For Fri, Sep 24:
- turn in 17.1 - 1,2,9,10,11,13,14 and 17.2 - 1,2
- Also, do and turn in the following: If C1 and C2 are real
constants, verify that the function
is a solution of the Hooke's Law spring equation
Show that this solution is actually a sinusoidal
for some constants A (the amplitude)
(the phase). What are A and
in terms of C1 and
C2? What is the geometric interpretation of A and ? Draw a
careful sketch in the case when C1 = C2 = 1, and do the addition
graphically to plot the graph of
Does the sketch agree with your analytic determination
of the amplitude and phase in this case?
- read 7.9 (only separable equations) and 17.2.
For Mon, Sep 27:
- turn in 17.1 - 15,17 and 17.2 - 3,6,7,8,14,21 and 6.1 - 2
and 6.2 - 1.
- Do and turn in the following problem: Newton's law of cooling
states that, if a hot object of temperature
T is exposed to an ambient environment of constant cooler temperature
A, then the rate of change of temperature of the object is proportional
to the difference between the temperature of the object and the
temperature of the ambient environment.
- Express this relationship as a differential equation. Does it
matter whether you use the centigrade or the Kelvin temperature scale?
Why or why not?
- A cross-country skier takes a pot of boiling tea (100 degrees C)
outside from the lodge. Five minutes after she leaves the lodge, the tea
has cooled to 50 degrees C. Ten minutes after her departure from the
lodge, the tea has cooled to 20 degrees C. How cold is it outside?
- read 6.1. Skim 6.2 and 6.3.
For Wed, Sep 29:
- turn in 6.1 - 3,7,8,12 and 6.2 6,8,13 and 17.4 - 1,2
- read 7.9 and 17.4
For Fri, Oct 1:
- turn in 6.1 - 20,23,30, 6.2 - 28,44, 6.3 - 4,11, 17.4 - 3,4,7,8,
17.6 - 1, Appendix I - 1,5
- read 17.6, Appendix I, and the handout on the complex exponential
For Mon, Oct 4:
- turn in 17.6 - 2,3,4, Appendix I - 15,23,31,32,40,53, 3.7 - 2
- read 3.7 and 17.7
For Wed, Oct 6:
- turn in 3.7 - 1,14,18,19,20,21,22,28, 7.1 - 2
- read 3.7 and 17.7 again. Also read 7.1
For Fri, Oct 8:
- turn in 7.1 3,5,6,9,13. Also:
- Prove that an overdamped or
critically damped freely vibrating spring passes through the equilibrium
position (the ``relaxed" natural rest position the spring would assume if
it were not vibrating) at most once.
- Consider an oscillatory motion of frequency
Show that x(t) can also be
for some complex constant A.
(HINT: recall an analagous problem from the first homework handout.)
- read 7.1 again.
For Mon, Oct 11
- turn in 7.1 - 16,17,19,21 and 4.8 - 1,2 and 9.2 - 2
- read sections 4.8 and 9.2
For Wed, Oct 13
- turn in 4.8 - 7,8,13,17,21,27 and 9.2 - 1,4,21 and 9.7 - 2 and 9.8 -
An object of mass m is thrown upward with initial velocity v0; the
gravitational acceleration is g (directed downward), and there is a
retarding force (due to air resistance) which is proportional to the
speed. In terms of the constants in the problem (the mass, the
gravitational acceleration, the initial velocity, and the constant of
proportionality describing the air resistance), determine the ascent time
(the time until the object reaches its highest point and begins its
- read section 9.3, beginning with the Ratio Test (p. 542) to the end
of the section, section 9.5 through the Radius of Convergence topic (skim
the rest), sections 9.7 and 9.8. Note: this appears to be a longer
reading assignment than it actually is, since the topics we wish to cover
are scattered about in the book. To help you focus on the important
points, note that we are mainly concerned with the application of the
Ratio Test to determining the radius of convergence of a power series, and
with the approximation of functions by Taylor polynomials. We will
combine these two ideas in the next reading assignment.
For Fri, Oct 15
- turn in 9.7 - 5,6,10 and 9.8 - 3,4,12,15 and 9.6 - 1
- read sections 9.6 and 9.9
For Mon, Oct 18
- turn in 9.6 - 6,7,9,16,26,27,31,33,34. Note: in the exercises for
section 9.6, try to find the Taylor series from known series
(sin, cos, exp, the geometric series, etc.) by using the operations on
power series we've discussed, not by calculating the coefficients
- read sections 10.1 and 10.2
For Wed, Oct 20
there will be no homework to be turned in.
For Fri, Oct 22
- read section 10.2 again and read sections 10.3 and 10.4.
- turn in 10.1 - 4,9,17,26,31,40 and 10.2 - 3,13,16,22,24
10.3 - 1.
- read sections 10.3 and 10.4 again.
For Mon, Oct 25
- turn in 10.2 - 26,30,31
and 10.3 - 3,4,6,14,15,21,22 and 10.4 - 15.
- read section 10.4 again.
For Wed, Oct 27
- turn in 10.3 - 23,25
and 10.4 - 2,4,5,14, and the following problem on determinants:
Note the set of row vectors in the form
form a n-dimensional vector space when addition is given
by addition of the components and cv means multiplying each component
by c. It is also useful to let
where the 1 is in the ith component, and note
Similarly, the vector space of n x n matrixes is given by all
vectors in the form
The determinant, det, can be viewed as a function on matrixes
satisfying the following four properties
If you are confused by the notation describing property one, it is
saying if you interchange neighboring rows in the matrix then
the determinant's value changes sign. Note that by property 1, properties
2 and 3
hold for any row.
It is a fact that the determinannt exist for any n and is uniquely
determined by these four properties. In the book you are given the
formulas for the 2 x 2 and 3 x 3 determinats,
- Check that the above formula for the 2 x 2
determinant satisfies properties 1-4.
- Check that the above formula for the 3 x 3
determinant satisfies properties 1-4.
(Hint: you may want to look at exercise 10.3 - 18 which is immediately
true due to the volume interpretation of the scalar triple product).
- read section 10.4 again and read section 10.6.
for Thurs, Oct 28
- turn in 10.4 - 10,11,16,17,18,21,25,27.
- read section 10.6.
for Mon, Nov 1
- turn in 10.4 - 26,28,29,30 from the textbook and 2.2 - 1,2,4,5 and 2.3 - 1,2 from the
- read section 2.3 from the handout.
for Wed, Nov 3
- turn in 2.3 - 3,4,5,6,7,9,11,12 and 3.1 - 1,2,3 from the handout.
- read section 3.1 from the handout.
for Fri, Nov 5
- turn in 2.3 - 8,10,13 and 3.1 - 4,5,6,7,9 and 4.1 - 1 from the handout.
- read section 4.1 from the handout.
for Mon, Nov 8
- turn in 3.1 - 8,10,11,12,13 and 4.1 - 2,3,4,5,6,7 and 4.2 - 1,2,3 from the handout.
- Study for exam.
for Wed, Nov 10
Study for the exam. Do the practice exam, you will not have to turn this
Here a few corrections.
- Number 4: the second plane should be
A2 x + B2 y + C2 z + D2 =0,
A1 x + B1 y + C1 z + D1 =0.
- Number 8: the point D should be D = (1,-1), not (-1,1).
- Number 6: first note that by the set of cubic polynomials
it is meant all polynomilas in the form
a3 x3 + a2 x2 + a1
x + a0 where the ai are real numbers. ( i.e. we are
allowing that any of the ai are zero. So for example, for the purposes
of this problem a quadradic polynomial is a degenerate example of
a cubic polynomia.)
Secondly, to do the the problem you are
required to chose a range of T there are
You may fail to be able to do part (c) if you don't make a finite
dimensional choice, so this you should do.
for Fri, Nov 12
- turn in 12.1 - 2,22,25,27,28,29,30 and 12.2 - 4,8,14 and 12.3 - 3
- Read 12.1, 12.2 (if you haven't already), and 12.3 and 12.4
for Mon, Nov 15
- turn in 12.3 - 7,8,13,15,19,23,30 and 12.4 - 3,5,11,12,17,18
- Read the handout.
for Wed, Nov 17
- turn in 12.5 - 2,3,6,8, (see Note) 16,18,19,24,35 (oops I
meant 34 so your off the hook dont do either) and 12.6 -
14,15,16. Note: You are required to do problems 2,3,6,8 from the
linear transformation point of view to receive full credit. In other
words a complete solution will include how to
view the function
being examined as a composition of mappings,
differentiating these mappings, and
multiplying out the corresponding derivative matrices in order to find
the needed derivative. By two ways you should use the matirx methed as
as constructing the needed function via compostion and
- Read 12.7.
for Fri, Nov 19
Note to Prof. Webb's students: on Friday, you are required to turn
in the assignment listed above as being due on Wednesday - not the
assignment listed below.
- turn in 12.5 - 1,4,7,10,11,12 As above: You are required to do
these problems from the
linear transformation point of view to receive full credit.
12.5 - 34, and 12.7 - 4,5,9
- Read 13.1
for Mon, Nov 22
- turn in 12.7 - 11,12,17,18,21,22,23,26,28
and 13.1 - 1,3,4
- Read 13.1
for Mon, Nov 29
- turn in 13.1 - 5,6,9,11,14,15,21 and 13.2 - 1,2,4
- Read 13.2
for Wed, Dec 1
- turn in 13.2 - 3,6,7 and 13.3 - 1,3,4,6
- Read 13.3
for Sunday Dec 4
Try this prictice exam (it is of course not due.)
Math 9 Practice Final
- Find the Taylor series about x=0 of
- What is the radius of convergence of this series?
- Find an equation for the tangent plane to the graph of
f(x,y) = xy + y2 + e2x at (0,2,5).
- Find an equation for plane parallel to
this tangent plane
containing the point (1,1,0).
- Where does the line perpendicular to this new plane and containing
(1,1,0) hit the original tangent plane.
- Let f(x,y) = xy and
Describe the points where the level sets of f and g are
- Find the minimum and maximum values of
f(x,y) = (x + y)e-(x2 + y2)
on the disk
- Express f as a composite of a function
- Find Df and Dg (Df refers to the derivative of f).
- Use the chain rule to compute
- Find the kernel of
is a stretch of the plane by
in the direction of the
denote the stretch by
in the direction
determined by the line inclined at an angle
with respect to the x-axis. Find a formula for
- Let V be the vector space of polynomials of degree
let A denote the linear transformation of V to V given by
- Express A as a matrix.
- In each of the following cases describe the rank of A, and find a
non-zero polynomial in A's kernel whenever such an element exists.
and c = 0.
and b = 0 and c = 0.
answer to part (a) if we let x3 corrspond to the first standard basis
vector, x2 the second, x the third, and 1 the fourth the anser to a
- Find the general solution to
- Use the previous problem (problem 7(a)) to find a polynomial solution to
- Find a solution to
f(x) = a e2x + b ex
Math 9 Fall 1999