Analytic Geometry and Calculus II

Math. 114, Sec. 003, Spring 2018

Homework and Announcements

Important Stuff

I have posted a review sheet for the final exam at Final Exam Info here below. I intend to go over that in fair detail for what remains in this course.

2nd Computer Assignment Please note that I have posted the second (and last!) MATHEMATICA assignment here below at Mathematica Info The due date is Monday April 30 in your RCT session.

Both midterm exams I still have a big pile of both the first and second midtem exams that many of you have not gotten back. You should get them back from me for your own records.!

General Info

For general information regarding this course, please read the Syllabus here below. It should answer all practical questions, like when tests are, what the grading policies are etc.

NB! All handouts for this class, listed below, will be labeled with an itemized icon.

Your teaching assistant (TA) is Long Nguyen. He will see you during the recitation sessions (RCT) on Mondays and discuss homework (HW) exercises and give you quizzes. His email is: "lnguye33 the-at-sign masonlive dot gmu dot edu".

Homework Info

Below are the homework assignments (HW) so far in the course. All HW is from the text book of this course. They includes exercises to be discussed in your recitation sessions by your TA. Be sure to attempt them before going to the sessions.

  1. Lectures and Homework (HW) from Week 1. HW for RCT January 29. : Be sure that you know how to differentiate any function, be it a product, quotient or composite of functions. In theory, you should be able to find the derivative of any function that is handed to you! This is the reason why differentiation is easier that integration, since it is not always possible to find a closed form of an indefinite integral. Review how to integrate the basic functions, like polynomials, exponential functions, logarithmic functions and the trig functions. -- The main thing about slicing method is that the total volume is always the integral of the area A(x) of the slice, where x goes from one bounding value to the other.
  2. Lectures and Homework (HW) from Week 2. HW for RCT February 5. : The main thing to remember is that the total volume V is always the integral of dV, whether dV is (1) a small circular disc with thickness dx (or dy), or (2) a cylindrical shell of height f(x), thickness dx and circumference 2 times pi times the radius of the cylinder. Similarly, the total arc length is always the integral of ds, whether we view it in terms of x and dx, or in terms of y and dy. Using some general principles of centers of mass, we can compute the CM of various regions and curves. We derived some of such formulas in class that can be applied to a variety of examples.
  3. Lectures and Homework (HW) from Week 3. HW for RCT February 12. : As we have now seen, we can use integration to compute volumes, arc lengths, center of masses (CM) and surface areas of regions obtained when rotating curves around either x-axis or y axis. The method is pretty much the same, that we integrate some designated "small piece", that is dM, xdM or ydM. Hence, we always use the same principle, as the various formulas for uniform regions or uniform strings indicate. Note! If the plates or strings are not uniform, then we cannot cancel out the density delta in the formulas. Also, keep an eye open for various geometrical symmetries when computing CM, since that can simplify many of your calculations. -- When computing CM for uniform plates or strings, you can cancel out the density and use the nice formulas that I presented in class. Never forget to keep an eye open for various symmetries of the geometrical figures! -- The precise definition of the natural logarithm is given by an integral. From this definition alone we are able to derive all the properties of the logarithm and also the exponential funtion as well. Note that this gives us a way to define the number e as the inverse of the natural logarithm at 1 (that is, the exponential function at 1). Also we have that the limit of (1+x) to the power of 1/x goes to e when x goes to zero. This is an important limit!
  4. Lectures and Homework (HW) from Week 4. HW for RCT February 19. : A consequence of the limit of (1+x) to the power of 1/x when x goes to zero, is that continuously compounded interest is always more profitable than any discrete one, no matter how often per yeat the discrete one is performed. -- Hyperbolic functions are defined in terms of the exponential function. They have properties that very closely resemble those of the trig functions. Many identities are almost the same as for trig functions (usually only differ in sign somewhere...) This can be very useful when integrating. The hyperbolic functions are yet another group of new functions that can be helpful for integration. The main hyperbolic functions are the hyperbolic cosine, the hyperbolic sine and the hyperbolic tangent. Hence, we have added some new functions to our tool bag of functions, the hyperbolic functions, that can be helpful for certain substitutions as we discussed in class. -- When we face an integral involving a square root of a second degree polynomial it is always a good rule of thumb to complete the square and then use the corresponding substitution.
  5. Lectures and Homework (HW) from Week 5. HW for RCT February 26. : When integrating by parts, look at which function you like to get rid of (which you label as "N") and which function you feel like integrating (that you label as "D"). -- When you want to integrate a rational function P(x)/Q(x) where the denumerator is linear (of degree 0 or 1), then you can always use substitution. It gets harder when the denumerator is a quadratic polynomial. In general, to integrate a rational function P(x)/Q(x) one must use the method of partial fractions (PF). This methods decomposes the rational function into a sum of basic rational functions that we then integrate term by term. The general theme is as follows: (i) Check if P(x) is of lower degree than Q(x). If that is not the case, then use long division. (ii) Factor Q(x) completely into linear and quadratic factors (where each quadratic factor cannot be factored further). (iii) Then use the partial fraction decomposition to determine the constants A, B, C, D, ... etc. This can be done in various ways: (a) The usual way of comparing the coefficients to each power of x. (b) Assigning specific values to x. (c) Differentiate with respect to x. (d) Use the Heaviside cover-up method if Q(x) is a product of distinct linear factors. (iv) Finally integrate each summand in the partial fraction decomposition. -- To integrate trigonometric functions must be treated on a case by case basis. However, the general theme is to find and isolate a term that is the derivative of another trig function and then use substitution to solve the integral or at least reduce it to lower powers.
  6. Lectures and Homework (HW) from Week 6. HW for RCT March 5. : There are three basic trigonometric substitutions, each getting rid of a square root of a specific type. Be sure to draw the triangle and the angle theta! Everything can then be read from this triangle. Trigonometric substituitions are important to get rid of a square root, but they can also be used to compute integrals where we need to integrate rational functions where the numberator is a constant and the denumerator is some positive power of a non-factorable quadratic polynomial. The we proceed just like when there was a square root, namely (i) complete the square of the quadratic polynomial and then (ii) use the substitution involving tangent of an angle (the new variable). This will yield an integral involving a power of a trig function that we have already discussed. -- Be sure to write the final outcome/answer in terms of x, the original variable. This can all be read from the triangle you wrote down initially. -- An improper integral is defined as a limit of a proper definite integral, like the ones we have been dealing with so far, and for simple cases we can proceed computing those limits.
  7. Lectures and Homework (HW) from Week 7. HW for RCT March 19. : The first thing to notice for an improper integral, is where exactly is it improper? It is clear that when the limits include an infinity (plus or minus), then that contributes to the improper part, but sometimes it is also improper at a number, say at x = 0, when the function f(x) we want to integrate is not defined at x = 0. So a good plan of attack is as follows: (1) Find out where the "improperness" of the integral is (2) Express the improper integral as a limit (3) Evaluate the integral within the limit by usual methods (4) Take the limit and check it it/they exist. Then the improper integral is convergent if, and only if, all the corresponding limits exist as finite numbers. Sometimes, when we cannot evaluate the integral as a limit, we still need to know if it is convergent or not. One thing we can use then is the comparison test (CT). -- Have a nice Spring Break! See you the week after that.
  8. Lectures and Homework (HW) from Week 8. HW for RCT March 26. : To determine whether or not an improper integral is convergent we must first determine where the problem is: At a discontinuity or at infinity? For integrals where the problem is at infinity there are several ways to determine the convergence without directly evaluating the limit. (i) By Comparison Test (CT) or (ii) By Limit Form of Comparison Test (LFCT). When using LFCT be sure that the limit is actually a number not equal to zero. Remember, you can only conclude convergence if (a) the integral is less than a known convergent integral or (b) the integral is greater than a known divergent integral. If (c) the integral is less than a known divergent integral or (d) the integral is greater than a known convergent integral, then there is NO CONCLUSION. -- A sequence is just a function from the natural numbers into the set of real numbers. Two sequences are equal if, and only if, the n-th term is the same in both sequences for all natural numbers n. Sequences will play an important role in series, something we will discuss soon. Just like real functions, a sequence is convergent when n tends to infinity, when the n-th term gets closer and closer to some number. Otherwise the sequence is divergent. There are many ways to see if a sequence is convergent or not. If the corresponding function of x instead of n has a limit when x tends to infinty, then the sequence is convergent. Most sequences are defined by a formula or by some rule in terms of the previous terms (like the Fibonacci numbers, where each term is the sum of the previous two terms.) Many similar facts hold for sequences when n tends to infinity, as for regular real functions when we take the limit as x goes to infinity. In particular, we can use L'Hospital's rule on the corresponding real function and deduce from that what happens to the sequence if the sequence is given by a formula/expression we are familiar with in terms of a real function. Other facts holds as well, like the Sandwich Theorem. To prove that a sequence is divergent it is very often convenient to show that it has two subsequences where one converges to L and the other converges to L' distinct from L.
  9. Lectures and Homework (HW) from Week 9. HW for RCT April 2. : A fundamental fact for convergent sequences is that (1) An increasing sequence bounded from above is convergent and (2) A decreasing sequence bounded from below is convergent. Pickard's Method is an appealing general method that works to solve an equation of the form g(x) = x when |g'(x)| < 1 on a chosen interval that includes the root/solution x = r. To show that a sequence is divergent it is sufficient to find two subsequences that tend to different limits as n tends to infinity. -- Every sequence yields a series, which then is convergent or divergent. A series is convergent if, and only if, the partial sums of it is convergent. One of the most frequent and easy series is the geometric series. We know exactly when it converges and what its value is, when it converges. Another type of series is the telescoping series. It converges if, and only if, the corresponding function f(n) has a limit when n tends to infinity. One of the most important test for divergence is the n-th term test: If the n-th term does not go to 0 then the series is divergent. Also, the sum of two convergent series is a convergent series. The sum of a convergent series and a divergent one is divergent. The sum of two divergent series can be either convergent or divergent. -- To check for convergence in series with positive terms, there are basically five tests: (1) The integral test. To be used if the n-th term can be integrated when n has been replaced by x. (2) The direct comparison test. To be used if you know that each term is strictly less than a known term, the seris of which is convergent, or if each term is greater than a known term, the series of which is divergent (Remember Less-Conv, or Great-Div, or LCGD (ala ACDC..!)) (3) The limit form of comparison test is basically a refinement of the direct comparison test. -- The two remaining test (ratio and root tests) will be discussed next time.
  10. Lectures and Homework (HW) from Week 10. HW for RCT April 9. : The ratio and the root tests are nice since they are purey mechanical; just compute the limit and conclude (unless the limit is one.) The ration test should be used when the n-th term involves a lot of factorial stuff. The root test should be used if the n-th term has the form of something to the power of n. Recall that when comparing with another series, it is not always clear what to compare with, kinda like when integrating it is not always clear what substitution to make. When using the ratio or root test however, you just have to be careful to make the computations correct, kinda like when differentiating: You just do the calculations. We have seen that there are two kinds of convergence: absolute convergence and conditional convergence. If a series is absolutely convergent, then it is convergent, and more over it is OK to do all kinds of manipulations on them, like rearrange the terms at will and the corresponding series would also be absolutely convergent and with the same sum as the original one. -- If an alternating series is not absolutely convergent, then on uses Leibniz Test to test for conditional convergence.
  11. Lectures and Homework (HW) from Week 11. HW for RCT April 16. : Power series allow us to write a function as a series on a given interval with a given center (the radius of the convergence is half the intervals length). Such functions can be differentiated infinitely often, term by term, and also integrated (as often as you like) term by term. This is the best one could have hoped for regarding these functions! They behave exactly like we want them to.
  12. Lectures and Homework (HW) from Week 12. HW for RCT April 23. : To express a given function as a power series is to find the Taylor series (or Maclaurin series) for the function. We have so far been able to find the Taylor series of some key functions at x = 0. These series we can then differentiate and integrate, and further multiply by x or any power of it, and also substitute in for x to obtain Taylor series for other functions as well. This is a powerful method to obtain Taylor series for known functions. The applications of Taylor series is nice, since they give us a way to see the hidden behaviours of them for small x and we can effectively use them to compute variuos limits that are basically impossible with other methods like L'Hospital's. The n-th Taylor polynomial of a function is then the unique polynomial that approximates the function best among all n-degree polynomials, just like the best linear approximation of a function is given by the tangent line through the point in question. In this way Taylor polynomials are a generalization of the best linear approximations that is covered in math 113. -- The last application of Taylor series is for differential equations. But in order to fully understand this applicatoin, we need some basic info on differential equations in general. The two types of 1st order differential equations we have so far briefly discussed are the (i) separable ones, and then (ii) linear first order differential equations (LFDE). Next week we will continue our discussion about differential equation and how Taylor series can be used to solve some of them.
  13. Lectures and Homework (HW) from Week 12.

Exam Info (Quizzes, Midterm and Final)


Here below are the handwritten solutions to the quizzes so far.


Below is a review sheet for the first midterm. The midterm is to be held in the classroom, Tuesday, March 6 in the latter half of the lecture. -- Please note that the review sheet is not a recipe for the midterm, but merely a collection of problems some of which could appear on the midterm. Basically, the topics for this midterm includes all the lectures from and including last Thursday February 22nd. This is roughly Chapter 6, sections 2 -- 6 plus center of mass (from Chapter 13, section 6), and theorems of Pappus, sections 8, 9 (finance model), section 10. Chapter 7, sections 1,2,3,5. -- The midterm will consist of 5 - 8 problems, none of which are multiple choice. I will give partial credit if I see the methods are correct, even if the answer is totally wrong. No calculators nor cheatsheets are allowed.

The median score for the 1st midterm exam was 51/100. Many of you are doing well, but a considerable portion still has troubles with basic algebra. Please review your algebra, as we will rely on it throughout the course (and beyond!) Note that the letter grade posted on the Patriot Web is only for this first midterm exam, and nothing else.

Below are handwritten solutions to the first midterm. Please print out a hard copy for your records. Make sure you understand the solutions.


Below is a review sheet for the second midterm. The 2nd midterm is to be held in the classroom, Tuesday, April 3rd in the latter half of the lecture. -- The format of the 2nd midterm is the same as for the first midterm. And, again, please note that the review sheet is not a recipe for the midterm. Basically, the topics for this 2nd midterm includes all the lectures from the first midterm to and including Thursday March 22. This is roughly Chapter 7: sections 3,4,8. Chapter 8, sections 1,2.

Below are handwritten solutions to the second midterm. Please print out a hard copy for your records. Make sure you understand the solutions.

The raw median score (of the elevated "boxed" score you got) was 43/100.


Below is a review sheet for the final exam. My plan is to go over most of these problems in class. However, it is important that you look them over and do what you can before the classes. -- Please note that this review sheet is not a recipe for the final exam.



Here below is the first MATHEMATICA assignment for you, along with a short info-sheet on the basics of MATHEMATICA. Read carefully the instructions and make note of the due date!

You are to hand in your solutions to your TA in your RCT session no later than April 9th, 2018. There should be plenty of time so there will be no acceptance of late turn-ins. Make sure that you read and answer all the questions, since some of them do not require any MATHEMATICA outputs whatsoever! You may work in groups up to 3 people in a group. In case you work in a group, you should turn in just one solution to your TA with the names of all in the group clearly printed and signed. Be sure you don't wait until the last day to do this. This assignment is supposed to be fun and not an added stress booster.

As the above guide indicates, you can get your own copy of MATHEMATICA at this GMU computing site. Also, this Wolfram site contains pretty much all you need to play with MATHEMATICA. From here you can click on ``Symbolic and Numeric Computation'' and get to ``Mathematical Functions'' or ``Calculus''. From ``Calculus'' you can go to ``Integrate'', then under ``Tutorials'' you can go to ``Simplifying Algebraic Expressions''.

For plots click on ``Visualization and Graphics'' and get to ``Functional Visualization''. From there you can to to ``ParametricPlot'' for plotting a parametrized curve.

A useful "how to do this and that" in MATHEMATICA can be obtained from Wolfram How Tos. Please note! The commands are case sensitive, and all functions are capitalized.


Below is the second (and last) MATHEMATICA assignment, along with an info-sheet on MATHEMATICA you need for this assignment. Read carefully the instructions and make note of the due date!

You are to hand in your solutions to your TA in your RCT session no later than April 30th, 2018.