Ma 241 section 3 Homepage
Welcome, please note the homework and sylabus are linked just below.

Useful Materials and Links:
  • Course Sylabus ( grading scheme office hours etc...)
  • Course Schedule ( homeworks assigned and test dates )
  • MAPLE
  • Webassign (required homework)


    • Course Notes:
    Please beware there are some errors in these notes, the best way to make sure they're correct is to come to lecture. I will try alert you to mistakes as we find them.

  • Basics of integration: (4.9-5.4) Defintion of the integral motivated by the left endpt, right endpt, midpt, and Riemann rules. Then antiderivatives defined and used to calculate definite integrals via the FTC. Finally some applications of the FTC examined. These topics should be review for you, sorry to be so fast on these topics if they are new to you.
  • U-substitution: (5.5) remember when changing the variable of integration you must convert the measure (dx) and the integrand to the new variable.
  • Trig Substitution: (5.7 kinda) 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) IBP is integration's analogue to the product rule. We notice that the heuristic rule "LIATE" is usuful to suggest our choice of U and dV.
  • Partial Fractions: (5.7 and Appendix G) 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 builing blocks. Additionally I explain explicitly how to integrate any of the basic rational functions that can result from the method.
  • Numerical integration: (5.9) Simpson's rule and trapezoid rule, discussion or errors
  • Improper integration: (5.10) Integrals to infinity and integrals of infinity. Both of these must be dealt with by limits. We examine how these integrals suggest some shapes that have infinite length can have a finite area.
  • Areas bounded by curves: (6.1) Graph, draw a picture to find dA, then integrate. We calculate the area of the triangle, circle and ellipse among others.
  • Finding volumes: (6.2) Graph, draw a picture to find dV, then integrate. We calculate the volume of the cone, sphere, torus and more.
  • Arclength and averages: (6.3-6.4) 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) We extend notions from highschool physics to the more general case. 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.
  • CORRECTIONS of calculus to physics (6.5) We correct the horrible mistakes in the hydrostatic force examples and give a hint about one of the 6.5 webassign problems.
  • Direction Fields and Euler's Method (7.1-7.2) basic terminology and graphing DEqs. Euler's method is then discussed. We explicitly see how to construct an approximate solution by the iteration of Euler's method.
  • Seperation of variables (7.3) seperate then integrate. A number of interesting examples given
  • Exponential and logistic models (7.4-7.5) models defined and solutions derived and analyzed.
  • 2nd order linear DEqns (7.7-7.8) 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. Then we move onto the nonhomogeneous case, we use the method of undetermined coefficients to find the particular solution.
  • Motion of Springs (7.9) we study the motion springs in a viscous media, three cases result (under/over/critical damping) just like in the last section, its the same math. Then we study springs that are pushed by an outside force, we encounter the interesting phenomenon of "resonance". Finally, we note the analogy between the RLC circuit and a spring with friction.
  • Sequences (8.1) 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.
  • Series (8.2-8.4) 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. The task of determining wether or not a given series converges or diverges is a delicate question and we try to develope some intuition by examining a number of examples.
  • estimating a series (8.3-8.4) we explain how close a particular partial sum is to the series. This is important because its not always possible to calculate the limit of the sequence of partial sums. The alternating series error theorem is especially nice.
  • power series (8.5-8.6) 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. We also discover that a power series allows us to differentiate and integrate term by term. This theorem along with the geometric series allows us to find power series expansions for a number of functions. Admitably this method is a bit ackward, however if you take a course in complex variables (I recommend Ma 513) you'll find these calculations are quite important later on.
  • Taylor series (8.7) the taylor series explicitly connects the power series expansion of a function to the derivatives of that function. We can with the taylor series method simply generate the power series by taking some derivatives. We establish the standard Maclaurin series and discuss some 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 ackward is quicker for the examples it touches.
  • binomial series (8.8) this beautiful theorem shows us how to raise expressions to irrational powers. It is very important to many applications in engineering and physics where it is enuf 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) I discuss error a bit more, we see why sin(x)=x upto about 20 degrees to an accuracy of a percent( this claim should be familar to you from the pendulum in freshman physics). Then we continue an example from calculus one, we see how to calculate the squareroot 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


    • Tests from this semester with Solutions:
  • Test one solution
  • Test two solution
  • Test three solution
  • Test four solution
  • final exam and solution


    • Old tests with Solutions:
  • Test one solution
  • Test two solution
  • Test three solution
  • Test four solution
  • final exam and solution


    • More study helps:

    WARNING: these are from last time I taught the course, they contain some extra materials not relevant to your test 4, mainly the comparison tests and the binomial theorem (which we'll cover on the final). So if some problem seems impossible to do with what I taught you then it is likely it is a problem that involves a comparison test. If in doubt just ask me, but first notice I've indicated which parts are of interest for the next test.
  • power series BONUS EXAMPLES( E6 thru E13 relevant to ma241-003 )
  • quizzes quizzes 8 and 9 (#'s 1,3,5,6,7).
  • flowchart of convergence/divergence
  • test 4 review overview of test 4 topics

  • Integrate this. EXTRA CREDIT !!!!
  • Solution to extra credit
  • Approximate integration thru Maple (handed out first day at end of class)
  • old 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, her work is very concise.The only integrals you should assume are the basic integrals, that is those on page 372 of your text.
  • webassign 6.1 solutions by me.
  • webassign 6.2 solutions by me
  • final exam overview final exam overview


    • This is my niece (on my wife's side) won won and her mom. She lives in Canton Province China and enjoys hotdogs and ketchup as you can see.

    Dangerously Cute

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    Last Modified: 12-10-05