(Back pocket, last updated 6/21)
HW 12 – Final Review
HW 11 – Parametric Equations Due Tuesday June 21st at 11:59 PM
HW 10 – Series Review
HW 9 – Series (Taylor's version) Due Friday June 10th at 11:59 PM
HW 8 – Integral Test, again, Power Series, Absolute/Conditional Convergence Due Tuesday June 7th at 11:59 PM
HW 7 – Series, Power Series, and Approximations Due Wednesday June 1st at 11:59 PM
HW 6 – Asymptotic Limit Comparison Test Due Friday May 27th at 11:59 PM
HW 5 – Sequences, Series (Geometric Series, Divergence Test, Integral Test) Due Tuesday May 17th at 11:59 PM
HW 4 (solutions) – Improper Integrals, Continuous Probability Theory and More! Due Friday May 20th at 11:59 PM
HW 3 (solutions) – All kinds of integration, Intro to Improper Integrals Due Tuesday May 17th at 11:59 PM
HW 2 (solutions) – Integration by Parts, Trigonometric Substitutions, and a little Partial Fractions Due Friday May 13th at 11:59 PM
HW 1 (solutions) – U-Substitution and the Fundamental Theorem of Calculus (pt 1 (1,2,3) Due Tuesday May 10th at 11:59 PM) (pt 2 (4,5,6) Due Wednesday May 10th at 11:59 PM)
Friday June 24th : Final
Thursday June 23rd (notes and recording): Final Review
Wednesday June 22nd (notes and recording): Final Review: Integration (u-sub, trig integrals, trig substitution, integration by parts, improper integrals and the asymptotic comparison test)
Tuesday June 21st (notes and recording): Last day of new material before the final. We learned how to integrate with polar coordinates (it's really a reparameterization!)
Monday June 20th : No class, Juneteenth
Friday June 17th (notes and recording):
Thursday June 16th (notes and recording):
Wednesday June 15th : In-class midterm
Tuesday June 14th : Midterm Review
Monday June 13rd (notes and recording):
Friday June 10th (notes and recording):
Thursday June 9th (notes and recording): Today we introduced complex numbers. We defined a quantity i := sqrt(-1). For any real numbers a and b, bi is called an imaginary number, and a+bi is a complex number. We saw that addition could be represented visually "adding vectors" on the complex plane. Multiplication adds the angles and multiplies the lengths of vectors. The length of the vector representing a complex number is called the modulus.
Wednesday June 8th (notes and recording): We did more examples of using the Taylor Approximation Theorem. We discovered that binomial series let us get a handle on inverse trig functions like arcsin.
Tuesday June 7th (notes and recording): We introduced Taylor polynomials and Taylor remainders. We found that if we can bound the (N+1)th derivative, then using the Taylor Approximation Theorem, we could find a bound for the entire remainder going off to infinity.
Monday June 6th (recording): Today we reviewed the Integral Test and did one of the homework problems concerning the integral test in class. The Back Pocket is updated with all the convergence tests.
Friday June 3rd (notes and recording): We did an example of using series to evaluate a limit that we could previously only evaluate using L'Hopital's Rule. We derived the Taylor Series for cos(x) by differentiating the Taylor Series of sin(x) term-by-term. There is a more complicated example in the notes we didn't do in class. Take a look at OH recording for an explanation of HW7 Problem 3.
Thursday June 2nd (notes and recording): We reviewed what we started last time with Taylor Series. Given a function we found out that we can describe all of the coefficients of its power series representation using derivatives of the original function. We derived the Taylor Series for e^x and sin(x) centered at zero.
Wednesday June 1st (notes and recording): We started by differentiating the geometric series to get a power series representation for 1/(1-x)^2 on the interval (-1,1). Next we integrated the geometric series to get a power series representation for -ln(1-x) on the interval [-1,1). We noticed that our partial sum estimates got less accurate as we moved further towards the ends of the interval and away from the center of the interval of convergence. Lastly we introduced Taylor Series i.e. a machine for turning a function into a power series.
Tuesday May 31st (notes and recording): Today we generalized our notion of power series to include power series that are centered at some point "a" instead of zero. We did series manipulations to shift the radius of convergence of the geometric series from (-1,1) to (1,5). The partial sums of this new series still approximate the function 1/(1-x). We noted that there are three distinct possibilities for convergence of a given power series (a) you just get convergence at a point, (b) you get convergence across the whole real line, or (c) you get convergence on a interval (in which case you have to manually check the endpoints. We noted that if we integrate or differentiate a power series term-by-term, then it doesn't change the radius of convergence, but it could change the behavior at the endpoints of the interval of convergence.
Monday May 30th : No class, Memorial Day
Friday May 27th (notes and recording): Today we talked about the Ratio and Root Tests and introduced Power Series.
Thursday May 26th (notes and recording): Today we focused on the alternating series test. If you have a sequence ( b_n ) that is monotone decreasing and whose terms go to zero, then the alternating series sum[ (-1)^n b_n] converges. The proof of the alternating series test gave us a bound for how far away a partial sum is from the true limit. We noticed that by the AST the alternating harmonic series converges while the harmonic series does diverges. Such a series that converges, but if we take the absolute value of each term is no longer convergent is called conditionally convergent. Alternatively if the absolute value |a_n| of (the terms of) a series converges, we call the series sum[a_n] absolutely convergent. We noticed that we could rearrange the terms of the alternating harmonic series to get that it converges to a different value. We actually able to get any real number, and this is true of all conditionally convergent series.
Wednesday May 25th : In-class midterm (solutions) / drop deadline
Tuesday May 24th : Midterm Review
Monday May 23rd (notes and recording): We introduced the comparison test and limit comparison test, two tools analogous to the comparison test for improper integrals and asymptotic comparison test improper integrals, for determining the convergence of series with nonnegative terms. The comparison test says that if a series is bounded above by a convergent series for all terms, then the bounded series is also convergent. Similarly, if all of the terms of a given series are less than a terms of a series we know to be divergent, then the original series diverges. The limit comparison test, on the other hand, says that if the limit of the ratio of the terms of two series exists and is not equal to zero, then the series either both converge or both diverge.
Friday May 20th (notes and recording): We started today with an ad-hoc proof of why the harmonic (a_n = 1/n) series diverges. Then we introduced the Integral Test, which relates improper integrals of continuous, positive, decreasing continuous functions to series. If the improper integral converges, then the series also converges. The integral test doesn't give exact values for convergent series but it does allow us to get ballpark estimates for convergent or divergent series. For instance it would take adding together over 10^43 terms (if each term was a year, this would be the time it takes from now for ordinary matter to disappear from the universe) for the harmonic series to get between 100 and 101. We also talked about two special series that will come up a lot: "p-series" (a_n = 1/n^p) and the geometric series (a_n = x^n). We proved the p series converges for p>1. The geometric series converges to 1/(1-x) for |x|<1.
Thursday May 19th (notes and recording): Our discussion of sequences concluded today with a few useful properties. First, if you have a sequence that is defined by a continuous function, then if the limit of the function as it approaches infinity exists, it is the same as the limit of the sequence as n approaches infinity. The converse is not true (just because a sequence converges, it does not tell us if the function is it defined by converges). Second, there was a theorem that allows us to "bubble in" limits through continuous functions. Lastly we acknowledged the Squeeze Theorem. Next, we began talking about infinite sums, which are really just the limit of partial (finite) sums, similar to improper integrals. We finished with talking about the Divergence Test (if the underlying sequence does not go to zero, then the series cannot possibly converge).
Wednesday May 18th (notes and recording): We began our discussion of the second of three parts in the course: the study of sequences and series. We started with sequences, ordered lists of real numbers (a_1, a_2, a_3, ... ) often denoted { a_n }. Examples of sequences include the natural numbers, a list of all prime numbers, and the Fibonacci sequence. We discussed the rigorous definition of convergence of a sequence which states that the limit as n approaches infinity of a_n converges to some limit L if for each espilon > 0 there is a natural number N such that the distance between a_n and L is less than epsilon whenever n is greater than or equal to N. That is, however small a threshold (epsilon) I am given, I will always be able to find a point in my sequence where all terms beyond it fit inside the epsilon threshold. We did several examples to practice using this definition. Next we found that convergent sequences are bounded, but bounded sequences are not necessarily convergent. Thankfully sequences that are bounded and monotone are convergent!
Tuesday May 17th (notes and recording): We talked about the hierarchy of growth (exponential >> polynomial >> logarithm >> constant) and the rules of algebraic manipulations of little-o and asymptotics. This allowed us to formulate a new version of the comparison test that tells us for nonnegative continuous functions f and g, if f ~ g then the improper integrals of f and g over the interval (a,infinity) either both converge or both diverge. We did several examples of ascertaining the convergence or divergence of an indefinite integral using little-o and asymptotics. This concludes our discussion of integration.
Monday May 16th (notes and recording): Today we started out with an extended example surrounding continuous probability theory. A probability density function is specific kind of improper integral whose total area is 1 and is governed by a probabilistic phenomenon, or random variable, X (think rolling a given number on dice). In the example we talked about in class in depth, our random variable was the event of an "ePhone" breaking, and we modeled it using a so-called exponential distribution. Next we did some calculations involving our integral like the mean time to failure for a given ePhone. We found that even if the mean time to failure is 5 years, only 37% of ePhones will last 5 or more years. Next we talked about asymptotic discontinuities and did an example using partial fractions. Lastly we introduced asymptotic and "little-o" notation. For f(x) and g(x) continuous and nonnegative we say f~g (pronounced f and g are asymptotic to each other if the limit as x approaches infinity of the ratio f(x)/g(x) is 1. In other words f~g means f and g grow at the same rate. On the other hand we say f=o(g) (pronounced f is "little-o" of g) if the limit as x approaches infinity of the ratio f(x)/g(x) is 0. This means that g grows much faster than f. (Here are some resources about Asymptotics and Little-o Notation resource 1, resource 2 because they are not in the textbook).
Friday May 13th (notes and recording): Today we started our discussion of improper integrals, integrals in which one, or both, of the bounds goes off to infinity. Infinity isn't a real number, so we had to define improper integrals in terms of the limit of a bounded integral as the bound goes to infinity. If such a limit exists, we say that the improper integral converges, otherwise we say that it diverges. We were surprised to see that as the area under a function that never crosses the x-axis can still converge. We noticed that the area under the graph of 1/x from 1 to infinity diverges, while 1/x^2 converges over the same interval. We proved that in general, the area under the graph of 1/x^p from 1 to infinity converges if and only if p > 1. Next we talked about the comparison tests for improper integrals. Basically the comparison test allows us to figure out if the improper integral of a nonnegative function converges or diverges. Such an improper integral will converge the function if the integrand is bounded above by a function whose improper integral converges. Likewise, an improper integral will diverge if the function in the integrand is bounded below by a function whose improper integral diverges. Lastly we defined improper integrals that integrate from negative infinity to infinity (you have to break them up).
Thursday May 12th (notes and recording): Today we did many examples of using partial fraction decompositions to integrate rational functions. We noted that the method of partial fraction decomposition essentially reverses the process of finding a common denominator, (and we can always find a common denominator to check our work. We noticed that there a solid, step-by-step approach of (1) Check if the degree of the denominator is lower than the degree of the numerator, if not do polynomial long division. (2) Factor the denominator. (3) Do partial fraction decomposition (a constant for each factor, repeated for all repeated factors, and leaving an x variable in the numerator if there is an irreducible quadratic). (4) Integrate. We can also sometimes use u-substitutions to get our integrand into a form that is useful for partial fractions.
Wednesday May 11th (notes and recording): First we started by correcting my mistake in the u-sub + integration by parts problem from yesterday. Then we did two more trigonometric substitution integrals. Next we introduced another integration technique for rational functions (polynomials divided by polynomials). This method first relies on being able to factor the denominator, for polynomials that have high degrees the roots are difficult to eyeball. It's helpful to note that any rational root of a polynomial is of the form +/- (factor of the constant term)/(factor of the leading term), we can then plug our limited list of options for the roots into the original equation to see if we ever get zero. After we find one root we can use polynomial long division to find the rest. After we factor the denominator we used the method of partial fractions to find a sum of rational functions with constants in the numerator equal to our original function, which we could more easily integrate. We saw that there was a different pattern when we had a repeated root.
Tuesday May 10th (notes and recording): Today we did many examples of integration by parts and solving trigonometric integrals. We added many trig identities to the Back pocket. We found several odd things that can happen in integration by parts like: our choice of u & dv just making it more complicated and we have to start over, iterated integration by parts, getting another copy of the integral we were originally trying to solve, or combining integration by parts with u-sub ((I made a mistake in this example but it's corrected in the updated notes)). Next we did examples of solving trigonometric integrals, where we had to use a u-sub to get rid of a factor of sine or cosine had odd powers, use trig identities, and found out that the integral of sec(x) = ln|sec(x) + tan(x)| + C (which is totally a back pocket thing and it was a hard integral). Last we introduced another method of "Trigonometric Substitutions" when we see integrals involving (+/- a^2 +/- x^2). We had to use a triangle to get our original integral back in terms of the original value after substituting.
Monday May 9th (notes and recording): Today we went over the Syllabus, reviewed several concepts from Calculus I, and introduced two important integration techniques. Riemann Integration allowed us to define definite integrals as in terms of areas under a curve. The fundamental theorem of calculus relates integrals and derivatives, and allows us to port important tools from derivative calculus into integral calculus for finding antiderivatives (aka indefinite integrals). We introduced analogs to the chain rule (aka u-substitution) and the product rule (aka integration by parts) and did a bunch of examples.