Math 142 (Calculus II)
Section 200, Spring 1998
1/12 - 1/15
- M: First day of class. Four handouts distributed - course info sheet,
student questionnaire, Calculus Excellence Workshop info sheet, my old
141 final exam. After general introduction, students worked in groups
on 141 final exam.
Homework - Work through 141 final exam by Wednesday. We will have
a quiz on these problems at the beginning of Wednesday's class.
- T: If you are new to this textbook, review sections 2.2, 2.3, 2.4,
3.1, 3.3, 3.4, 6.3.
Students continued working in groups on old final exam.
Homework - Prepare for tomorrow's quiz which will be taken directly
from the old final exam. Read 6.1 - 6.3.
- W: I answered questions from old final exam. Students took quiz 1.
I did mini-review about the definition of a definite integral as a
limit of Riemann sums, the Fundamental Theorem of Calculus, and the
derivative rules that we know.
Homework - Read 6.1-6.4. Do #1,3,5,10 from 6.1 and #1,2,3,4,7,9,
12,15,16 from 6.3.
- Th: Tomorrow is the last day to change a course schedule or drop a
course without a grade of "W" being recorded.
We returned quizzes (almost all 9's and 10's!). I went over problems from
6.3 and started 6.4. We came up with rules for the indefinite integral
of sin(x), cos(x), ln(x), e^x, a^x, C, 0, x^n, and sec^2(x). We had
to do the "guess and check" method for a^x. We then used this
same approach for harder integrals involving the chain rule.
Homework - Look back at #16 from 6.3. Do #4,6,10,13,16,24,27,29,30,31,
36,41,45,49,50,52,53 from 6.4.
For Monday, turn in #12 from 6.3, #4,#27 from 6.4, and do the following
assignment: How fast can you throw a baseball upwards? In class,
we had guesses of 65 MPH, 50 MPH, 30 MPH, and 15 MPH. Calculate
how high the ball would actually go in each of these cases and then
decide whether these are reasonable guesses.
1/19 - 1/22
- M: I did lots of problems from 6.4 but still need to do some problems
with position, velocity, and acceleration.
Homework - Finish all the assigned problems from 6.4 and go to the
Math Lab if you are having difficulty.
- T: We will have a quiz on Thursday covering 6.3, 6.4, and 7.1. I answered questions from 6.4. I wrote down list of integrals which students must
know. Namely, the integral of x^n, 1/x, e^x, cos(x), sin(x), a^x,
sec^2(x), and 1/(1+x^2), as well as the integral of f(x)+g(x), f(x)-g(x),
and cf(x). Students then worked in groups on the following homework.
Homework - Read 7.1. Do #4 from 6.5, #8,42 from Chap 6 Review, #9,14,16,
23,27,30,36,43,46,49,51,54,55 from 7.1.
- W: Students presented problems at board. Long quiz tomorrow on 6.3, 6.4,
7.1, and all the integration rules we learned yesterday.
- Th: We learned to use substitution in order to compute integrals. A long
quiz was given during the last 35 minutes.
Homework - Read 7.2. Do #12,20,21,23,24,25,28,29,30,35,36,40 from 7.2.
1/26 - 1/29
- M: I worked through quiz and answered questions from 7.2.
Homework - Read 7.3. Do #2,3,5,9,10,12,14,17,18,19,21,29,30.
For tomorrow, turn in #20,30,36 from 7.2.
- T: I passed back homework #1 and discussed baseball problem. I answered
questions from 7.3 and discussed integrals of sin(x), cos(x), tan(x),
cot(x), sec(x), and csc(x). I discussed "integration by parts" from
Homework - Try the substitution u = e^(x^(1/3)) in #3 from 7.3.
Read 7.4. Do #3-28 odd from 7.4. For tomorrow, turn in #4 and #30
- W: How do you know which method of integration to try first?
I usually try things in the following order.
In class, we did definite integrals of (sin(x))^3, (x^3)ln(x), and
arctan(x). Exam will be one week from tomorrow. Tomorrow's quiz will
cover 7.2, 7.3, and 7.4.
- Refer to your list of memorized integrals.
- Try rewriting the integrand.
- Try substitution.
- Try integration by parts.
Homework - Do #40,41,42,43,45,46 from 7.4.
- Th: When using integration by parts, the mneumonic device "ILATE" can
often help. We did problems which involved using integration by parts
multiple times. Tabular integration can sometimes make this process
easier. Neither of these approaches is foolproof so it is extremely
important to understand integration by parts and when it can help.
Quiz #3 is given at end of class.
Homework - Finish problems from 7.4 and evaluate integrals of x*sqrt(x-1),
(x^2+1)/sqrt(x+3), e^sqrt(x), (1-x)/(3+x), and 1/(4+x^2). Bonus points
will be given to those who present correct solutions at the board on
Monday. Read 7.5 and do #5,11,17,23,29,35,41,47 from 7.5.
2/2 - 2/5
- M: Pavel has eliminated his Thursday office hours but added office hours
on Friday from 10:30 am - noon. Thursday's exam will cover up to
and including section 7.5.
To integrate x*sqrt(x-1), we set w=x-1. To integrate
(x^2+1)/sqrt(x+3), we set w=x+3. To integrate e^sqrt(x), we
set w=sqrt(x) and after the substitution we do integration by
parts. To integrate (1-x)/(3+x), we set w=3+x. The substitution
used in these problems is a little different from the other
substitutions we have done. It is not the only way to solve
these problems but it is a very useful technique. We discussed
using polynomial division and completing the square to rewrite
the integrand in a form which is easier to integrate.
Homework - From 7.5 do #64,65(word problems), #45,46,50(polynomial
division), #44,46,47(completing the square), #37,38,42,45,48,49,50
(partial fractions), and #10,12,24,34(powers of sine and cosine).
- T: Thursdays Exam will cover derivative rules, 6.1-6.4, 7.1-7.5.
Know the derivative of sin(x),cos(x),x^n,tan(x),arctan(x),ln(x),f(x)+g(x),
cf(x),f(x)g(x),f(x)/g(x),f(g(x)). Know integrals of x^n (including
n=-1),sin(x),cos(x),sec^2(x),1/(1+x^2),f(x)+g(x),cf(x). Know integration
by substitution, integration by parts. To simplify your integrand be
prepared to rewrite the integrand by doing basic algebra or completing the
square or doing long division of polynomials. Be able to use the graph
of f'(x) to draw a graph of f(x). Handle problems involving position,
velocity, and acceleration. Compute areas between curves and average
values of functions. Know the fundamental theorem of calculus.
Be able to use the table of integrals from 7.5.
- W: Review for exam. There will be two help sessions tonight in LC412.
Mrs. O'Leary will be there from 6-8 pm and I will be there from 8-10 pm.
- Th: Exam 1.
2/9 - 2/12
- M: The tables in 7.5 are not necessary for computing many integrals as
long as you know a few techniques. The methods of substitution and
integration by parts are extremely important techniques which we have
already discussed at length. We may also rewrite the integrand by
polynomial division, completing the square, or using partial fractions.
We rewrote the integrand to compute the integral of 5/(x^2-4x+7)dx by
completing the square and doing a substitution to make the integrand
1/(1+w^2)dw. We rewrote the integrand to compute the integral of
1/(x^2-4x+3) by the method of partial fractions. How does a calculator
approximate a definite integral so quickly? We've used Riemann sums
for approximating definite integrals but there are more efficient
ways to do this. Sections 7.6 and 7.7 discuss two other approaches
referred to as the Trapezoidal Rule and Simpson's Rule.
Homework - Compute the integral of 3/(x^2+2x+6) and the integral
of (6x-32)/(x^2-9x+14). Read 7.6 and 7.7. There will be a quiz
tomorrow based solely on the reading from 7.6 and 7.7. Be sure to
read the summary at the end of section 7.7.
- T: I went over homework and discussed more general methods of partial
fractions. We discussed how to handle the case where the denominator
has distinct linear factors, has distinct quadratic factors, or has
repeated factors. This won't be stressed that much but you should be
aware of the ways to handle these cases. I introduced 7.8 by computing
the definite integral of 1/x^2 from 2 to 5, then from 2 to 100, then more
generally from 2 to b. This led to the notion of integrating from 2
to infinity. Quiz 4 was given at the end of class.
Homework - Read 7.8. To turn in tomorrow - Find the integral of
(19-x)/(x^2-3x-10)dx and the integral of 7/(x^2-6x+13)dx. Show all
your work and do not use the tables in 7.5 for this.
- W: I passed back quiz 4 and exam 1 and then went over solutions to the
exam. The exam was worth 150 regular points plus 8 bonus points.
The highest exam scores were 147, 141, and 135. These are very good
scores but the scores were mostly very low. In fact, the average was
only 88.4. Much of this was due to insufficient preparation but part of
it was due to my writing an exam which was too long. For this reason, I
have added 13 points to each student's exam score. This puts the top score
at 160 out of 150. For those whose adjusted scores are still less than
90, this puts you in the 'F' range. You may want to consider dropping
the course. Please see me if you wish to discuss your progress in more
- Th: I went over the quiz on Trapezoidal Rule and Simpson's Rule.
A couple of days ago we discussed the integral of 1/x^2 from 2 to
infinity. Today we blindly computed the definite integral of 1/x^2
from -2 to 3 and got a negative number - why does this not make sense?
We looked at the picture and knew that a negative answer was impossible.
One student remembered that there was a problem with applying the
Fundamental Theorem of Calculus when the function had a discontinuity.
I then discussed improper integrals but only dealt with those where
one of the limits of integration was infinity. We looked at the definite
integral of 1/x^p from 1 to infinity for various values of p and discussed
the comparison techniques of section 7.9. If we found convergence
when the integrand was 1/x^3. What does that say about the integral
of 1/(x^3+4) or 1/(x^3-4) from 1 to infinity? I want you to be able
to look at an integrand like (x+4)/(x^3+4x-2) and immediately see
that the integral from 1 to infinity will converge. This is because
the integrand is defined for x>=1 and for large values of x,
the integrand grows roughly like x/x^3 = 1/x^2 and we know that the
integral of 1/x^2 from 1 to infinity converges.
Homework - Read 7.8 and 7.9. Do #1,2,3,7,10,17,20,29,30 from 7.8
and #1,2,3,4,5,6 from 7.9.
2/16 - 2/19
- M: We discussed the 3 useful integrals for comparison given on page 403.
To determine convergence or divergence of an improper integral, we
can sometimes compare the integral to these three. The integral of
(2x^2+3x+4)/(5x^5+10x+1) from 4 to infinity converges since the integrand
is defined for all values of x>=4 and it behaves like (2x^2)/(5x^5) =
(2/5)(1/x^3) for large values of x. It follows upon comparison to our
3 useful integrals that our original integral convergences. Sometimes
we can simply look at the graph of the integrand to determine convergence
or divergence. The integral of e^(-2x) from -infinity to 0 clearly
diverges since the area under the curve is infinite. If the picture
doesn't seem to help or we have trouble comparing to some of the basic
integrals, we can hope to find the answer by using limits. In fact, you
should try to determine convergence or divergence in as many ways as
possible to get a better feel for what is going on. Whenever an integral
converges, it would be nice to know what number it converges to. For
easy integrals, just use limits to find the exact value of the integral.
If you can't determine an antiderivative, then try comparing the
integrand to known functions which are easier to integrate and bound
the value of your convergent integral between two other values.
For example, since the sine function oscillates between -1 and +1, we see
that (sin(x^2)+4)/(x^3) is between 3/(x^3) and 5/(x^3) for positive values
of x. Thus the integral of (sin(x^2)+4)/(x^3) from 4 to infinity converges
to a number between the integral of 3/(x^3) from 4 to infinity and the
integral of 5/(x^3) from 4 to infinity. Using limits, we compute these
to be 3/32 and 5/32. So our integral converges to some number between
3/32 and 5/32. We will have a quiz tomorrow on 7.8 and 7.9.
Homework - Do #4,8,9,14,15,16,24,25 from 7.8 and #15,16 from 7.9.
- T: I answered questions from 7.8 and 7.9 and then gave quiz #5.
I then began discussing chapter 8. Given that v(t) is the velocity of an
object at time t, find the total distance traveled between times t=a
and t=b. At this point we know that the answer is the definite integral
of v(t)dt from a to b. But thinking back to our original approach,
we divided the interval from a to b into subintervals of equal width and
approximated the distance traveled on each subinterval by something of
the form v(t)*(delta t) (i.e., rate*time). We added up the approximate
distances traveled over each subinterval to get something like
TOTAL DISTANCE is approximately equal to v(t0)*(delta t) +
v(t1)*(delta t) + v(t2)*(delta t) + ... +
This is just a Riemann sum for v(t). We then defined the definite
integral of v(t)dt from a to b as being the limit of these Riemann
sums. This is why we can now say immediately that TOTAL DISTANCE =
the definite integral of v(t)dt from a to b. All of chapter 8 deals
with first approximating some quantity by a Riemann sum for some
function. Figuring out what function to do a Riemann sum for is the
hard part and certainly the most important part. Once you know which
function to do a Riemann sum for, you can write that the exact answer
is the appropriate definite integral of this function. If you know an
antiderivative of this function, you can now evaluate this integral
exactly by using the Fundamental Theorem of Calculus. If you don't
know an antiderivative, you can approximate the value of the definite
integral in a number of different ways (calculator, Riemann sums,
Trapezoidal Rule, Simpson's Rule, etc.) I then gave the following problem
but did not have time to discuss it:
The air density (in kg/m^3) h meters above the earth's surface is given
by the function g(h). Find the mass of a square-based column of air
30 kilometers high given that the square base has sides of length
Homework - Read 8.1 very carefully being sure to understand each
example. Do #3,4,5,7,8 from 8.1. We will have some homework from
8.1 and 8.2 due on Thursday.
- W: I discussed yesterday's air density problem then had students work
in groups on the homework from 8.1.
Homework - Read 8.2. Do #4,6,10,12,14,15,16 from 8.2.
- Th: Tomorrow is the last day to drop a course or withdraw without a
grade of "WF" being recorded. You may want to consider dropping if you
are currently failing the class.
I passed out my record of all quiz, homework, and exam scores and
discussed grades. Quiz #5 was returned and discussed. I then worked
#8 from 8.1 at the board. I also derived the definite integral
which represents the length of a curve. At the very end of class, I asked
how to compute the volume of a very strange looking sweet potato. A
couple of students said to put it in a container of water to see how
much water it displaced. Another said to cut it into slices so I
did that in class. We saw that the cross-sections looked liked
circles for which we know how to compute area, so we approximate the
volume of a slice as being the cross-sectional area multiplied by the
width of the slice. We add up the volumes of each slice to approximate
the volume of the whole sweet potato. This is the key idea of section
8.2. Unfortunately we had no time to do any examples.
Homework - Read 8.3. Do #9 from 8.1, #18,19,21 from 8.2. On Monday,
turn in #4 from 8.1 and #12,#21 from 8.2.
2/23 - 2/26
- M: I collected and discussed the homework. I also went over #10 from
8.2. We will have a quiz on Wednesday covering 8.1, 8.2, and 8.3.
We will have our second exam one week from this coming Thursday.
Homework - Redo #12 from 8.2 using semicircles for the cross-sections
and then equilateral triangles for the cross-sections. Also do
#3,5,6,7,11,12 from 8.3.
- T: I discussed #3,5 from 8.3, and #16,12(using equilateral triangle
cross-sections) from 8.2. Students gave me their heights in inches
which we will use when discussing 8.5 and 8.6. Group projects were
passed out and students had 10 minutes to find group members and
look through the questions on the group project. Tomorrow's quiz
will cover 8.1, 8.2, and problems from 8.3 involving WORK. The exam
next week will cover 7.6,7.7,7.8,7.9,8.1,8.2,8.3,8.5,8.6, as well as
the ability to compute derivatives and integrals successfully.
- W: I did #6,7,11 from 8.3. These took a long time so I decided to
make the quiz a take-home quiz (resulted in thunderous applause!)
- Th: I introduced 8.5 and 8.6 with the example on age distribution
found in the text. We then used the density function p(x)=.4*e^(-.4x)
for x>=0 to model the distribution of the measurements of earthquakes
on the Richter scale. Now what is the probability that a random
earthquake will measure between 2 and 4 on the Richter scale?
less than 5? more than 5? I then solved #2 from the quiz. Students
had about 20 minutes at the end of class to work on group projects.
The group projects are difficult and students should have already
started working on them. If you have questions about it, I will
have extra office hours tomorrow from 9 am until noon in LC101.
Homework - Read 8.5, 8.6. Do #4,5,6,10 from 8.5 and #3,5,6,7,8 from
3/2 - 3/5
- M: Students worked on group projects.
- T: I passed out practice problems for exam 2 and solved #6 and #7 which
may help for the group project. I also did #6 from 8.5 and talked a little
about probability. I will have a help session tomorrow night from 7:30 pm
until 9:00 pm in LeConte 412.
Exam 2 will cover sections 7.6-7.9, 8.1-8.3, 8.5-8.6. You should look
over old homeworks and quizzes as well as the practice problems distributed
today. For the points on this exam, I expect to approximately have 30%
from sections 7.6 - 7.9, 50% from sections 8.1 - 8.3, and 20% from
sections 8.5 - 8.6. You will need to be able to do integration by
substitution and integration by parts. For this exam, I will not
have any questions involving completing the square or partial fractions.
For 7.6-7.7, you need to know how to approximate definite integrals using
left, right, and midpoint Riemann sums as well as the Trapezoidal and
Simpson's Rule. You should know what areas these represent in a picture
and thereby determine whether they are underestimates or overestimates
for the definite integral. You should also know which methods are
generally the best approximation techniques.
For 7.8-7.9, you should be able to determine whether a given improper
integral converges or diverges. This can be done by working out the
integral or by comparing the integrand to 1/x^p or e^(-ax) as on
page 403. If the integral does converge, then you will be expected to
compute the value of the integral.
For 8.1-8.3, you will be expected to know how to approach a new problem
for which you need to calculate some quantity. This can be done in the
(1) Divide the problem into small "slices" for which you can approximate
the contribution that each slice makes.
(2) Add up the contributions for the individual slices to get an
approximation for the quantity in question.
(3) Take the limit as the width of the slices approaches 0 (i.e. take a
definite integral which will give the exact value of the quantity in
For some problems you will probably need to go through the thought process
embedded in parts (1) and (2). In fact, I may ask you to explain how
you are solving the problem in this way. For other problems you will be
able to jump immediately to the required definite integral in part (3).
For example, I would suggest that you be able to quickly set up the
required definite integrals to compute area, volume, or length.
For 8.3, the only formula that I expect you to memorize is
Work = Force * Distance.
For 8.5-8.6, we want to understand how some quantity is distributed
throughout a population. This is done with probability density functions
and cumulative distribution functions. You should know the definition of
these two functions, be able to draw the graph of one given a graph of
the other, and answer questions such as "what percentage of the population
exhibits a certain property?", "what is the probability that a certain
property is exhibited?", etc. Given a formula for the probability
density function, you should be able to compute a formula for the
corresponding cumulative distribution function and vice-versa. You
should also be able to compute the mean and median given formulas for
either of these two functions.
- W: Help session tonight from 7:30 pm - 9:00 pm in LeConte 412.
- Th: Exam 2.
3/9 - 3/12
- M: Spring Break - No Class.
- T: Spring Break - No Class.
- W: Spring Break - No Class.
- Th: Spring Break - No Class.
3/16 - 3/19
- M: I returned and discussed most of the exam. We will have some homework
from chapter 9 due Wednesday. We will have two quizzes on Thursday -
one on chapter 9 material and one on techniques of integration.
Homework - Read 9.1 and do #1,2,3,4,5,9 from 9.1.
- T: I finished going over the exam and then introduced chapter 9.
I stated what a differential equation was and asked the students to
find algebraic solutions to dy/dx = 2, dy/dx - x = 0, dy/dx - y = 0, and
dP/dt = P(1-P). The students, of course, could find general solutions to
the first two. After one incorrect answer to the third, many of the
students were able to find a particular solution of y = e^x. When I
asked for a more general solution, one person said e^x + C which we
checked and found did NOT work. Another student suggested y = Ce^x
which we found to work. Are there other solutions? We will discuss
later how we know if we have found the most general solution. For
dP/dt = P(1-P), nobody came up with a solution. So I gave a solution of
P = 1/(1+Ce^(-t)) which we verified. We will learn later in this
chapter how we can sometimes find algebraic solutions. Those taking
Math 242 will learn many techniques for finding algebraic solutions of
differential equations. I talked a little about why we wish to model
real world situations and how this usually leads to differential equations.
In particular, I mentioned the population of two species, as well as
the spread of contaminants in the groundwater. I then gave the following
problem: A yam is placed inside a 200° oven. The yam gets hotter at
a rate proportional to the difference between its temperature and the
oven's temperature. When the yam is 120°, it is getting hotter at
a rate of 2° per minute. Write a differential equation that models
the temperature, Y, of the yam as a function of time t. We discussed how
to translate this paragraph into the differential equation dY/dt =
.025(200-Y). I then gave the solution as Y = 200-Ke^(-.025t). We
discussed why we should expect this constant K to show up. I then gave
the added information that the temperature of the yam started out at
20°. We used this to find that K = 180. I mentioned what an initial
value problem was and gave dy/dx = e^(.01x^2), y(0)=2 as an example. Since
we can not write a simple formula for y, I started to discuss graphical
solutions to a differential equation. We got a graph out and also started
Euler's Method for numerical solutions but it was too much to cover
in a short time.
Homework - Read 9.2 and 9.3. Do #12 from 9.1, #3 from 9.2, and #5 from
9.3. Turn in #12 from 9.1 tomorrow.
- W: Group projects will not be returned until Monday. There will be a
quiz tomorrow on 9.1 - 9.3. For the techniques of integration (and
differentiation), I've decided to make it a homework which I will
pass out tomorrow. Today we compared the different methods of handling
the initial value problem y'=2x and y(0)=1. We drew the slope field
for y'=2x, we used Euler's Method, and we found the exact solution
y=x^2+1. We often cannot find an exact solution so the first two
methods are very imporant techniques. We also drew slope fields for
y'=3(4-y) and y'=1+y^2. From these slope fields we could determine
the end behavior of the various solution curves (i.e. what happens
as x approaches infinity.) This may often depend on your starting
Homework - Do #6-14 in 9.2 and #2,3,6 from 9.3.
- Th: I wrote 6 differential equations on the board asking which ones
were the easiest to solve. They were (1) dy/dx = x, (2) dy/dx = y,
(3) dy/dx = x-4, (4) dy/dx = y-4, (5) dy/dx = x+y, (6) dy/dx = xy+y.
Students picked (1) and (3) where the derivatives were written in
terms of the independent variable. These were easy since we only need
to antidifferentiate simple functions. (2) was chosen as the next easiest
because we've seen before when a function equals its derivative. For
(2) we get the solution y=Ce^x, where C is an arbitrary constant. Someone
then guessed that y=x+Ce^x was a solution to (5). We checked this
and found that dy/dx=1+Ce^x while x+y=x+(x+Ce^x). These are not the
same so we have not found a solution to the differential equation.
Instead of guessing anymore for this one, we looked at (4) and stepped
through the process of "separation of variables". I then asked them to
try separating variables for (5) and (6). They couldn't do it for (5)
so I explained that we can still understand what's going on with this
differential equation graphically using slope fields, and numerically
using Euler's method (if we had a starting value). We will not learn
how to handle (5) exactly in this course but those going into Math 242
will learn a technique for solving it. Students were successful in
separating variables for (6) but we didn't go through the next
step of integrating both sides. I wanted to leave plenty of time for
the quiz. I then answered questions from 9.1-9.3 before giving the quiz.
Homework - Read 9.4. Do #2,3,6 from 9.3 and #1,2,6,10,11,12,14,16 from
9.4. For Monday, turn in homework #7 (the green sheet) which was passed
out during class today.
3/23 - 3/26
- M: I passed back group projects and then answered questions from 9.4.
I also gave one example from section 9.5. We will have a quiz tomorrow
on problems from 9.4 and the reading of 9.5.
Homework - Do #20,22,26 from 9.4. Read 9.5.
- T: I passed back sample group project solutions taken from those submitted
by the students. I gave quiz #8 and then worked through the separation
of variables problem. We then discussed #10 from 9.5 and #1 from 9.6.
Homework - Read 9.6. Do #6,7,10,12,14,16,18 from 9.5 and #1,2,5,8,11,13,
16 from 9.6. For tomorrow, turn in #12 from 9.5 and #5 from 9.6.
- W: The plan for the rest of the semester is to finish 9.7 by Monday and
jump right into chapter 10 at that time. If we have time at the end of
the semester, we will return to the remaining sections of chapter 9.
Today I answered a few questions from 9.5 and let them work on 9.5/9.6
problems at the end of class. We will have a quiz tomorrow similar to
the problems in 9.5. You should know what compounded continuously means
as well as knowing Newton's Law of Cooling. If you get stuck on problems
in 9.6, it may help to think about the units involved (i.e. ft/sec, lbs
per cubic ft, etc). It will also help to use the fact that rate =
rate in - rate out.
Homework - Read 9.7. Finish all of the problems in 9.5 and most of the
problems in 9.6 for tomorrow.
- Th: I answered questions from 9.5. I did a Newton's Law of Cooling
problem where I wrote something like dT/dt=k(68-T). They were bothered
by my doing this instead of dT/dt=k(T-68) or perhaps dT/dt=-k(T-68).
I don't remember exactly. All of these possibilities above are showing
the same thing: "the rate of change of the temperature of an object is
proportional to the difference between its temperature and that of the
environment". They seem to be memorizing some differential equation and
if what I write deviates even slightly from what they memorized, then
they are confused. I also mentioned for another problem (determining
time of murder) that they could let time t=0 at 9 AM so that t=1 at 10 AM,
or let time t=0 at 10 AM so that t=-1 at 9:00 AM. More students accepted
this but some were still bothered by it. I then briefly discussed 3
kinds of growth: linear, exponential, and logistic. In chapter 1, we
learned easy ways to recognize linear growth vs. exponential growth from
a table of values and come up with a formula for each type. Logistic
growth is harder to recognize from a table. We should know the basic
shapes of graphs of functions exhibiting each type of growth. We should
also recognize differential equations for each type of growth.
If k and L are constants, then dP/dt = k represents linear growth for P,
dP/dt = kP represents exponential growth for P, and dP/dt = kP(1-P/L)
represents logistic growth for P. In each case we can find a solution
for P by doing separation of variables. Of course, we know solutions
for linear and exponential growth already so this separation of variables
is really only necessary for logistic growth. I then gave the quiz.
Homework - Do #14 from 9.6 and #1,2,3,4,5,16 from 9.7. For Monday,
turn in #14 from 9.6 and #4 from 9.7. Finish all assigned problems from
9.1-9.7 by Monday. I will answer any questions from chapter 9 on Monday
and then we will begin chapter 10.
3/30 - 4/2
- M: We began class late since our room was locked. After collecting
the homework, we spent almost the entire class period discussing it.
Students are having difficulty writing down the correct differential
equation when the needed information is given in paragraph form.
Therefore we will spend tomorrow's class going over problems like those
in 9.5, 9.6, and 9.7 and then setting up the appropriate differential
equations. The reading of 10.1 will not be given for homework until
Homework - go back over 9.5, 9.6 and 9.7, trying to fully understand
the examples, and then reattempt the homework problems from those
- T: Passed out a sheet of practice problems from chapter 9. We spent all
but 5-10 minutes on setting up the differential equations (and initial
values) given in the first 7 word problems on this practice sheet.
None of these problems will be turned in, but students should work out
complete solutions to all of these practice problems before the next exam.
I introduced the idea of chapter 10 very quickly.
Homework - Read 10.1. Do #1,3,7,8,10,12,13,14,23,24,27,30,34 from 10.1.
- W: Given an arbitrary function f(x), we wish to find a polynomial,
Pn(x), of degree n which "looks like" f(x) near x=a. By "looks
like" we mean: f(a)=Pn(a), f'(a)=Pn'(a),
f''(a)=Pn''(a), f'''(a)=Pn'''(a), . . . ,
f(n)(a)=Pn(n)(a). Of course all these
derivatives of f have to exist at x=a. The polynomial which works is
called the Taylor Polynomial of degree n centered at x=a and it is
found to have the form given on page 597 of our text. We found Taylor
Polynomials for a couple of simple functions by hand. A student mentioned
that the calculator can do it as well. We found the command under
SYMBOLIC but saw that it could only find those Taylor polynomials which
were centered at x=0. It also had trouble with some functions -
the 7th degree polynomial for sqrt(x+1) was taking too long to be of
any real worth. I expect everyone to be able to find Taylor
Polynomials by hand! I then displayed a graph of e^x along with an
animation of its first 10 Taylor Polynomials centered at x=0. We saw
that the polynomials were getting closer and closer to e^x as long as
we stayed near x=0. The hope is that we can get a good approximation
of e^x for ANY x-value as long as we use a Taylor polynomial of large
enough degree. This turns out to be the case for the function e^x but
we will see that we run into difficulties for other functions if we
get too far away from the point at which we centered our polynomials.
Our Taylor Polynomials had some finite degree. A Taylor Series can
be thought of as a Taylor Polynomial of infinite degree. Finding
Taylor Polynomials and Taylor Series is a very mechanical process which
everyone will learn with practice. Next time we'll talk about what these
are actually good for.
Homework - Read 10.2. Do #6,8,9,12,15,19 from 10.2.
- Th: I passed out a sheet of practice problems for chapter 10.
From this sheet we did #1, #2a, #4, #5b, part of #6, #8b.
We also approximated sqrt(0.9) with the first couple of Taylor
polynomials for sqrt(x) centered at x=1. We also discussed the
interval of convergence. The Taylor series for 1/(1-x) was found
to be 1+x+x^2+x^3+... Let Pn=1+x+x^2+x^3+...+x^n. We found
that this is equal to (1-x^(n+1))/(1-x) and since this limit, as n
approaches infinity, exists and is equal to 1/(1-x) only for x between
-1 and 1, we
say that 1/(1-x)=1+x+x^2+x^3+... on the interval (-1,1). This is called
the interval of convergence. Finding the interval of convergence is a bit
more involved for other functions and we will not discuss methods for
determining this interval in general. I do, however, expect you to
remember the interval of convergence for the Taylor series of sin(x),
cos(x), e^x, 1/(1-x), and ln(1-x) all centered at x=0.
Homework - Finish the problems from 10.1, 10,2, and all the problems on
the "Practice Problems for Chapter 10" sheet. For Monday, turn in
#14,24 from 10.1; #8 from 10.2; #2d,5a,8c from the "Practice
Problems for Chapter 10" sheet.
4/6 - 4/9
- M: I answered a few questions from the last homework and discussed two
techniques found in 10.3 - finding Taylor series by antidifferentiation
and solving differential equations using Taylor series. We will have
a long quiz tomorrow on 10.1 - 10.3. You should also be able to quickly
find the Taylor series centered at 0 for e^x, sin(x), cos(x), 1/(1-x),
and ln(1-x). For each of these you should also know the interval of
convergence. The interval of convergence is (-infinity, infinity) for
e^x, sin(x), and cos(x); it is (-1,1) for 1/(1-x); it is [-1,1) for
ln(1-x). No calculators will be allowed on the quiz.
Homework - Read 10.3. Do #13,14,17,26,27,28,30 from 10.3.
- T: I answered questions from 10.2, 10.3, and the worksheet. I also
used Taylor series to solve the initial value problem dy/dx=x+y, y(0)=0.
We obtained y = (x^2)/2! + (x^3)/3! + (x^4)/4! + .... We recognized
this as the Taylor series for y = e^x - 1 - x. We then verified that
this was indeed a solution by checking that y(0)=0 and dy/dx=x+y.
I gave a 35 minute quiz at the end of class. We will have another
quiz this Thursday including topics from 10.4
Homework - Read 10.4.
- W: I went over the quiz and then talked about 10.4. In previous
sections we started with a function and found its Taylor series.
Can we also start with a Taylor series and decide what function
(written in closed form) has this series for its Taylor series?
We should recognize that 1+x+x^2/2!+x^3/3!+... is equal to e^x for
all x values. Thus we should hopefully recognize that
1-1+1/2!-1/3!+1/4!-1/5!+... is equal to 1/e since this is obtained
by plugging x=-1 into the series for e^x. In 10.4 we'll only deal with
recognizing one particular series - a geometric series. The Taylor
series centered at x=0 for a/(1-x) is a+ax+ax^2+ax^3+ax^4+... We call
this a geometric series.
We proved earlier (with a=1) that the finite series
a+ax+ax^2+ax^3+...+ax^n = (a-ax^(n+1))/(1-x), and by letting n approach
infinity, we found that the infinite series a+ax+ax^2+ax^3+... is equal
to a/(1-x) when |x|<1 but diverges otherwise. I then spent time in class
looking at examples and determining if the series is geometric. If it is
geometric then does the series diverge or does it converge (i.e. have an
exact finite value for its sum)? If it converges then what is its sum?
Homework - Do #1,2,3,5,6,9,11,14,15,17,18,19,23 from 10.4. For tomorrow
turn in #2,18 from 10.4.
- Th: I passed out group project #2 and solutions to the worksheet on
chapter 9 problems. We will have a quiz next Tuesday on 10.4. I used
sigma notation and wrote lots of series on the board.
Some of the series were finite and some were infinite. Some of
the series were geometric and some were not. I stressed that any
finite series converges - recall that 'converges' basically means that the
sum adds up to some specific finite value. I had examples like
the harmonic series 1+1/2+1/3+1/4+... which diverges (the sum here is
infinite) even though each term gets closer and closer to 0. I gave the
example of 1+1/4+1/9+1/16+1/25+... which happens to converge. Leonhard
Euler showed that the sum was (pi^2)/6. Neither of those two series
was geometric so the simple techniques learned in 10.4 could not help
us in determining convergence or divergence. I then gave examples like
1+2+(1/3)+(1/3^2)+(1/3^3)+(1/3^4)+... or the sum of (2^n+3^n)/(4^n) as
n goes from 1 to infinity. The first series was 1+2+(a convergent
geometric series) so the series converges to 1+2+(1/3)/(1-1/3) = 3.5.
For the second series, we summed the series from n=1 to n=k,
rearranged terms to get the sum of two different geometric series,
and then let k approach infinity to get the sum of the infinite series.
The infinite series converges since each ratio is less than 1 in absolute
value. I then gave lots of repeating decimals like 0.3333..., 0.77777...,
0.99999..., 0.23232323..., 0.314314314..., 0.2151515..., etc. which
we converted to series and evaluated. We got 3/9, 7/9, 9/9, 23/99,
314/999, (2/10)+(1/10)(15/99). Students noticed the pattern pretty
quickly for decimals that start repeating right after the decimal point.
Homework - Read 10.6. Do #27,28,29,33,34 from 10.4.
4/13 - 4/16
- M: Easter Holiday - No Class.
- T: Thursday's exam will cover 9.1-9.7, 10.1-10.4. Look at homeworks 6-11,
quizzes 7-11, the chapter 9 practice problem sheet, and the chapter 10
practice problem sheet. I will have an early help session tomorrow -
hours are not yet decided. Today I answered questions from 10.4 including
problems like finding the sum and interval of convergence for
3+9(x-2)+27(x-2)^2+81(x-2)^3+... I then gave the quiz.
Homework - prepare for exam.
- W: There will be help sessions today from 4:30 pm - 7:30 pm in Gambrell
258. I will be there from 4:30 pm - 6:00 pm and Mrs. O'Leary will be
there from 6:00 pm - 7:30 pm. Tomorrow's exam will cover 9.1-9.7 and
10.1-10.4. None of the financial applications (present value, future
value, etc.) will be included on this exam.
- Th: Exam 3.
4/20 - 4/23
- M: I went over exam. Students will be allowed to ask questions about the
group project until Wednesday. There will be a quiz tomorrow based upon
the problems in appendix C.
Homework - Read appendix C and work the problems from that section.
- T: I distributed two handouts on polar coordinates and discussed the
basics. Students then did a group quiz - it was #1b,2b,3b,6b, and 8b from
the polar coordinates problem sheet. I briefly described how to graph
polar equations on the HP48.
Homework - Learn to use the calculator to graph polar equations. See
the manual or ask me if you are having difficulties. Do #18, 16 from
- W: We discussed the use of the calculator for plotting polar equations.
We then investigated the graph of r=sin(n*theta) for n=1,2,3,4, and 5.
The students described these as butterflies and flowers and were able
to predict the graph of sin(n*theta) for any positive integer n.
I also showed how to qualitatively find the shape of sin(3*theta).
As theta goes from 0 to pi/6, we see that r goes from 0 to 1;
As theta goes from pi/6 to pi/3, we see that r goes from 1 to 0;
As theta goes from pi/3 to pi/2, we see that r goes from 0 to -1;
We were able to use this information to graph r=sin(3*theta) without
a calculator and without explicitly making a large table of values.
Students are allowed to ask questions about the group project
up until tomorrow night. I then passed out a sheet of practice problems
for the final exam.
Homework - For tomorrow, turn in #16 from the Polar Coordinates worksheet,
and turn in #15 and #20 from the Practice Problems for the Final Exam.
- Th: I passed out solutions to the practice problems for the final exam.
These solutions were typed up late last night so there may be a couple of
typos. In particular, for #15, the volume of cedar should have been
written as the integral from 2 to 4. Students worked during class on the
group projects. Group projects were supposed to be turned in tomorrow by
5:00 PM. Actually, as long as it is put in my mailbox any time on Friday,
I will not deduct any points. The building is usually locked around
10:00 PM but I am not positive of the exact time. It is your
responsibility to get it to me before the building closes or to finish
your project in the building so as to turn it in by midnight.
Homework - prepare for final exam. Come to class on Monday and receive
10 BONUS points for filling out a questionnaire.
4/27 - 4/30
- M: Last Day of Classes. I passed back group projects and homeworks.
I passed out a sheet of paper listing the scores that I have on record
for each student as well as the breakdown of points needed for various
letter grades. Brian and I left the classroom for 25 minutes as the
students filled out the standard evaluation forms as well as a
supplemental questionnaire (worth 10 bonus points). I then had time
to answer only a couple of questions. If you need help today, look
for Brian. Tomorrow I'll be available from 4:00 PM - 6:00 PM in the
- T: Reading Day.
- W: Cumulative Final Exam, 9 am - noon, Humanities 201.
If you have any questions, please send e-mail to