Class Diary
Week 1
August 22: After taking care of administrative stuff (grading schemes, course structure etc), we introduced the main topics of the course and illustrated the perspective we will take with a classical example of a physical system, the simple pendulum. We saw that the analysis of such a system proceeds by forming a set of (differential, integral...) equations, finding numerical or analytical solutions, describing them by functions and then analysing the (extremal, infinitesimal, long-term) behaviour of these solutions using the tools of calculus.
Homework problems: no problems today, but you may want to go over the first chapter to be ready for tomorrow's class. If you do not have the book, you can use any online resource to brush up the following: real numbers and inequalities, the Cartesian coordinate system, real valued functions and their graphs, and the basic trigonometric functions.
August 23: Today we introduced the basic properties of real numbers, focusing on the order properties and the fact that all bounded rational sequences have limits. We practised manipulations with inequalities and defined open and closed intervals (bounded and unbounded). Then we saw the two-dimensional Cartesian coordinate system and gave an algebraic description of the simplest curve on the plane, the straight line.
Homework problems: there are no problems due next Tuesday since this material is preliminary, but remember you must be able to do all problems from the preliminary chapter! if there are problems you cannot do, please come and see me immediately, because chapter 0 is assumed known to you.
August 24: Today we finished our discussion of the Cartesian coordinate system with the circle, and how the Pythagorean theorem provides an algebraic equation for its coordinates. Then we moved on to functions, their definition and basic features, and made a list of the most important classes of functions and their uses in describing quantitative phenomena. We saw how to produce new functions from old using the arithmetical operations and by composing two or more functions.
Homework problems: as per the previous note. Make sure you can reproduce the graphs in page 27 and be able to do all the variations that the arithmetic operations allow (plus composition with f(x) = x-a to shift the x-axis left or right). Also make sure you can breeze through the trig functions section and you have no problem doing any of the exercises in 0.7. To verify your understanding, do as many of the problems from 0.8 as you can.
August 26: Today we listed some patterns of behaviour of functions around a point, introduced limits, the notion of nearness and neighbourhood, and computed limits of some functions intuitively and using the definition. We also saw how to show a function does not have a limit by finding an x violating the definition. Finally we stated the main limit theorem which allows the computation of limits of all polynomials, most rational functions and some functions involving roots.
Homework problems (due Tuesday Sept 6): from section 1.1 do problems 8 and 34. From 1.2 do problems 4, 14 and 24.
Scanned pages from the book for homework.
August 29: Today we elaborated on the main limit theorem, part of its proof and what it allows us to compute (limits of all rational functions in their domain of definition). Then we moved on to trigonometric functions, their continuity and some limits that cannot be computed by substitution. To that end, we used the geometric definition of sine and cosine plus the squeeze theorem to show the existence and compute the value of the limit of sinx/x as x goes to zero.
Homework problems (due Tuesday Sept 6): from section 1.3 do problems 6, 24 and 50. From section 1.4 do problems 4 and 12.
August 30: Today we talked about infinite limits and limits at infinity, and linked them to vertical and horizontal asymptotes respectively. We gave an example of asymptotes of a rational function, and then introduced continuity of functions and its main properties: all polynomials, rational functions, trigonometric functions (and as we will see, the exponential and logarithm as well) are continuous in their domain. Furthermore, continuity is preserved under addition, multiplication, rescaling, division, taking powers and n-th roots.
Homework problems (due Tuesday Sept 6): from section 1.5 do problems 4 and 42. From section 1.6 do problems 10 and 28. Use the convention that if a function is not defined at some point, it is discontinuous there (this statement makes no sense and contradicts the correct notion of continuity of a function - unfortunately we have to live with it in calculus I). From 2.1 do problems 10 and 12.
September 2: Today we practiced the limit definition of derivative and stated the main theorem about differentiating sums, products and ratios. We computed the derivative of sine and listed derivatives for all trigonometric functions. Then we stated and illustrated the chain rule for composition of two or more functions. Finally we talked about higher order derivatives and computed the high order derivatives of monomials.
Homework problems (due Tuesday Sept 13): see Sept 6 entry.
September 6: First problem solving session, mostly about limits and derivatives by hand.
Homework problems (due Tuesday Sept 13): from section 2.2 do problems 30 and 64. From 2.4 do 26. From 2.5 do 38 and 76.
September 7: Today we talked about implicit differentiation, functions that describe curves locally, related rates and linear approximations. In particular, the concept of linearization of a function solidifies our intuition of the tangent line as a local approximation to a function (or any smooth curve, really) and the role of the derivative in getting this.
Homework problems (due Tuesday Sept 13): from section 2.7 do 18, 42. from 2.8 do 18 and 20. From 2.9 do 32 and 46.
September 9: Continuing the theme of derivative as a microscope into the behavior of a function, we dealt with critical points and extreme values today (the existence of extreme values is guaranteed for continuous functions on closed intervals). The derivative detects stationary and singular points, which together with endpoints give the only candidate locations for extrema. Between critical points we have intervals of monotonicity, where the function increases or decreases, and can also be detected using the sign of the derivative. After monotonicity we talked about concavity, telling us how fast and how nicely a function increases or decreases. Again, concavity can be detected by a derivative sign, this time that of the second derivative of the function.
Homework problems (due Tuesday Sept 13): from section 3.1 do problem 15 (try to do it on your own before looking at the solution - then compare) and 24. From 3.2 do problems 14, 36 and 50.
September 12: Today we mostly saw applications of the derivative in the analysis and graphing of functions, and illustrated the machinery with a practical problem (note: if your engineering problem at some point leads to fitting a cylinder in a cone, you know you took a wrong turn at Albuquerque). Then we stated and proved the mean value theorem for derivatives.
Homework problems (due Tuesday Sept 20): from section 3.4 do problems 32 and 38. From 3.6 do problems 24, 28 and 45 (as always, try on your own before looking at the solution for odd numbered problems).
September 13: Second problem solving session.
September 14: Today we saw some examples of the mean value theorem and went on to anti-differentiation: the reverse process of differentiating a function. We saw that the family of anti-derivatives of a function on an interval consists of translates of a single function by arbitrary real numbers. We also saw that this fails for functions not on intervals (the constants may be different on different intervals). We derived the change of variables formula from the chain rule and applied it to find anti-derivatives of some complicated functions. Then we introduced the concept of a differential equation, where the unknown is a differentiable function and the constraint is an equation involving derivatives of the unknown function. We gave some simple examples, showed the link between differential equations and anti-differentiation, and stated the concept of an initial value problem.
Homework problems (due Tuesday Sept 20): from section 3.8 do problems 10, 36 and 43. From 3.9 do 4, 10 and 22.
September 16: Today we started with an introduction to differentials and their properties, especially how these properties are abstracted from intuitive notions of infinitesimal changes in given directions. Then we solved some separable differential equations using both differential notation and classical change of variables. We gave the abstract pattern of separation of variables and the two main stumbling blocks that need to be overcome to use the method. Finally we saw the example of dy/dx=y/x which we solved not by some step by step method, but by investigating the geometry it describes, guessing the solutions, and finally showing that the solutions we found are the only ones.
Homework problems (due Tuesday Sept 20): from section 3.9 do 26. From outside the textbook, consider the differential equation dy/dx = -x/y. Find all solutions by separation of variables and interpret the result geometrically. Trickier: try to guess the solutions before using separation of variables.
September 19: Today we introduced the concept of aggregate and average value of a function for step functions, and linked it to the "area under the graph" of said functions. This motivated the notion of integral for an arbitrary function, which comes from aggregates of step functions via a limiting process (which is complicated and we will elaborate on Wednesday). We spent the rest of the time practising the discrete version of the integral, the sum of a sequence. We saw constant sequences, the sequences a_k = k and b_k = k^2, telescoping sequences and computed the sum of 1/k(k+1) by writing it as a telescoping sum. Finally we stated the formula for the sum of the geometric series (exercise).
Homework problems (due October 4, but my advice is to start now! these will be featured in the exam): from section 4.1 do problems 26 and 28. From 4.2 do problem 4 and 8.
September 20: Problem solving session 3.
September 21: Today we defined partitions, Riemann sums and the notion of integrability. We saw that continuous functions are integrable, which allowed us to conveniently choose partition sequences for computing integrals. We illustrated this with integrals of some linear functions. Then we stated and proved some theorems about comparing integrals of different functions, and finally saw that the mean value of a function lies between its extreme values.
Homework problems (due Oct 40): from 4.2 do problems 12 and 20. From 4.3 do problems 4 and 10.
September 23: Today we stated the linearity of the integral and then stated and proved rigorously the two fundamental theorems of calculus. We illustrated them with some examples, but emphasized the concepts and structure of the results. Along the way, we defined the accumulation (aggregate) function and showed that it is an antiderivative of the original function. We used the second fundamental theorem to get the change of variables formula ("u-substitution") for definite integrals.
Homework problems (due Oct 4): from section 4.3 do problems 40 and 42. From 4.4 do problems 30 and 46. Furthermore, find the integral, on [-1,3], of the function f(x) = (1+x)^100 (the hundredth power).
Practice exam. This has more problems than the actual exam for more practice. Expect 5 problems instead of 6 in the exam.
September 27: First midterm.
September 28: Today we stated and proved the mean value theorem for integrals, which states that the mean value of a continuous function is a value of the function; this fails spectacularly even for the simplest discontinuous function. Then we saw how to compute areas of some planar regions, namely those that are bounded (in some coordinate system) by the graphs of one or more functions. The idea is to understand the relative positions of these boundary functions by getting their points of intersection, and strategically cutting up the region into subregions, each of which has an area computable by a simple integral. Then we moved on to defining the length of a curve as a limit of polygonal approximations by sums of lengths of small line segments; this (surprisingly? integration is the continuous version of summation) led to a Riemann sum of a certain function, turning the limit into an integral. We will continue this investigation with parametric curves later in this class.
Homework problems (due Oct 4): from 4.5 do problems 10 and 22, From 5.1 do 12 and 36. From 5.4 do problem 4.
Exam 1 solutions.
September 30: Today we explored two topics, volumes of solids of revolutions and parametric curves. First we used the formulas we know for areas of disks to reduce the volume computation down to single integrals, and then computed some of these volumes using our tools from calculus. Next we introduced parametric curves on the plane, the notion of parametrization, parameter space and motion on a curve. We saw that a parametrized curve is more than just the geometric locus of its points, but comes with a way of traversing that locus. Changing the way of traversing the curve leads to the concept of reparametrization.
October 3: Today we mostly talked about the exam; I emphasized common mistakes and observed weaknesses and solved some of the problems on the board. Also make sure to read the solutions I gave carefully. In the remaining time we talked a little more about parametric curves, the velocity and speed of a curve traversal in terms of the parameter function and some examples.
Homework problems (due Oct 18): see Oct 4 entry.
October 4: Today we continued our discussion of curves, deriving the formula for the arc length of a parametrized curve, and viewing it as a function of the parameter. This allows an important reparametrization, the arc length reparametrization, which comes from inverting the arc length function (we will talk more about this when we do inverse functions). Then we introduced the logarithm function as the integral of 1/x and derived its basic properties: it is an increasing, concave differentiable function which satisfies log(ab) = log(a) + log(b) for all positive a, b. We used the logarithm together with the chain rule and implicit differentiation to differentiate some complicated functions using what is known as logarithmic differentiation. Finally, we introduced inverse functions for monotone functions on an interval and gave some examples.
Homework problems (due Oct 18): from section 5.4 do problem 22 b). From section 6.1 do problems 10, 34, 43 and 44. From 6.2 do problems 10 and 22.
October 5: Today we concluded our discussion on inverse functions, proved the inverse function theorem (assuming the existence of the derivative of the inverse function) and illustrated it with a polynomial example whose inverse we cannot find explicitly. Then we defined the exponential function as the inverse of the logarithm, and used the properties of the latter to analyze it. We proved it turns sums into products, is differentiable, and crucially is its own derivative (we proved that differently from the book, by showing exp' is a two-sided inverse of the logarithm, and so by uniqueness of the inverse equals exp). Finally, we computed derivatives of composite functions involving the exponential, to prepare for the method of integrating factors in differential equations.
Homework problems (due Oct 18): from section 6.2 do problems 30, 40 and 44. From 6.3 do problems 8, 32, 42 and 51.
October 7: Today we talked about exponential growth and decay and illustrated them with bacteria populations and half-life of radioactive isotopes respectively. Then we talked about the general first order linear differential equation and its solutions via integrating factors. An example we saw was Newton's law of cooling for an object in a (stable, relatively large) environment. Finally, we talked about the possibility of defining inverses for trigonometric functions.
Homework problems (due Oct 18): from section 6.5 do 24 and 34. From 6.6 do 4.
October 17: Today we introduced the inverse trigonometric functions, the algebraic and differential relations they satisfy and the antiderivatives they allow us to compute. Then we moved on to hyperbolic functions, saw that they parametrize a hyperbola on the plane and satisfy similar differential equations to the trigonometric functions. Now using the trigonometric functions we have the general solution to y''+y=0 and using the hyperbolic functions we have the general solution to y''-y=0.
Homework problems (due Nov 1 - the canvas date is wrong): from 6.8 do problems 4, 30, 34, 70. From 6.9 do 11, 12 and 22.
October 18: problem session.
October 19: Today we started with the logistic equation (both with equilibrium and extinction parameters) and interpreted it in terms of populations in a limited resource environment. This gave an opportunity to talk about partial fractions in the simplest setting (two linear factors). Afterwards we talked about the inverse hyperbolic functions, derived a formula for the inverse hyperbolic sine and hinted at a relationship between inverse hyperbolic functions and trigonometric integrals (see for example homework problem 30 from 6.8, involving an expression very similar to what is in the inverse hyperbolic sine formula). Finally we reviewed the two basic techniques of integration, substitution and integration by parts.
Homework problems (due Nov 1 - the canvas date is wrong): from 7.1 do problem 56. From 7.2 do problems 75 (the Fourier coefficient problem) and 84. In general you should be able to do all problems from those two sections.
Practice exam 2.
October 21: Today we continued with integration techniques, computing the integrals of powers of the sine and cosine functions, powers of the secant, functions involving square roots of linear functions (rationalizing substitutions), and partial fraction decomposition which we applied to systematically solve equations like the logistic equation.
October 24: exam review.
October 25: second midterm.
October 26: Today we reviewed, at a deeper level than high school, the notion of indeterminate forms (so called 0/0, infinity/infinity, infinity-infinity forms etc.). We saw what it means for a function to have this form as it approaches a limit point and stated L'hopital's rule; we then saw how to apply L'hopital's rule rigorously and illustrated some of the pitfalls of careless invocation of this theorem. After many examples and non-examples, we went on to consider alternate indeterminate forms and manipulate them so that L'hopital's rule is applicable.
Homework problems (due Nov 8): see Oct 28 entry.
October 28: Our main topic today was the notion of improper integral: a process that allows us to integrate over unbounded regions of the domain, and to integral unbounded functions like x^{-p}. Following an intuitive consideration, we defined improper integrals as limits of regular Riemann integrals as the endpoints reach a singularity of the function (for unbounded functions) or infinity. If these limits converge, we say the improper integrals converge. We saw several convergent improper integrals as well as some divergent ones, and especially saw when the integrals of x^{-p} converge for limits 0 and infinity (p<1 and p>1 respectively). Using this and some basic inequalities we were able to prove the convergence and divergence of many other integrals of positive functions without being able to compute their exact values.
Homework problems (due Nov 8): from section 8.1 do problem 12, from 8.2 problem 22, from 8.3 problem 26 (partial fractions) and from 8.4 problem 38.
October 31: Today we finished talking about improper integrals with several more examples and went on to introduce the notion of sequence: a function from the natural numbers to the real numbers, or in other words, a discrete arrangement of real numbers like the sequence of digits or the sequence of values of an accumulation function at integer points. Then we stated the definition of the limit of a sequence and compared it to that of functions. Using the definition we computed the limit of 1/n^p for p>0 and found it to be zero.
Homework problems (due Nov 8): from 8.4 do problem 42; from 9.1 do problem 8.
November 1: Problem solving session.
November 2: Today we started with a review of the limit of sequences, stated the main limit theorem (whose proof is identical to that for functions) and used it to compute some simple limits. For more complicated ones, we stated the squeeze theorem and applied it to the limit of sin^3(x)/x and to several technical results like the implication |a_n|->0 implies a_n->0. We then introduced monotone sequences (our main example was accumulation functions of positive integrands) and stated the monotone convergence theorem: a monotone increasing (resp. decreasing) and bounded above (resp. below) sequence has a finite limit. This concluded the justification for the integral comparison test we saw last week.
Homework problems (due Nov 8): from 9.1 do problems 18, 32 and 53.
November 4: Today we spent most of the time reviewing and discussing the exam; afterwards we did one final example of a limit of a sequence defined recursively (I made a mistake in the monotonicity so I will complete the proof on Monday) by u_1 = sqrt(3) and u_(n+1) = sqrt(3+u_n).
Homework problems: none, but assuming the limit of the sequence above exists, try to find it.
November 7: Today we introduced a special kind of sequence formed by consecutively summing the elements of another, given one. If a_n is the sequence, s_n = a_1+a_2+...+a_n is the sequence of partial sums of the previous one. For some a_n, we were able to find explicit ("closed form") formulas for the partial sums; in general this is not possible. The crux of this construction is that s_n, being a sequence, may have a limit as n goes to infinity. When it does, we define the limit to be the infinite sum of the terms a_n, called the series of a_n. We saw examples where the limit obviously does not exist (or is infinite) like a_n = 1 and a_n = n, and some examples where the limit exists and is computable. Most notably, we studied the geometric series for various parameters r and computed limits. Finally, we saw that the harmonic series diverges, even though the sequence a_n converges to zero.
Homework problems (due Nov 15): from section 9.2 do problems 6, 14, 24 and 26.
November 8: Today we listed and proved some criteria for convergence of infinite series, starting with the sequence convergence criterion: if s_m is to converge, a_n must converge to zero as n goes to infinity. We used this to show some series are divergent and emphasized the falsity of the converse. Our next criterion was the integral test, applicable to sequences of the form a_n = f(n) for f positive, non-increasing and continuous. After sketching a picture-proof, we used it to show that the sum of reciprocal squares of integers converges.
Homework problems (due Nov 15): from 9.3 do problems 6, 14 and 34.
November 9: Today we applied the integral test to prove the p-series test, showing that the sum of 1/n^p converges if and only if p>1. We then stated the general series comparison test and combined it with previous tests to show convergence or divergence of various series. We finished our discussion with the limit comparison test, whose proof contains an important idea about how to compare series: if individual terms are comparable for all large n, the series are comparable. This is what the limit comparison test formalizes.
Homework problems (due Nov 15): from 9.4 do problem 2 using the limit comparison test, and then using the series comparison test (this is why the limit comparison test is not really important, it can always be replaced by the series comparison test, the second being more flexible and thus more useful).
November 11: Today we started by introducing and proving the ratio test, the first general test that does not require comparing different series or inventing a series for comparison; we saw cases where it is applicable and where it is not. Then we talked about non-positive series and their convergence; we focused on the case of alternating series, stated and proved the alternating series test, and contrasted this situation to positive series using the alternating harmonic series as an example.
Homework problems (due Nov 22): from section 9.4 do problems 28, 32. From 9.5 do 6, 12 and 42 (the last one is hard; give it a try but it is ok if you cannot crack it).
November 14: After finishing the alternating series section with absolute and conditional convergence, we diverged from the book and talked about sequences of functions: a collection of functions indexed by positive integers. For example we saw f_n(x) = x/n. We spent some time getting familiar with this concept; in particular we saw that if we fix x, what we get is a sequence of numbers. From this we found a way to talk about limits of sequences of functions: f_n will converge to f if for every fixed x in the domain, the sequence of numbers f_n(x) converges to f(x). This is what we called point-by-point convergence, or pointwise convergence. We saw f_n(x) = x/n converge point-by-point to f(x)=0 and g_n(x)=nx does not converge to any function. Next time, we will talk about the more subtle notion of uniform convergence, and introduce series of functions.
Homework problems: see next entry. The problems for these two entries will not come from the book.
November 15: problem solving session.
November 16: Today we elaborated on pointwise convergence and saw an unsavory feature: a sequence of continuous functions can converge pointwise to a discontinuous one (the example we gave was of the sequence x^n on [0,1]). This is because in pointwise convergence there is no mention of nearby points: we fix each point individually and consider the limit of the resulting sequence of numbers. So we introduced uniform convergence, a notion that does understand 'nearby points' by requiring the entire graph of f_n to be close to that of the limiting function f for large n. We saw some uniformly convergent sequences, some that were not uniformly convergent, and stated the important theorem about uniform convergence: if f_n converge uniformly to f and all the f_n are continuous, then f is continuous.
Homework problems: see this pdf.
November 18: Today we finished talking about uniform convergence and introduced several methods to compute the sums of series of functions. Starting with the geometric series (the most important base case of a power series) and its convergence properties, we were led to power series expansions of several other functions like the logarithm and the exponential function (the latter, by verifying that the given power series satisfies the same initial value problem as the exponential function). We studied where those series converge and how to prove the expansions by algebraic manipulation, differentiating and integrating term by term (without proof of when we can do that).
November 21: Continuing our study from Friday, we emphasized the role played by differential equations, antiderivatives and complex number identities in deriving power series expansions for known functions. Along the way, we proved the uniqueness of solutions to the equation y'=y with given initial condition y(0)=a using a differencing and squaring trick to avoid questions about negative values. We then introduced the general notions of center, coefficient sequence, radius and interval of convergence. We stated that a power series centered at a point x_0 converges absolutely for x at a distance smaller than the radius of convergence R, and diverges for distance bigger than the radius. Within the radius, we saw the series can be differentiated and integrated term by term, which is a consequence of uniform convergence. At the boundary, we saw several behaviors illustrating the complications that arise there.
November 22: problem solving.
November 23: Today we took the opposite perspective of previous days: we now focus on a general principle for expanding arbitrary smooth functions into power series, if possible. To do this, we investigated the relationship between the coefficient sequence and the values of the function f. From the humble starting point, a_0 = f(x_0), we were led to show that if a function is thus expandable, the n-th term of the coefficient sequence is the n-th iterated derivative of f at the center point, divided by n factorial. This we called the Taylor expansion of f, and finished with some examples of Taylor expansions around 0 and around 1.
Homework problems (due Dec 6): from section 9.6 do problems 6, 32 and 34. From 9.7 do 4, 24 and 30 (do not use the Taylor characterization of a_n and b_n, follow the hint in the book). From 9.8 do problems 4 and 28. A Maclaurin series is nothing but a Taylor series centered at x_0=0 (I did not mention this name in class).
Practice midterm.
November 25: Today we continued our discussion of Taylor expansions focusing on the Taylor formula with remainder together with Taylor's theorem: every function that has derivatives of all orders on an interval can be written as f(x) =T_n(x) +R_n(x) where T_n(x) is the order n Taylor polynomial of f and R_n the remainder; then T_n converges to f pointwise on the interval if and only if the remainder R_n tends to the zero function as n tends to infinity. We observed the book's careless approach to R_n(x) regarding the point where f^(n+1) (X)is evaluated, and saw it generally depends on every other variable: n, x and x_0, so the question of convergence is not entirely straightforward. We saw how to bound R_n(x) by using the fact that c is in (x-x_0,x+x_0) for some basic functions like the exponential (at x_0=0) and the logarithm (at x_0=1).
November 28: exam review.
November 29: third midterm.
November 30: Today we devoted the class to reviewing the results of the midterm and making a general review of power series, up to Taylor and Maclaurin series (without remainder).
Post midterm material (Dec 1 - Dec 5): After the midterm review, we focused on questions of approximation and convergence of Taylor expansions. We saw an example of a function whose Taylor series around zero converges to the zero function, but the function itself is not the zero function. This apparent paradox was traced to the fact that the remainder R_n does not tend to zero, in fact all the information about the function is contained in the sequence of functions R_n. However, we saw a theorem saying that if we start with functions whose Taylor series converge to them and perform any algebraic operations, we still get a function represented by its Taylor series, on the largest domain where all the algebraic operations were defined (for example, we cannot divide a function by (x-1) and expect to have a Taylor series at x=1). We also talked about sources of error in numerical approximation and the compromises they mandate: errors of method require us to use many terms of the Taylor expansion to get close to the correct value, but errors of rounding become larger and larger as we perform more computations, forcing us to restrict the number of terms we can use! Finally, we saw the problems that arise when subtracting two nearly equal numbers, and demonstrated how the loss of accuracy can ruin a simple computation for the derivative of x^2.
Review topics: We devoted the last two days to a review of earlier important material, especially differential equations and improper integrals (including various indeterminate forms). The most crucial notions we will need from differential equations are the initial value problem (going from a family of solutions to a unique solution in the family), the method of separation of variables (which is the only method we saw for solving non-linear equations), the method of integrating factors (which gives in principle a complete solution to the problem y' +P(x)y = Q(x)) and the asymptotic properties of their solutions (which is the most important question about the analysis of differential equations). For improper integrals, the most important concept was the comparison test with p-integrals (not to be confused with the similar but distinct comparison test between series with p-series). This is the only tool we have when we cannot find a formula for the antiderivatives of the integrand. Finally, we saw some trickier problems with indeterminate forms and saw how to use Taylor series to make educated guesses as to what the limits of indeterminate forms should be (which then need to be demonstrated rigorously by other means).
Practice final exam.
Solutions to the practice final.