] [ 1]

[6 7 8] [ 1] [1 5 9] x = [-2] [8 3 4] [ 3]

Find the line y=mx+b that best fits the data below in the least-squares sense

x_i y_i

0 -3

1 0

2 3

3 5

4 7

5 12

6 14

Find the least-squares solution of
[2 1] [ 1] [4 2] x = [ 0] [1 1] [ 0]

Find the least-squares solution of the system of equations
x + 2 y = 0 2x + y + z = 1 2y + z = 3 x + y + z = 0 3x + 2 z = -1

H², section 2.10, #2.6.1, 2.6.2
Find the matrix which expresses the coordinates (x,y,z) in terms of the following basis for R³:
[ 1] [ 2] [ 0] [ 2] [ 0] [ 1] [ 1] [-1] [ 1]
Find the inverse of that matrix and explain what it does.
Same as the previous problem, but use other three component vectors such as your personal "telephone code" and those of two linearly independent friends. Show by a calculation that you and your friends are linearly independent.
Find the matrix which expresses the coordinates (x,y) in terms of a basis which has been rotated from the standard basis by pi/6 (counterclockwise)
Show that the following sets of vectors are linearly dependent and express one of them as a combination of the others:

[ 9] [ 8] [ 7] [ 3] [ 2] [ 1] [ 6] [ 5] [ 4]

[ 1] [ 2] [ 4] [ 2] [ 0] [ 4] [ 1] [-1] [ 1]

[ 1] [ 2] [ 0] [ 6] [ 2] [ 0] [-4] [ 4] [ 1] [-1] [-3] [ 0] [ 1] [ 3] [ 1] [ 8]

Due Thursday, 12 April.

Reading:

Review the fourth test

green version
white version
with solutions by Maple:

green version
white version

Note: the quiz on Thursday will be a rerun of a problem from the test.

Prof. Harrell's classnotes on overdetermined systems and underdetermined systems.
Prof. Harrell's classnotes (Maple worksheet) on column and null spaces. Note: Some graphics are missing in the text portion of this worksheet due to a Maple bug (we're working on it). For some reason the graphics which are missing from the html version of the worksheet are different from the ones missing from the Maple version. Weird.
H², section 2.5
Prof. Spingarn's notes, Describing the Solution Set of a Linear System
Demko (optional, but on reserve at the library), 3.3, 3.4, 3.5
Prof. Carlen's notes, Subspaces, Bases, and Coordinates and Geometry of Linear Transformations and Equations.

Exercises

Redo all the problems from the previous homework and last week's test, especially the problems marked on this page with ***

Due Thursday, 19 April.

Reading:

Prof. Harrell's classnotes on overdetermined systems and underdetermined systems.
Prof. Harrell's classnotes on eigenvectors and eigenvalues.
Demko (optional, but on reserve at the library), 4.3, 5.1-5.4

Exercises
Many of this week's homework problems are from Demko. Please check back later for more.
Find the "smallest solution" to the following underdetermined systems:

A z = g, where
[1 1 1 0] [2] A = [ ], g = [ ] [0 1 1 1] [0]

A z = g, where
[1 1 1 1] [0] A = [2 0 1 0], g = [1] [0 2 1 0] [2]

x₂ + 2 x₃ + x₄ = 1 x₁ + x₂ + x₃ + x₄ = 2

Find the function of the form a + b x² + c x⁴ that satisfies f(0) = 2, f(1) = 4, and minimizes a² + b² + c². Hint: Plug in and convert this to an underdetermined linear problem.
Find the affine function that rotates the plane by pi/2 counterclockwise about the point (x,y) = (0,2).
Find the affine function that rotates the plane by pi/4 counterclockwise about the point (x,y) = (1,2).
Find an affine function that takes the rectangle -1 < x < 2, 0 < y < 1 onto the rectangle 0 < x' < 10, 0 < y' < 5.
Find the affine function that reflects the points in the plane through the line y = 4 x - 3.
Find the affine function that reflects the points in the plane through the line y = - 3 x + 4.
Find the affine function that reflects the space R³ through the plane x + y + 2 z = 4
Find the affine function that rotates the space R³ about the axis (x,y,z) = (1 + t, 2 + t, 3 + t) by angle alpha.
Find the eigenvalues and eigenvectors of the following matrices

[1 2] [3 4]

[4 2] [8 1]

[1 2] [1 1]

[1 0 0] [2 2 3] [3 2 2]

[6 1 8] [7 5 3] [2 9 4]

the projector onto the plane x + y - 2 z = 0

[1 1 2] [1 1 2] [2 2 4]

Due Thursday, 26 April.

Reading:

Prof. Harrell's classnotes on eigenvectors and eigenvalues.
Demko (optional, but on reserve at the library), 5.1-5.4
Prof. Carlen's classnotes on eigenvectors and eigenvalues.

Exercises

Diagonalize the following matrices, or show that they are not diagonable:

[3 1] [2 4]
[0 1] [1 2]
[7 5] [5 7]
[1 1] [0 1]
The 2 by 2 matrix that rotates vectors by pi/4 radians counterclockwise about the origin.
The 2 by 2 matrix that rotates vectors by pi/3 radians counterclockwise about the origin.
[3 -2 0 ] [-2 0 0 ] [1 0 2 ]
[1 2 3 ] [2 4 6 ] [3 6 9 ]
[0 0 1] [0 1 0] [1 0 0]
The 3 by 3 matrix that projects onto the plane x - 2 y - z = 0.
The 3 by 3 matrix that reflects through the plane x - 2 y - z = 0.
The 3 by 3 matrix that rotates about the line through the origin parallel to
[1] [1] [2]
by angle pi/4 (counterclockwise as seen with the vector pointing at you).
The 3 by 3 matrix that rotates about the line through the origin parallel to
[ 1] [ 1] [ 2]
and which sends the vector
[ 1] [-1] [ 0]
to the vector
[ 0] [ 2] [-1]

[4 3 8] [9 5 1] [2 7 6] Hint: This matrix is a magic square, which means that all rows and columns (and even diagonals) sum to the same thing. This tells us something about an eigenvector.

Use diagonalization to calculate the sixth powers of the matrices in the previous exercise, when possible.
Use diagonalization to calculate the inverses of the matrices in the previous exercise, when possible.
Use eigenvectors to find the general solutions of the following systems of differential equations:

x' = 3 x + y y' = 2 x + 4 y
x' = 7 x + 5 y y' = 5 x + 7 y
x' = 3 x - 2 y y' = - x z' = x + 2 z
x' = 5 x - 2 y + z y' = - 2 x + 2 y + 2 z z' = x + 2 y + 5 z

Extra Credit Possibilities

Produce a creative model, with a beautiful and literate exposition, for the ant problem, and discuss the solutions. Deadline: 19 March
Beginning with Prof. Herod's Maple worksheet, " Area by Chance," find some creative calculations of integrals using the Monte Carlo method. As usual, the exposition must be beautiful and literate. Implement the calculations with Maple, Mathematica, Java, Excel, or other software of your choice, approved in advance by Prof. Harrell. EXCEPT that whatever you produce should run on all platforms (Linux, Mac, Solaris, and Windows) without any installations. It should be absolutely user-friendly and operate within 20 seconds of launching. It will not be considered as submitted unless this is the case. Deadline: 19 March.

Link to:

The 1502 home page
Evans Harrell's home page
The School of Math on-line resources

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