Ma 241 section 3 Homepage
Welcome, please note that the homework and syllabus are linked just below.
Useful Materials and Links:
Course Syllabus for ma 241-003 ( grading scheme office hours etc...)
Course Schedule ( homeworks assigned and test dates )
ma 241-003 homework forum ( message board for homework discussion )
Homework translated into 2nd Edition (see 3rd edition for certain problems)
Stewart's Calculus and Concepts additional topics download the pdf's linked below.
I have invented section numbers, they do not appear on the pdf's, but I may refer to the numbers in my notes and in lecture. I am adding the integrating factor method, variation or parameters and solving via
series because they are very useful methods and are worth knowing about before you take sophmore engineering courses. I will likely scale back my treatment of 7.8 in view of these additions.
Integrating Factor Method (7.3+)
2nd Order Homogeneouos DEqns (7.7)
2nd Order Nonhomogeneouos DEqns (7.8)
2nd DEqn applications (7.9)
Solving DEqns with series (8.10)
Test reviews and practice tests:
The tests will follow closely the material emphasized in the reviews below. The best defense is to understand the homework, meaning you can do it cold without looking at a solution. Most of you will find the test is long so it is important that you learn how to calculate the problems both correctly and quickly. You can also look at the old exams to get a better idea of the typical length of my tests.
Test I overview
Test II practice test
Test III practice test
Test III practice test solution
Test IV practice test and solution
Test solutions from this semester:
Test I solution
Test II solution
Test III solution
Test IV solution
Homework quiz solutions from this semester:
These are solutions to the quizzes. There will be about 10 of these.
U-subst. , 7-3-07
Trig-subst./Partial Fractions , 7-6-07
Areas and Volumes. , 7-12-07
Applications to Physics and more Volumes , 7-13-07
first order ODEs
2nd order ODEs
Sequences and series, basics
Power Series, Interval of Convergence and Geometric Series Techniques
Bonus Project (due July 19):
These are not required. It is entirely possible to
make an A+ w/o ever doing this. Grades aside, I would recommend
doing the project because some of the ideas in it will likely continue
to be useful to you long after this course ( well for some of you anyway,
my apologies to the non-technical majors )
fun with complex variables worth 10pts
bonus points may also be gained by pointing out errors in the notes. If you find an error
then email me a description of where it is and how to fix it. Bonus points are added to your test
Posted below are links to my lecture notes from Ma 141 and Ma 241. I have tried to eliminate all the errors
from the ma 241 material, but it's likely there is something wrong somewhere. If you find an error in the notes somewhere in pages
98-235 and you are the first to email me about it then I will give you a bonus point. Generally, the material in these
notes matched up with the text, however, in several places, the text is not as detailed as I would like, so I add
examples and discussion. I have intentionally tried to avoid duplicating examples that are given in your text.
I do try to follow these notes fairly closely in lecture, modulo questions about homework.
Basics of functions: (Chapter One)
[1-9] Review of functions, their properties, and graphs. We will use these ideas thoughout the course.
Motivations and limits: (2.2-2.5)
[10-23] Need for derivative, finite limits, and limits involving infinity.
Tangents and the derivative: (2.1 & 2.6-2.9)
[24-34] Definition of tangent to curve, definition of derivative at a point, and the derivative as a function.
Finally, the derivatives of the basic functions are calculated.
[35-50] Having established all the basic derivatives, we learn
how to differentiate most any function you can think of. This is accomplished through the application of several rules and techniques: product rule, quotient rule, chain rule, implicit differentiation, logarithmic differentiation. These calculational tools form the heart of this course.
Parametrized curves and more:(3.8-4.1)
[51-58] Parametric curves discussed (please ignore 52, it is wrong), Approximation of a function by the "best linear approximation"
explained, finally a number of related rates problems worked out.
L'Hopital's Rule: (4.5)
[59-64] We return to the study of limits again. With the help of differentiation, we are able to calculate many new limits through L'Hopital's Rule. We discuss a number of different indeterminant forms and see how to determine each.
[65-75] We study local and global maxima and minima of functions. A number of geometric ideas are introduced, increasing, decreasing, concave up, concave down, critical points, inflection points, local maximum, local minimum, absolute(global) maximum, absolute(global) minimum. These ideas are studied through the revealing and powerful lense of calculus. The first and sencond derivative tests are presented and applied to some examples.
[76-81] We apply differential calculus to a number of interesting problems. The first and second derivative tests are applied to some real-life problems.
Basic Integration: (4.9-5.5)
[82-103] We define the definite integral and see how it gives the "signed" area under a curve.
While it is intuitively clear that the definite integral defined by the limit of the Riemann sum
gives the area under a curve, it is almost impossible to directly calculate that limit for most examples.
Fortunately, we find that the fundamental theorem of calculus (FTC) allows us to avoid the messy infinite limit.
Instead of finding the limit of the Riemann sum, we merely must find the antiderivative of the integrand
and use the FTC. Finding antiderivatives (aka indefinite integration) is a nontrivial task in general. We
only begin the study listing the obvious examples from the basis of our study of differentiation.
Then, we conclude our study by considering the technique of U-substitution. U-substitution is the most useful technique of integration since it is basically the analogue of the chain rule for integration.
( **** Ma 241 lecture starts here **** )
WARNING: these notes contain material that is in places beyond
the difficulty of the course. The best way to learn which parts are most important is to attend
[98-104] U-substitution: remember when changing the variable
of integration that you must convert the measure (dx) and the integrand to the new variable.
Integrating powers of sine or cosine (5.7)
[105-106] Integrating powers of sine and cosine. Odd-powers amount to an easy u-substitution while
even powers require successive applications of the double-angle formulae.
[107-111] Trig-substitution: same game as U-substitution except that the
substitutions are implicit rather than explicit. Typically, trig substitutions are used to remove unwanted
radicals from the integrand.
Integration by parts (5.6)
[112-115] Integration by parts: IBP is integration's analogue of the product rule. We
note that the heuristic rule "LIATE" is useful to suggest our choice of U and dV.
Partial Fractions (5.7 & Appendix G)
[116-122] This special technique helps us to rip apart rational functions into easily digestable pieces. I explain how to
break up a rational function into its basic building blocks. Additionally, I explain explicitly how to integrate
any of the basic rational functions that can result from the method. The overall result is that we can
integrate any rational function with the help of the partial-fractal decomposition. Besides being useful for integrating
rational functions, the algebra introduced will be useful in later course-work ( for example Laplace Transforms...)
Numerical integration (5.9)
[123-125] Simpson's rule and trapezoid rule, brief discussion of errors.
Improper integration (5.10)
[127-131] Integrals to infinity and integrals of infinity. Both of these must
be dealt with by limits. We examine how these integrals suggest that some shapes that have infinite length can have
a finite area.
Areas bounded by curves (6.1)
[132-139c] Graph, draw a picture to find dA, then integrate. We calculate the
area of the triangle, circle and ellipse and more.
Finding volumes (6.2)
[140-146g] Graph, draw a picture to find dV, then integrate. We calculate
the volume of the cone, sphere, torus and more.
[147-150] How to find the length of an arbitrairy curve and how
to take the average of a function.
Average of a function (6.4)
[151-153] How to find the length of an arbitrairy curve and how
to take the average of a function.
Applications of calculus to physics (6.5)
[154-165] We see how notions from highschool physics are generalized with the help of calculus.
We calculate work done by a non-constant force - the net force applied by a non-constant pressure.
We conclude with a discussion of how to find the center of mass for a planar region of constant density.
[165b-165c] The definition for a probability distribution of a continuous variable x is given. Then the
probability for a particular range of values is defined. Finally, the
mean and median are defined and discussed briefly.
Introduction to differential equations (7.1)
 Differential equations are described in general. The definition of the terms: ordinary,
order, autonomous, linear, homogeneous, nonhomogeneous and solution, are given and discussed as
they apply to differential equations.
Direction Fields and Euler's Method (7.2)
[167-170] Basic terminology and graphing DEqs. Euler's method is then discussed.
We explicitly see how to construct an approximate solution by using Euler's method. In essence,
this is nothing more than tracing out a solution curve in the direction field. Equilibrium solutions are also
defined and discussed in several examples.
Separation of variables (7.3)
[171-177] Separate then integrate. A number of physically interesting examples given.
Exponential growth (7.4)
 Perhaps the most naive growth model, yet it is a good approximation of many
real world processes. We study the differential equation that it arises from and derive
the solutions by math we've learned in earlier sections.
Logistic growth (7.5)
[179-182] The logistic equation is slightly less naive than simple exponential growth. We analyze
the logistic differential eqn. in two ways. First, we study implications of the DEqn directly
and find some rather interesting general conclusions about any solution. Second, we solve the
logistic DEqn directly and find the general form of the solution. An example of how you might
try to apply it is then given (realistic modeling of population growth has not proven to
be very reliable historically, so I'll abstain from anything but the math here... )
Homogeneous 2nd order linear ordinary DEqns (7.7)
[183-187] We begin by carefully analyzing the possible solutions
to the homogeneous case. We find three possiblities corresponding to the three types of solutions to the quadratic
characteristic equation. In each of the cases I, II, and III we find two linearly independent fundamental solutions.
The general solution is then formed by taking a linear combination of the fundamental solutions.
Nonhomogeneous 2nd order linear ordinary DEqns (7.8)
[183-187] We begin by reviewing the possible solutions to the homogeneous case. Next, we explain the form of the
general solution to a nonhomogenous DEqn is the sum of a complementary and particular solution. The complementary solution
is found by the same technique as in section 7.7. Then, we see how to find the particular solution through the
method of undetermined coefficients. We begin with several examples, next a general algorithm describing
the method is given (hopefully exposing some of the subtleties avoided in the first few examples), and after that
yet more examples are given. Finally, we conclude by explaining how the complementary and particular solutions
combine to make the general solution (a proof long overdue at this point in the notes).
Springs and RLC circuits (7.9)
[192-195] We study the motion springs in a viscous media and three cases
result (under/over/critical damping), just like in the last section. It is the same math. Then we study springs
that are pushed by an outside force and we encounter the interesting phenomenon of "resonance". Finally, we
note the analogy between the RLC circuit and a spring with friction.
[196-200] Definitions and examples to begin. Then we discuss how
to take the limit of a sequence using what we learned about limits in calculus I. The squeeze theorem and absolute
convergence theorem help us pin down some otherwise tricky limits.
[201-203] The series is a sum of a sequence. We give a careful definition of this - a series
is the limit of the sequence of partial sums. When the sequence of partial sums converges, we say the series converges.
We discuss telescoping and geometric series - these two are the easiest series to actually calculate. The n-th term
test is given - this is likewise the easiest and quickest of all the convergence tests to apply. Finally, we give
some general properties of convergent series.
Convergence Tests (8.3-8.4)
[204-210] The task of determining whether or not a given series converges or diverges is a delicate question and we
try to develop some intuition by examining a number of examples. We go over a number of tests which can be used
to prove that a series converges or diverges. Lastly, we summarize the tests at our disposal in this course.
Estimating a series (8.3-8.4)
[211-213] We explain how close a particular partial sum is
to the series. This is important because it's not always possible to calculate the limit of the sequence of partial sums.
The alternating series error theorem is especially nice.
Power series (8.5)
[214-216] A power series is a function which is defined pointwise
by a series. We study a number of power series and discover what elementary functions they correspond to. The set
of real numbers for which the power series converges is known as the "interval of convergence" (IOC). We learn that the IOC must be
a finite subinterval of the real numbers, or the whole real line. This means the IOC can be described by its center point and the
radius of convergence (R). We also discover that a power series allows us to differentiate and integrate term by term.
Geometric series tricks for power series (8.5-8.6)
[217-218] The term by term calculus theorem (from the last section of notes) along with the geometric
series allows us to find power series expansions for a number of functions. Admitably, this method is a bit awkward, however,
if you take a course in complex variables (I recommend Ma 513), you'll find that these calculations are quite important later on.
Taylor series (8.7)
[219-226] The Taylor series explicitly connects the power series expansion
of a function to the derivatives of that function. The Taylor series method simply generates the power series
by taking some derivatives and evaluating them at the desired center point for the power series expansion.
We establish the standard Maclaurin series and discuss how to generate new series from
those basic series. While this section allows us to generate the power series in a straightforward fashion, it is not always
the case that this is the most efficient method. The last section, while awkward, is quicker for the examples it touches.
Binomial series (8.8)
[227-228] This beautiful theorem shows us how to raise binomials to irrational powers.
It is very important to many applications in engineering and physics where it is enough to keep just the
first few terms in the binomial series. Since the series expansion is unique (if it exists), we find some of the same results
as we did with Taylor series and before.
Applications of Taylor series (8.9)
[229-235] I discuss error a bit more and we see why sin(x)=x up to
about 20 degrees to an accuracy of 0.01 radians(this claim should be familar to you from the pendulum in freshman physics).
Then we continue an example from calculus I and we see how to calculate the square root using the power series expansion of
sqrt(x). Then we calculate power series solutions to some otherwise intractable integrals. Then we conclude the course by
examining how some series we've covered are used in physics.
Bonus Point: First person to email me the identity of the scientist pictured below:
The minigrader says do your homework early and ask good questions. You can't argue with the minigrader, she's always right.
Integration indefinite and definite integrals.
Partial Fractions how to check your algebra with Maple
Improper Integrals surprise, surprise - Maple will do these to.
Finite sums how to use Maple to
compute Simpson's, Trapezoid, Left, Right and Midpoint Rules.
Direction Fields pretty Maple plots of DEqns
Solving ODEs Maple finds general solution for ODEqns
Taylor Series Maple finds the Taylor series to finite order
Wild world of maple sheets:
I have gathered together some simple applications of Maple to aid you in completing your homework (these are also hyperlinked
in the homework schedule itself). Of course, you can find much more in the Maple help, but this ought to get you
started. We can add to this list as the semester goes on with your help. I want to post more very simple Maple
sheets that get straight to the point of how to do this or that with Maple. So if you know something
else that would be good to add here, email me your idea ( and attach the sheet).
More study helps:
As you view the materials below remember that previous semesters had a slightly different focus. Generally
the integration in our course will be more sophisticated. However, I will omit certain topics from this
course which I have included previously. I'd guess that 95% of the materials below are still
relevant. If in doubt, ask.
Test solutions Spring 2006:
Test I solution
Partial fractions make-up quiz (add's maximum of 5+1 points to test one)
Test II solution
Test III solution
Test IV solution
Homework quiz solutions Spring 2006:
These are solutions to the homework I asked the previous ma 241 course
to work out in class. I plan to ask some even problems this time around.
U-subst. in-class exercise, 1-10-06
Trig-subst. in-class exercise, 1-17-06
Solution to 6.1 #15
Homework quiz on 5.3, 1-10-06
Homework quiz on 5.5, 1-13-06
Homework quiz on trig. substitution, 1-20-06
Homework quiz on IBP(5.6), 1-23-06
Homework quiz on partial fractions(5.7 and App. G), 1-26-06
Homework quiz on areas(6.1), 2-9-06
Homework quiz on volumes(6.2), 2-15-06
Homework quiz on arclengths(6.3), 2-20-06
Homework quiz on averages(6.4), 2-21-06
Homework quiz on work(6.5), 2-23-06
Homework quiz on center of mass and probability(6.5 and 6.7), 2-24-06
Homework quiz on separation of variables, 3-15(?)-06
Homework quiz on homogeneous 2nd order ODEs with constant coefficients, 3-20-06
Homework quiz on nonhomogeneous 2nd order ODEs with constant coefficients, 3-24-06
Homework quiz on convergence and divergence(8.1-8.4)
Tests from fall 2005 with Solutions:
Test one solution
Test two solution
Test three solution
Test four solution
Final Exam and solution
Tests from spring 2005 with Solutions:
Test one solution
Test two solution
Test three solution
Test four solution
Final Exam and solution
More Integration Examples:
Solution to old extra credit project
Older extra credit project Solutions by Ginny. Warning, she's my wife so she
doesn't have to show all her work. You do not have this privilige on the tests. Please ask me if you don't
understand some step she made since her work is very concise.The only integrals you should
assume are the basic integrals, that is, those on page 91 of my lecture notes.
More sequences and series:
WARNING: These are from a previous time I taught the course. They may contain some extra materials not relevant to this current course.
convergence/divergence test guidewith box to put in an example of your own
convergence/divergence test guidewith box filled with the wise guidance of a fellow TA and a warning about what my focus is on for our section of ma 241.
geometric series trick chart my graphical representation of how to use the geometric series indirectly. This trick should be combined with some common sense to solve problems in section 8.6. Of course some problems are just plain-old geometric series, so try that first when attacking 8.6 questions.
a selection of homework problems worked I work out a few of your homework problems. I deal with the endpoints for an example or two, but then focus on the main part which is the open interval of convergence. That just requires careful application of the ratio test. The endpoints require more thought sometimes.
a selection of homework problems worked mostly even numbered problems, I hope these help you understand how to get started. These sort of problems reflect what I think is the most important element of chapter 8. This is the part that you can use in other courses ( convergence and divergence is fine and all but if you can't calculate the power series representation of a function then the question of where it converges seems somewhat pointless ). Also the fact we can integrate almost anything with power series is just fantastic.
power series extra examples ( E6 thru E13 relevant to ma241-006 )
quizzes quizzes 8 and 9 (#'s 1,3,5,6,7).
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Last Modified: 8-01-07