Course Title: | Calculus II | Days: | Monday & Wednesday |
Course Code: | MAT 1224 | Time: | 8:00 am - 9:40 am |
Credits: | 4 | Place: | McKinney Humanities Building |
Prerequisites: | MAT 1193 or MAT 1214 | Room 3.03.16 | |
Date Range: | Jan 17, 2023 - May 10, 2023 | ||
Instructor: | Dr. Josué Tonelli-Cueto | Type: | Traditional in-person |
Office: | FLN 1.04.04 | ||
E-mail: | josue.tonelli-cueto@utsa.edu | Exceptions: | February 6 & 8, 2023 |
Office Hours: | Monday, 10.00 am - 12.05 pm | April 17 & 19, 2023 | |
Also by appointment and by Zoom |
Each lecture will have a short pitch (3-5 minutes), called MathInContext, where I will introduce different aspect of the context of mathematics: the people who have done and do math, the history of math or the applications of math to everyday life.
Recall that for a continuous function $f:[a,b]\rightarrow\mathbb{R}$, \[\int_a^b\,f(t)\,\mathrm{d}t\] is the signed area under $f$ in $[a,b]$. Intuitively, the positive areas comes from those regions where $f$ is positive and the negative ones from those where $f$ is negative, but the rigorous definition of this is done through Riemann sums where we use limits and that this notion is clear for locally constant functions (see sect. 1.1 and 1.2 for the details). The Fundamental Theorem of Calculus asserts that \[\int_a^b\,f(t)\,\mathrm{d}t=F(b)-F(a)\] where $F$ is an antiderivative of $f$, i.e., $F$ is a continuous function $F:[a,b]\rightarrow \mathbb{R}$ such that for every $x\in (a,b)$, $F'(x)$ exists and satisfies $F'(x)=f(x)$.
In another formulation of the Fundamental Theorem of Calculus, we just say the function $F:[a,b]\rightarrow\mathbb{R}$ given by \[F(x):=\int_{a}^{x}f(t)\,\mathrm{d}t\] is an antiderivative of $f$ on $[a,b]$, i.e., a continuous function derivable in $(a,b)$ and such that for all $x\in(a,b)$, $F'(x)=f(x)$. Note that this is equivalent to the above version due to the Mean Value Theorem (see Theorem 4.7 in Vol. 1). However, this version asserts succintly that the antiderivative of a continuous function exists, which is not necessarily clear from the version above.
The reason we call this theorem 'fundamental' is because it relates the two fundamental notions of calculus: the derivative—how does the function vary?—and the integral—how does the change accumulate?—. This theorem establishes a direct connection between these two fundamental notions of calculus, allowing us both to compute and study deeper how the functions that we encounter in the world change. A main focus of this course will be in how to use this theorem for computing areas and other physical quantities.
MathInContext: Gottfried Wilhelm Leibniz, the other father of calculus.
Additional materials to learn more: Gottfried Wilhelm von Leibniz by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Newton and Leibniz: Crash Course History of Science #17 by CrashCourse in youtube. Gottfried Wilhem Leibniz by Strayer in youtube. Philosophy of rationalism—Descartes and Leibniz by Philosophy Animal in youtube.
Media to Watch: Integration and the fundamental theorem of calculus by G. Sanderson in 3Blue1Brown in youtube. What does area have to do with slope? by G. Sanderson in 3Blue1Brown in youtube.
Learning Objectives: a) Use the Fundamental Theorem of Calculus to evaluate derivatives of integrals. b) Use the Fundamental Theorem of Calculus to evaluate definite integrals. c) Explain the relationship between differentiation and integration.
The chain rule allows us to compute the derivative of a composition of functions, the analogue when we integrate is the so-called integration by substitution, change of variables or $u$-substitution. Change of variables is based on the following formula: \[ \int_a^bf(g(u))g'(u)\,\mathrm{d}u=\int_{g(a)}^{g(b)}\,f(v)\,\mathrm{d}v \] where $f$ is continuous on the range of $g:[a,b]\rightarrow\mathbb{R}$ whose derivative $g'$ is continuous. This formula allows to transform the functions that we have to integrate. Unfortunately, unlike the chain rule, integration by substitution is not automatic. Because of this, we need to practice in order to identify and come up with the correct change of variables.
MathInContext: Gloria Conyers Hewitt, an African-American mathematician specialized in group theory and abstract algebra and the 4th African-American woman to receive a PhD in Mathematics in the USA.
Additional materials to learn more: Gloria Conyers Hewitt by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Gloria Conyers Hewitt, Black History Month 2018 Honoree in Mathematically Gifted and Black. Gloria Hewitt in Biographies of Women Mathematicians (by Shannon Hensley).
Media to Watch: Video-Lecture on Integration by Substitution by Prof. Robert Ghrist in youtube.
Learning Objectives: a) Recognize when to use integration by substitution. b) Use substitution to evaluate indefinite integrals. c) Use substitution to evaluate definite integrals.
The definite integral is the signed area under a function. Because of this, we can use definite areas also to compute the area between two curves. More concretely, we want to find the area of the region bounded by the two curves given by $y=f(x)$ and $y=g(x)$ and the two vertical axis at $a$ and $b$. If the curves do not cross between $x=a$ and $x=b$ and $g(x)\leq f(x)$ for all $x\in [a,b]$, then the area is given by \[\int_a^b\,(f(x)-g(x))\,\mathrm{d}x.\]
MathInContext: Katherine Johnson, African-American mathematician whose calculations of orbital mechanics where fundamental for the sucess of NASA's first crewed spaceflights.
Additional materials to learn more: Katherine Coleman Goble Johnson by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Katherine Johnson: An American Hero by NASA in youtube. The Woman Behind Project Mercury (Outlier) by Timeline in youtube. Computer (occupation) in Wikipedia, with additional readings of Harvard Computers and West Area Compiuters. Film to Watch: Hidden Figures. Accuracy: How Accurate Is Hidden Figures? by L. Noboru Lima in Looper.
Learning Objectives: a) Determine the area of a region between two curves by integrating with respect to the independent variable. b) Find the area of a compound region. c) Determine the area of a region between two curves by integrating with respect to the dependent variable.
To determine a volume, we can just slice the object and integrate along the areas of the different slices. The so-call Slicing Method or Cavalieri's Principle states that for a solid $K\subset\mathbb{R}$, we have \[\mathrm{vol}(K)=\int_a^b\,A(x)\,\mathrm{d}x\] where $A(x_0):=\mathrm{Area}(\{(y,z)\in\mathbb{R}^2\mid (x_0,y,z)\in K\})$ is the area of the intersection of $K$ with the plane $x=x_0$, and $a,b\in\mathbb{R}$ are such that $X$ lies between the planes $x=a$ and $x=0$—i.e. $[a,b]\supseteq \{x\in\mathbb{R}\mid A(x)\neq 0\}$. This method is a higher dimensional version of how we compute the area of regions on the plane: we integrate the length of the slices of the two-dimensiuonal regions with a vertical line along the $x$-axis.
For a special kind of solids, the solids of revolution, produces by the rotation of a shape around an axis, the slicing method can be easily applied. Under the name of Disk Method, we have the formula \[\int_a^b\,\pi(f(x))^2\,\mathrm{d}x\] for computing the volume of rotating the region bounded by $y=f(x)$, $x=a$, $x=b$ and the $x$-axis around the $x$-axis. Under the name of the Washer Method, we have the formula \[\int_a^b\,\pi\left[(f(x))^2-(g(x))^2\right]\,\mathrm{d}x\] for computing the volume of rotating the region bounded above by $x=a$ and $x=b$, and above by $y=f(x)$ and below by $y=g(x)$ around the $x$-axis. However, these two formulas are just easy consequences of the Slicing Method or Cavaleri's Principle of the above paragraph.
MathInContext: Bonaventura Cavalieri, Italian mathematician and Jesuate, known among other things as a precursor to calculus.Additional materials to learn more: Bonaventura Francesco Cavalieri by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Cavalieri's principle in 3D by Khan Academy in youtube. The War against Disorder: The Jesuit Victory over Indivisibles by Amir Alexander. The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods of Continuity and Infinitesimals in Standford Encyclopedia of Philosophy.
Learning Objectives: a) Determine the volume of a solid by integrating a cross-section (the slicing method). b) Find the volume of a solid of revolution using the disk method. c) Find the volume of a solid of revolution with a cavity using the washer method.
A cylindrical shell is the figure obtaine by rotating the region delimited by $x=a$, $x=b$, $y=f(x)$ and the $x$-axis around the $y$-axis. We can see that the volume of this figure is given by \[\int_a^b\,2\pi\,x\,f(x)\,\mathrm{d}x.\] This allows us to compute further volumes which have rotational symmetry. Note that if we rotate with respect a different vertical axis, then we don't have $2\pi\,x$, but $2\pi\,\mathrm{distance}(x,\text{axis of rotation})$.
MathInContext: Archimedes, the greatest mathematician of anchient history. His work was a precursor or calculus.
Additional materials to learn more: Archimedes by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. How taking a bath led to Archimedes' principle by Mark Salata in youtube. The real story behind Archimedes' Eureka! by Armand D'Angour in youtube.
Learning Objectives: a) Calculate the volume of a solid of revolution by using the method of cylindrical shells. b) Compare the different methods for calculating a volume of revolution.
If we consider $x\mapsto (x,f(x))$ as a curve in the plane, its velocity is given by $x\mapsto (1,f'(x))$ and, therefore, its speed by $x\mapsto \sqrt{1+(f'(x))^2}$. Hence the length of the graph of $f$ between $x=a$ and $x=b$ is given by \[\int_a^b\,\sqrt{1+(f'(x))^2}\,\mathrm{d}x,\] which is the integral of the infinitesimal length $\sqrt{1+(f'(x))^2}\,\mathrm{d}x$ that we move along the vurve after an infinitesimal increase $\mathrm{d}x$ of $x$.
Similarly, we can deduce that the surface area of the surface of revolution obtained by rotating the graph of $f$ between $x=a$ and $x=b$ along the $x$-axis is given by \[\int_a^b\,2\pi\,f(x)\sqrt{1+(f'(x))^2}\,\mathrm{d}x.\]
These formulas allow us to compute the length and surface area of many curves and surfaces that we encounter both in mathematics and applications. However, these integrals might be difficult to solve exactly.
MathInContext: Schwarz lantern is an example of how one should be careful when one defines the arc-length and surface area a limit of polyhedral approximations. This example was produced by the German mathematician Hermann Schwarz. Not all approximations are good.
Additional materials to learn more: How to lie using visual proofs by G. Sanderson in 3Blue3Brown in youtube.
Learning Objectives: a) Determine the length of a plane curve between two points. b) Find the surface area of a solid of revolution.
Integrals appear in many applications in Physics. For example, they play a role in the computation of the mass of an object by integrating the density, the computation of the work of a force along a tranjectory and the hydrostatic force that the liquid does over an object submerged in the liquid.
Among the applications, the one covered in more detail will be about the computation of the work. The importance of the formula \[W=\int_a^b\,F(x)\,\mathrm{d}x\] lies in that it allows us to determine the amount of energy needed to move a given object, such as an object attached to an spring or pumping the water out of a tank up to a given height.
MathInContext: Robert Hooke, although widely known only for his law regarding springs, he was an English scientist with many achievements. For example, he is also credited with the discovery of microorganisms (together with Antoni van Leeuwenhoek) and he was a pioneer in gravitation theory before Newton—he was the first to hypothesize that the gravitational force should follow an inverse square law.
Additional materials to learn more: Robert Hooke by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. DicKtionary — M is for Mathematics — Newton and Hooke by TimeGhost History in youtube. Newton and the equations of Nature by OpenMind. Micrographia: turning the pages of Robert Hooke's masterpiece by The Royal Society in youtube.
Learning Objectives: a) Determine the mass of a one-dimensional object from its linear density function. b) Determine the mass of a two-dimensional circular object from its radial density function. c) Calculate the work done by a variable force acting along a line. d) Calculate the work done in stretching/compressing a spring. e) Calculate the work done in lifting a rope/cable. f) Calculate the work done in pumping a liquid from one height to another. g) Find the hydrostatic force against a submerged vertical plate.
The center of mass of an object allows us to treat a non-punctual object as a punctual object located at the center of mass for many applications in physcis. This allows us to simplify many computations. For example, Pappus' theorem allows us to compute the volume of rotating a planar region along a line. More concretely, let $K\subset \mathbb{R}^2$ be some region and $L\subset \mathbb{R}^2$ a line not intersecting $K$, then the volume of the object of revolution $S$ obtained by rotating $K$ around $L$ is \[\mathrm{vol}(S)=2\pi\, \mathrm{distance}(L,\mathrm{centroid}(K))\, \mathrm{Area}(K)\] where $\mathrm{centroid}(K)$ is the center of mass of $K$ assuming that the density of $K$ is uniform.
MathInContext: Pappus of Alexandria, one of the last great Greek mathematicians of Antiquity. His work was fundamental in the development of analytical geometry by François Viète and René Descartes.
Additional materials to learn more: Pappus of Alexandria by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Descartes’ Mathematics by Mary Domski in Standford Encyclopedia of Philosophy. Descartes and the birth of analytic geometry by Eric G. Forbes in Historia Mathematica. The Invention of Analytical Geometry by Carl B. Boyer in Scientific American (January 1949).
Learning Objectives: a) Find the center of mass of objects distributed along a line. b) Find the center of mass of objects distributed in a plane. c) Locate the center of mass of a thin plate. d) Use symmetry to help locate the centroid of a thin plate.
Integration by parts is the "product rule" of integratrion, although we cannot apply it automatically without making a choice. This technique of integration is usually summarizes as \[\int\,u\,\mathrm{d}v=uv-\int\,v\,\mathrm{d}u,\] which is a shorthand for \[\int\,f(x)g'(x)\,dx=f(x)g(x)-\int\,g(x)f'(x)\,\mathrm{d}x\] where we take $u=f(x)$ and $dv=g'(x)\,\mathrm{d}x$. Note that both of the above equalities are up to a constant, but for definite integration this is not the case, since \[\int_a^b\,f(x)g'(x)\,dx=\left[f(b)g(b)-f(a)g(a)\right]-\int_a^b\,g(x)f'(x)\,\mathrm{d}x\] is a full equality.
The power of integration by parts is that it allows us to integrate many functions, including some of the well-known ones: $\ln x$, $\arcsin x$, $\arccos x$, and $\arctan x$. Additional functions that we can integrate easy products such as $t^k\mathrm{e}^x$, $t^k\sin x$, $t^k\cos^x$, $\mathrm{e}^x\sin(ax)$ and $\mathrm{e}^x\cos(bx)$. Because of this, integration by parts is a fundamental tool in integration.
MathInContext: Alexandria, is a city in the north of Egypt. This city was founded by and named after Alexander III of Macedon (more commonly known in the Western world as Alexander the Great) after conquering Egypt. This city became the cultural center of the Ancient Greek World. This is the reason that people from Alexandria in Ancient times are described as Greek and not as Egyptian. Among the famous intellectual figures that were based in Alexandria, we find the geometer Pappus of Alexandria, the philospher Hypatia (of Alexandria), the geometer Apollonius of Perga, and the so-called 'father of geometry' Euclid (of Alexandria).The book Elements of Euclid was a compilation, synthesis and organization of all the mathematical knowledge of his time, becoming the main mathematical reference for the next millenia. When the king of Egypt at the time, Ptolemy I, asked Euclid if there was an easier and quicker path to learning geometry than reading his Elements, Euclid just answered:
There is no Royal Road to Geometry.
As of today this sentence exmplifies the difficulty of learning mathematics, and how one needs to do the work in order to learn them.
Additional materials to learn more: Euclid of Alexandria by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Hypatia of Alexandria by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Euclid: Biography of a Great Thinker by Socratica in youtube. Ancient Greece and Euclid's Elements | Math and the Rise of Civilization | Documentary by Earth Planet in youtube. How to prove a mathematical theory by Scott Kennedy in TED-Ed. The murder of ancient Alexandria's greatest scholar by Soraya Field Fiorio in TED-Ed's youtube channel. Film to Watch: Agora. Accuracy: An Historian Goes to the Movies ~ Exploring history on the screen by Aelarsen.
Media to Watch: Video-Lecture on Integration by Parts by Prof. Robert Ghrist in youtube.
Learning Objectives: a) Recognize when to use integration by parts. b) Use the integration-by-parts formula to evaluate indefinite integrals. c) Use the integration-by-parts formula to evaluate definite integrals. d) Use the tabular method to perform integration by parts. e) Solve problems involving applications of integration using integration by parts.
Many of the integrals we find in science are of the form \[\int\,R(\cos x,\sin x)\,\mathrm{d}x\] where $R$ is some function depending on $\cos x$ and $\sin x$. In general, we can try the so-called Weierstrass substitution (aka the tangent half-angle substitution), but we will focus in trigonometric integrals that can be solved through more specialized—and faster—methods. These integrals are of the form \[\int \cos^a x\,\sin^bx\,\mathrm{d}x\] and of the form \[\int \sec^a x\,\tan^bx\,\mathrm{d}x.\] Additionally, we study how to compute integrals of the form \[\int \cos ax\,\cos bx\,\mathrm{d}x,\,\int \sin ax\,\sin bx\,\mathrm{d}x\text{ and }\int \sin ax\,\cos bx\,\mathrm{d}x\] which are very common in Fourier analysis.
MathInContext: David Blackwell, an African-American mathematician famous for his contributions in probability and statisitics. For example, the Rao-Blackwell theorem in statistics is named after him; and his textbook Basic Statistics was one of the first textbooks on Bayesian statistics. About his life as a researcher, Blackwell commented the following:
Basically, I'm not interested in doing research and I never have been. I'm interested in understanding, which is quite a different thing. And often to understand something you have to work it out yourself because no one else has done it.
Additional materials to learn more: David Harold Blackwell by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Remembering David Blackwell in The Qualcomm Institute's youtube channel. Conversation with David Blackwell in visionaryproject's youtube channel. A tribute to David Blackwell coordinated by George G. Roussas in Notices of the American Mathematical Society.
Learning Objectives: a) Evaluate integrals involving products and powers of $\sin(x)$ and $\cos(x)$. b) Evaluate integrals involving products and powers of $\sec(x)$ and $\tan(x)$. c) Evaluate integrals involving products of $\sin(mx)$ and $\cos(nx)$ for $m,n$ integers. d) Solve problems involving applications of integration using trigonometric integrals.
In this lecture, we focus on integrals of the form \[\int\,R\left(x,\sqrt{q(x)}\right)\,\mathrm{d}x\] where $q(x)=\alpha x^2+\beta x+\gamma$ is some quadratic polynomial and $R$ is some function depending on $x$ and $\sqrt{q(x)}$. After a change of variables, we can assume, without loss of generality, that $q(x)$ is either of the form $a^2-x^2$, $a^2+x^2$ or $x^2-a^2$. In each one of these cases, we perform substitutions satisfying, respectively, $x=a\sin\theta$, $x=a\tan\theta$ and $x=a\sec\theta$. After these substitutions, we obtain trigonometric integrals that we can integrate.
MathInContext: J. Ernest Wilkins, Jr., an African-American mathematician famous for his work in the Alamos and in nuclear energy. Together with Eugene Wigner, he developed the Wigner-Wilkins approach for estimating the distribution of neutron energies within nuclear reactors. The latter is still used today in nuclear reactors.
Additional materials to learn more: Jesse Ernest Wilkins Jr by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Fragment of Ronald Mickens's Interview about J. Ernest Wilkins, Jr. in AtomicHeritage. Biography of J. Ernest Wilkins, Jr. in Atomic Heritage Foundation.
Learning Objectives: a) Evaluate integrals involving the square root of a sum or difference of two squares. b) Solve problems involving applications of integration using trigonometric substitution.
Polynomials are easy to integrate. Quotient of polynomials—_rational functions_—are not so easy to integrate, but we can still integrate them easily by writting them as sums of partial fractions. In theory, every rational fraction can be written as \[ P(x)+\sum_{i=1}^r\sum_{j=1}^{m_i}\frac{a_{i,j}}{(x-\zeta_i)^j}+\sum_{k=1}^s\sum_{l=1}^{n_k}\frac{b_{k,l}x+c_{k,l}}{(x^2+\beta_k x+\gamma_k)^l} \] where $P(x)$ is a polynomial, the $\zeta_1,\ldots,\zeta_r$ are the real zeros of the denominator with multiplicities $m_1,\ldots,m_r$, and $x^2+\beta_1 x+\gamma_1,\ldots,x^2+\beta_s x+\gamma_s$ are the irreducible factors of degree $2$ of the denominator with multiplicites $n_1,\ldots,n_s$—note that these come from the $2s$ conjugate pairs of complex roots of the denominator.
Using computational algebra, we can efficiently construct the above decomposition for a rational function and so efficiently integrate rational functions. For doing so, we will employ polynomial division, the Ruffini-Horner method, and logarithmic derivatives.
MathInContext: Emmy Noether, a German (and Jewish) mathematician usually referred as “the mother of modern algebra” due to her influence in the development of the area within and across mathematics. About her, one of the founders of algebraic topology, Pavel Alexandrov, said:
It was she who taught us to think in terms of simple and general algebraic concepts—homeomorphic mappings, groups and rings with operators, ideals—and not in terms of cumbersome algebraic computations; and thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some complicated special situation which was not at all suited for the accustomed approach of the classical algebraists.In addition to this, she is well-known in physics due to Noether's theorem which states, roughly, that symmetries and conservation laws of a physical system are in correspondence.
Additional materials to learn more: Emmy Amalie Noether by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. The most beautiful idea in physics - Noether's Theorem by Mithuna Yoganathan in her youtube channel Looking Glass Universe. Emmy Noether: The Greatest Forgotten Mathematician in History by Biographics in youtube. Emmy Noether: La madre del álgebra abstracta (in Spanish) by myself in the Magazine $\pi$kasle. Emmy Noether, alphabetic portrait at my blog Ill-said Throughts (Also in Spanish at the blog mujeres con ciencia).
Learning Objectives: a) Integrate a rational function whose denominator is a product of linear and quadratic polynomials. b) Recognize distinct linear factors in a rational function. c) Recognize repeated linear factors in a rational function. d) Recognize distinct irreducible quadratic factors in a rational function. e) Recognize repeated irreducible quadratic factors in a rational function. f) Solve problems involving applications of integration using partial fractions.
Up to now, we have handle integrals over bounded intervals of bounded functions, but what happens when the interval or the function becomes unbounded? These are the so-called improper integrals which arise naturally in many branches of science—such as physics and statistics. Among these integrals, we encounter two form of improper integrals: integrals of the form \[\int_a^{\infty}\,f(x)\,\mathrm{d}x,\,\int_{-\infty}^b\,f(x)\,\mathrm{d}x\,\text{ and }\,\int_{-\infty}^{\infty}\,f(x)\,\mathrm{d}x,\] and integrals of the form \[\int_{a}^b\,f(x)\,\mathrm{d}x\] where for some $c\in [a,b]$, $f$ is discontinuous (and possibly undefined) at $c$. In the latter case, the most important case is when $\lim_{x\to c^-}f(x)=\pm\infty$ and/or $\lim_{x\to c^+}f(x)=\pm\infty$.
In this lecture, we will learn how to handle rigourously these improper integrals, i.e., we will leanr when we can assign a well-defined valuye to these integrals—when we say that the improper integral converges—and how to compute this value if this is the case.
MathInContext: Georg Cantor, a German mathematician who is one of the founders of set theory and who developed the first mathematical treatment of infinity. The importance of Cantor's mathematics in the 20th century cannot be downplayed, being highly controversial at the time his contributions. In the midst of these controversies, the mathematician David Hilbert said:
No one shall expel us from the paradise which Cantor has created for us.
Additional materials to learn more: Georg Ferdinand Ludwig Philipp Cantor by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive. Integers & Rationals are both infinite but is it the SAME infinity? by Dr. Trefor Bazett in youtube. Integers & Reals have different, infinite sizes! **Cantor Diagonlization** by Dr. Trefor Bazett in youtube. The Life and Works of Georg Cantor by Rica Lopez in youtube. Georg Cantor and His Opposition by moderndaymath in youtube.
Learning Objectives: a) Recognize improper integrals and determine their convergence or divergence. b) Evaluate an integral over an infinite interval. c) Evaluate an integral over a closed interval with an infinite discontinuity within the interval. d) Use the comparison theorem to determine whether an improper integral is convergent or divergent.
In this lecture, we will review the techniques of integration and types of integrals seen so far in the course: $u$-substitution (or change of variables), integration by parts, trigonometric integrals, trigonometric substitutions, integrals of rational functions and improper integrals.
Learning Objectives: a) To consolidate the techniques of integration seen so far. b) Structure the techniques of integration seen so far in an unified way.
Differential equations play a central role in science, as they allow us to describe precisely how the different variables that we measure change in terms of the other variables that we measure. In general, understanding how a variable of interest—such as position, heat, concentration of a certain molecule—means understanding the underlying differential equation that describes how it varies. Now, solving differential equations \[\dot{x}=f(x,t),\] where $\dot{x}$ denotes the derivative of $x$ with respect to $t$, is hard in general—note that a more propper formal way to write the differential equation is $\dot{x}(t)=f(x(t),t)$ since we are trying to find $x$ as a function $t\mapsto x(t)$ of $t$. Even when we are just interested in the evolution of a single variable $t\mapsto x(t)\in\mathbb{R}$ and not a vector function $t\mapsto x(t)\in\mathbb{R}^n$, finding the solution of these differential equations is hard.
In this introductory lecture, we will explore some of the uses and philosophies underlying the use of differential equations in science. This is a side lecture before the Spring Break to motivate the material seen until now in the course.
MathInContext: Where do differential equations appear? Although the differential equations in physics are widely taught in many curricula, differential equations are not limited to physics. We can find them in all the sciences: in marketing, describing the efficacy of publicity; in economy, describing how price varies in terms of the offer and the demans; in chemistry, describing how the concentrations of the different molecules change during a chemical rreaction; in biology, describing how populations grow and decrease in an ecosystem; and in epideomology, describing how a dissease is transmitted across a population.
Additional materials to learn more: Chapter 1 of Amelkin's book Differential Equations in Applications, to explore the mentioned applications in an accessible way. The talk Taking Mathematics to Heart by Alfio Quarteroni, to see how partial differential equations—a more general type of differential equation where we don't only measure variation with respect to time—can be applied to model the human heart. [The point of watching the talk is not to understand everything, but to get a taste how compteporary applied mathematics look like.]
Learning Objectives: a) Recognize the meaning of a differential equation. b) Be aware of some of the applications and uses of differential equations. c) Develop and analyze elementary mathematical models.
In these lectures, we introduce sequences, their terminology and their properties. A sequence $\{a_n\}$ is an ordered infinite list of numbers of the form \[ a_1,a_2,\ldots,a_n,\ldots~\text{ or }~a_0,a_1,a_2,\ldots,a_n,\ldots \] where we call each number $a_n$, the $n$th term of the sequence and the subscript $n$ the index variable of the term $a_n$.
As we cannot give an infinite list of numbers by specifying each of the terms of the sequences. We have to give a sequence $\{a_n\}$ by specifying how to compute the general term $a_n$ out of the index variable $n$. To do this, we usually use either formulas or recurrence relations. For example, the sequence of odd numbers \[ \{x_n\}_{n=0}^{\infty}:=\{1,3,5,7,9,11,\ldots\} \] can be given by a formula as \[ x_n=2n+1 \] or by a recurrence relation as \[ x_n=\begin{cases}1&\text{if }n=0\\a_{n-1}+2&\text{if }n\geq 1.\end{cases} \] In particular, we introduce arithmetic and geometric sequences which are respectively of the form $\{an+b\}$ and $\{cr^n\}$.
As the index variable $n$ goes to infinity, the $n$th term $a_n$ may approach a limit value $L$. We formally define the limit, $\lim_{n\to\infty}a_n$, of a sequence $\{a_n\}$ as the unique number $L$ for which for all $\varepsilon>0$, there is a number $N_{\varepsilon}$ such that for every (index variable) $n\geq N_{\varepsilon}$, \[L-\varepsilon< a_n< L+\varepsilon.\] In other words, no matter how small $\varepsilon>0$ is, the sequence will eventually lie between $L-\varepsilon$ and $L+\varepsilon$. A sequence with a limit is called convergent if it has a limit, and divergent otherwise. For certain divergent sequences $\{a_n\}$, we write $\lim_{n\to\infty}a_n=\infty$ or $\lim_{n\to\infty}a_n=-\infty$ to indicate that the limit does not exist because the sequence grows or decreases without a bound.
The first example, we study is that of a geometric series showing that \[\lim_{n\to\infty}r^n=\begin{cases}\infty&\text{if }r>1\\1&\text{if }r=1\\0&\text{if }|r|<1\\\text{does not exist}&\text{if }r<-1.\end{cases}\] After this example, we study the algebraic laws of limits, showing that for convergent sequences $\{a_n\}$ and $\{b_n\}$, \[\lim_{n\to\infty}(a_n\pm b_n)=\lim_{n\to\infty}a_n\pm\lim_{n\to\infty}b_n,\,\text{ and }\,\lim_{n\to\infty}(a_n b_n)=\left(\lim_{n\to\infty}a_n\right)\left(\lim_{n\to\infty}b_n\right),\] and, if $\lim_{n\to\infty}b_n\neq 0$, \[ \lim_{n\to\infty}(a_n/b_n)=\left(\lim_{n\to\infty}a_n\right)/\left(\lim_{n\to\infty}b_n\right). \] We also study the continuity law which states that for $f:I\rightarrow \mathbb{R}$ continuous and a convergent sequence $\{a_n\}$, \[ \lim_{n\to\infty}f(a_n)=f\left(\lim_{n\to\infty}a_n\right). \] Finally, we study the sandwich theorem or squeze theorem that states that for sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ such that $a_n\leq b_n\leq c_n$ and that $\lim_{n\to\infty}a_n=\lim_{n\to\infty}c_n=L$, we have that $\{b_n\}$ is convergent with \[\lim_{n\to\infty}a_n=L.\]
To end this lesson, we learn how to compute the limit of quotients. To do so, we estudy the orders of growth of different sequences, so that, by identifying the fastest grwing terms in the numerator and the denominator, we can compute the limit. In particular, we study the growth rates of sequences of the form $\{\ln^{\alpha}n\}$, $\{n^{\beta}\}$ and $\{r^n\}$, by showing that for $\alpha,\beta>0$ and $r>1$, we have \[ \lim_{n\to\infty}\frac{\ln^{\alpha}n}{n^{\beta}}=\lim_{n\to\infty}\frac{\ln^{\alpha}n}{r^n}=\lim_{n\to\infty}\frac{n^{\beta}}{r^n}=0. \] In other words, powers of logarithms grow qualitatively slower than powers which in turn grow qualitatively smaller than exponentials.
We end up considering the notions of boundedness and monoticity for sequences. We shows that every convergent sequence is bounded, although the converse is not true as the alternating sequence $\{(-1)^n\}$ shows. However, the monotone convergence theorem shows that a bounded monotone sequence is convergent. This theorem is useful to determine the existence of limits and compute them assuming that they exist.
MathInContext: Karl Weierstrass (written Karl Weierstraß in German), a German mathematician who, together with Bernard Bolzano, is considered gave the foundations of calculus, giving birth to modern mathematical analysis. Because of this major contribution, he is known as the “father of modern analysis”. Interestengly, Weierstrass dropped out of university and became a high school teacher, becoming an university professor due to his scientific publications on the side that allowed him to obtain an honorary doctor's degree.
Additional materials to learn more: Karl Theodor Wilhelm Weierstrass by J.J. O'Connor and E.F. Robertson in MacTutor History of Mathematics Archive.
Learning Objectives: a) Find a formula for the general term of a sequence. b) Find a recursive definition of a sequence. c) Determine the convergence or divergence of a given sequence. d) Find the limit of a convergent sequence. e) Determine whether a sequence is bounded and/or monotone. f) Apply the Monotone Convergence Theorem.
A series is an infinite sum. To define the value of a series $\sum_{n=1}^\infty a_n$ we consider the finite sums $\sum_{n=1}^Na_n$ and consider the limit as we add more and more summands, i.e., \[ \sum_{n=1}^\infty a_n:=\lim_{N\to \infty}\sum_{n=1}^Na_n. \] It is important to observe that the initial index does not have to start with $n=1$, but it also can start with $n=0$ or with any other integer value. In general, we can always rewrite $\sum_{n=c}^\infty a_n$ as $\sum_{n=0}^\infty a_{n+c}$. Also, we note that when all the terms of a series are non-negative, then, by the Monotone Convergence Theorem, the limit sum is either a finite real number or $\infty$.
After introducing series, we show the algebraic laws of limits, which state that \[ \sum_{n=0}^{\infty} ca_n=c\,\sum_{n=0}^{\infty}a_n~\text{ and }~\sum_{n=0}^{\infty}(a_n\pm b_n)=\sum_{n=0}^{\infty}a_n\pm \sum_{n=0}^{\infty}b_n \] for $\sum_{n=0}^{\infty}a_n$ and $\sum_{n=0}^{\infty}b_n$ convergent. We note that $\sum_{n=0}^{\infty}(a_n+b_n)=\sum_{n=0}^{\infty}a_n+\sum_{n=0}^{\infty}b_n$ is still true for series of non-negative terms.
We finish the lesson by introducing some examples, among which the most important ones are the harmonic series, the geometric series and the telescoping series. The harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ is a divergent series for which we can show that $\sum_{n=1}^{2^N}\frac{1}{n}\geq 1+\frac{1}{2}(N-1)$ and so \[\sum_{n=1}^\infty \frac{1}{n}=\infty.\] The geometric series is a series of the form $\sum_{n=0}^\infty r^n$ for which we can obtain its exact value by noting that for the partial sum $\sum_{n=0}^Nr^n$ we have \[(1-r)\sum_{n=0}^Nr^n=\sum_{n=0}^N(r^n-r^{n+1})=(1-r)+(r-r^2)+\cdots+(r^N-r^{N+1})=1-r^{N+1}.\] Using this identity, we obtain that the geometric series converges if and only if $|r|< 1$, and that then \[ \sum_{n=0}^{\infty}r^n=\frac{1}{1-r} ~\text{ and }~ \sum_{n=1}^{\infty}r^n=\frac{r}{1-r}. \] Finally, a telescoping series is a series that can be written in the form \[ \sum_{n=0}^\infty (b_n-b_{n+1}). \] For such a series, we have that $\sum_{n=0}^\infty (b_n-b_{n+1})=b_0-\lim_{n\to\infty}b_{n}$ and so it is convergent if and only if $\{b_n\}$ is convergent. One should observe that a telescoping series might not be recognizable at first sight, such as \[\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right).\]
MathInContext: Zeno's paradoxes are paradoxes related to infinity posed by the Greek philosopher Zeno of Elea. Calculus can be seen as a solution to the so-called Zeno's paradoxes of motion. Funnily, while Zeno was stating his paradoxes and affirming the movement didn't exist at all, the Greek philosopher Diogenes stood up quietly and went away walking. This led to the expression
Movement is proven by Walking.
and the expression solvitur ambulando (it is solved by walking) which is used to emphasize practical experience in the solution to certain theoretical problems. One of the aparadoxes, known as Achilles and the tortoise, is, as retold by Aristotles, as follows:
In a race, the quickest runner (Achilles) can never overtake the slowest (the tortoise), since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
To solve the paradox, we can use the geometric series. Assume that Achilles runs at a constant speed $V$ and the tortoise at a constant speed $v$ having the tortoise a distance advantage of $s$ over Achilles. Initially, Achilles traverse this distance $s$ in a time of $s/V$. However, in that time the tortoise has moved $sv/V$. Again, Achilles has to traverse this distance, which reach after a time of $sv/V^2$, but in this time the tortoise has moved $(sv/V^2)v=s(v/V)^2$. Continuing so an so, we see that Achilles will reach the tortoise in a time of \[ \frac{s}{V}+\frac{sv}{V^2}+\frac{sv^2}{V^3}+\frac{sv^3}{V^5}+\cdots=\sum_{n=0}^{\infty}\frac{sv^n}{V^{n+1}}. \] Now, since Achilles is faster than the tortoise, $v< V$, and so this a converging geometric series with a value of \[ \frac{s}{V}\frac{1}{1-\frac{v}{V}}=\frac{s}{V-v}. \]
Additional materials to learn more: Zeno's Paradoxes by N. Huggett in Stanford Encyclopedia of Philosophy. Zeno's Paradox - Numberphile in Numberphile in youtube. What is Zeno's Dichotomy Paradox? - Colm Kelleher by C. Kelleher in TED-Ed in youtube. 3 Paradoxes That Gave Us Calculus by J. Tan-Holmes in Up and Atom in youtube.
Learning Objectives: a) Write an infinite series using sigma notation. b) Find the nth partial sum of an infinite series. c) Define the convergence or divergence of an infinite series. d) Identify a geometric series. e) Apply the Geometric Series Test. f) Find the sum of a convergent geometric series. g) Identify a telescoping series. h) Find the sum of a telescoping series.
The divergence test states that for a convergent series $\sum_{n}a_n$ we have that \[\lim_{n\to\infty}a_n=0.\] In other words, if a series is convergent, then the general term convergences to zero. However, although this is a necessary condition, it is not sufficient, since there are series, such as the Harmonic Series, which satisfy the divergence test, but that are diverging.
The integral test is one of the most powerful criterions for convergence. In one of its forms, it states that: given a series $\sum_{n=0}^{\infty}a_n$ of non-negative terms and decreasing function $f:[0,\infty)\rightarrow [0,\infty)$ such that $a_n=f(n)$, then $\sum_{n=0}^{\infty}a_n$ converges if and only if $\int_0^{\infty}f(x)\,\mathrm{d}x$ converges. Moreover, one can refine this result to obtain error bounds for the approximation of the series in terms of the partial sums. Unfortunately, finding the function $f:[0,\infty)\rightarrow [0,\infty)$ in the statement of the integral test is the most difficult part for applying the test. This can be seen in the series \[ \sum_{n=1}^{\infty}\left(\frac{1}{n^2}+\sin^2(\pi n)\right) \] which is convergent, but for which the naive choice $f(x)=\frac{1}{x^2}+\sin^2(\pi x)$ gives a function for which $\int_1^\infty$\,f(x)\,\mathrm{d}x$ diverges—note that $f$ is not decreasing in this case.
The most important case of application of the integral test are $p$-series. A $p$-series is a series of the form \[ \sum_{n=1}^\infty \frac{1}{n^p}, \] which, for $p=1$, we recover the Harmonic Series. For these series, using the integral test, we have that $\sum_{n=1}^\infty \frac{1}{n^p}$ converges if and only if $p>1$.
MathInContext: Leonhard Eulers, a Swiss mathematician who is the most prolific mathematician of all history. His works do not cover all branches of mathematics, but his works are pioneering in areas such as graph theory or topology, which are central in contemporary mathematics.
For example, Euler was a pioneer in graph theory, by solving the so-called problem of the seven bridges of Königsberg. This problem ask whether was possible to walk in the city of Königsberg crossing all the birdges of the city once and only once. Euler provided a negative solution, showing why this was not possible. Since then graph theory has evolved a branch of mathematics not only of theoretical interest, but of great practical importance. Nevertheless, this is only a small sample of the impact of the work of Euler.
Additional materials to learn more: How the Königsberg bridge problem changed mathematics - Dan Van der Vieren by D. Van der Vieren in TED-Ed in youtube. Leonhard Euler by J.J. O'Connor and E.F. Robertson in the MacTutor History of Mathematics Archive. ¿Por qué LEONHARD EULER es mi matemático favorito? (Spanish) [Why is Leonhard Euler my favorite mathematician?] by E. Sáenz de Cabezón in Derivando in youtube. A (very) Brief History of Leonhard Euler in moderndaymath in youtube.
Learning Objectives: a) Use the Divergence Test to determine whether a series diverges. b) Use the Integral Test to determine whether a series converges or diverges. c) Estimate the sum of a series by finding bounds on its remainder term.
The comparisson test allows us to determine the convergence of a non-negative series by comparing it with another non-negative series. This test is as follows: Let $\sum_na_n$ and $\sum_nb_n$ be series of non-negative terms such that for all sufficiently large $n$, $a_n\leq b_n$. If $\sum_na_n$ diverges, then $\sum_nb_n$ diverges; and if $\sum_nb_n$ converges, then $\sum_na_n$ converges. This test is used by comparing with well-known series such as the geometric series and the $p$-series.
The limit comparisson test allows us to determine the convergence/divergence of a series by assymptotically comparing the series with another one. In its most simple form, it states the following: Let $\sum_na_n$ and $\sum_nb_n$ be series of non-negative terms. If $\lim_{n\infty}\frac{a_n}{b_n}=L\in (0,\infty)$, then $\sum_na_n$ converges if and only if $\sum_nb_n$ converges. However, if $\lim_{n\infty}\frac{a_n}{b_n}=0$ and $\sum_nb_n$ converges, then $\sum_na_n$ converges; and if $\lim_{n\infty}\frac{a_n}{b_n}=\infty$ and $\sum_nb_n$ diverges, then $\sum_na_n$ diverges.
To use the limit comparisson test the best way is to identify the order of growth of the general term of the series. For example, to determine the convergence of \[\sum_{n=1}^\infty \frac{6n-5}{2n^3-n},\] we note that the fastest growing term in the numerator is $6n$ and in the denominator is $2n^3$. Hence, we obtain that \[ 1=\lim_{n\to\infty} \left(\frac{6n-5}{2n^3-n}\right)/\left(\frac{6n}{2n^3}\right)=\lim_{n\to\infty} \left(\frac{6n-5}{2n^3-n}\right)/\left(\frac{3}{n^2}\right). \] Hence $\sum_{n=1}^\infty \frac{6n-5}{2n^3-n}$ converges if and only if $\sum_{n=1}^\infty \frac{3}{n^2}$ converges. However, the latter is a $2$-series, and so it converges. In this way, the limit comparisson test allows us to determine converge by looking at the order of convergence to zero of the general term of the series.
MathInContext: Sofia Kovalevskaya, a Russian mathematician who is known by her work in partial differential equations and mechanics, particularly for the Cauchy-Kovalevskaya theorem and the so-called Kovalevskaya top. After being privately letured by the German mathematician Karl Weierstrass, she was the first woman to earn a doctorate (in the modern sense) in mathematics and to become a full professor in Europe.
Additional materials to learn more: Sofia Vasilyevna Kovalevskaya by J.J. O'Connor and E.F. Robertson in the MacTutor History of Mathematics Archive. Remembering Sofya Kovalevskaya by M. Audin published by Springer in 2011. Sofia Kovalevskaya, retrato alfabético by M. Audin in mujeres con ciencia. Sofya Kovalevskaya: Unsung Heroes of Science 2020 by Almudena in Hertford College, Oxford in youtube.
Learning Objectives: a) Use the Direct Comparison Test to determine whether a series converges or diverges. b) Use the Limit Comparison Test to determine whether a series converges or diverges.
We study why the Alternating Harmonic Series, \[ \sum_{n=1}^\infty \frac{(-1)^n}{n}, \] converges. After thinking how this happens, we arrived to the so-called Alternating Series Test (also known as Leibniz's Criterion) which states that for a decreasing sequences $\{a_n\}$ of non-negative numbers converging to zero, i.e., for all $n$, $a_n\geq a_{n+1}\geq 0$ and $\lim_{n\to\infty}a_n=0$; the alternating series \[ \sum_n(-1)^na_n \] converges.
By considering re-orderings of the Alternating Harmonic Series, we see that \[\sum_{n=1}^\infty \frac{(-1)^n}{n}~\text{ and }~\sum_{n=1}^\infty\left(\frac{1}{4n-2}+\frac{1}{4n}-\frac{1}{2n+1}\right)\] have different sums. In other words, infinite sums might not be commutative. Because of this, we say that a series $\sum_na_n$ is absolutely convergent if \[\sum_n|a_n|\] converges, and conditionally convergent if it is convergent, but not absolutely convergent. We can easily see that absolutely convergent series are convergent, which allow us to use the tests we know for non-negative series for general series.
Finally, we introduce Riemann's Rearrangement Theorem which motivates why we differentiate between absolute and conditional convergence. The theorem states the following: If $\sum_na_n$ converges absolutely, then, no matter how we re-arrange its terms, the series will converge to the same limit. However, if $\sum_na_n$ converges conditionally, then we can re-arrange the terms of the series in such a way that the series converges to any real number, or that it diverges to $\infty$, $-\infty$ or no value at all.
MathInContext: Bernhard Riemann, a German mathematician known for his work in analysis, number theory and differential geometry. In the latter, Riemann's lecture “On the Hypotheses on which Geometry is Based”; gave rise to the so-called Riemannian Geometry which became foundational in the creation of General Relativity by Albert Einstein.
Additional materials to learn more: Georg Friedrich Bernhard Riemann by J.J. O'Connor and E.F. Robertson in the MacTutor History of Mathematics Archive. A (very) Brief History of Bernhard Riemann in moderndaymath in youtube. The Riemann Hypothesis - Numberphile in Numberphile in youtube. Visualizing the Riemann zeta function and analytic continuation in 3Blue1Brown in youtube.
Learning Objectives: a) Use the Alternating Series Test to determine the convergence of an alternating series. b) Estimate the sum of an alternating series. c) Explain the meaning of absolute convergence and conditional convergence.
The ratio and root tests are tests based in the assymptotic comparisson with a geometric series. For the ratio test, we consider \[ \rho=\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}, \] and for the root test, \[ \rho=\lim_{n\to\infty}|a_n|^{\frac{1}{n}}. \] In both cases, we have that if $\rho< 1$, then the series $\sum_n a_n$ is absolutely convergent; and if $\rho>1$, then $\sum_n a_n$ is divergent. However, if $\rho=1$, we cannot conclude anything.
In general, the root test is stronger than the ratio test. If both $\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}$ and $\lim_{n\to\infty}|a_n|^{\frac{1}{n}}$ exist, then they are equal. However, for some series, such as \[\sum_{n=1}^\infty 3^{-n+\sin(n)},\] the ratio test gives absolute convergence, while the ratio test is inconclusive.
MathInContext: Alan M. Turing, an English mathematician known for his foundational contributions to computer science. He is considered the father of theoretical computer science and artificial intelligence. Among his most well-known theoretical contributions, we find the Turing machine, which is the first theorical model for a general-purpose computer; the matrix condition number, which marks the foundation of Numerical Analysis; and the Turing test, which is a fundamental thought experiment in the study of Artifical Intelligence.
Additionally, Alan M. Turing was a pioneer in cryptography, being his work fundamental in the victory of the Allies in Second World War. In this war, he and his team (where we should mention Gordon Welchman and Harold Keen among many others) in Bletchley Park constructed one of the first computers, the Bombe, to decode the transmissions of the German Enigma machine. Despite this, after the discovery of his homosexuality in 1952, he was condemned and submitted to inhuman treatment. It was not until half a century later, in 2013, when Alan Turing was pardoned posthumously for the crimes due to his homosexuality. Fortunately, this law was followed by the so-called Alan Turing Law that pardoned posthumously all men condemned for participating in homosexual acts.
Additional materials to learn more: Alan Mathison Turing by J.J. O'Connor and E.F. Robertson in the MacTutor History of Mathematics Archive. The Life and Death of Alan Turing in Weird History in youtube. Turing Machines - How Computer Science Was Created By Accident by J. Tan-Holmes in Up and Atom in youtube. The Halting Problem - An Impossible Problem to Solve by J. Tan-Holmes in Up and Atom in youtube. The Turing test: Can a computer pass for a human? - Alex Gendler by A. Gendler in TED-Ed in youtube. J.H. Wilkinson's 1970 Turing Award Lecture Some comments from a numerical analyst. Film to Watch: The Imitation Game. Accuracy: How Accurate Is The Imitation Game? by L.V. Anderson in Slate.
Learning Objectives: a) Use the Ratio Test to determine absolute convergence or divergence of a series. b) Use the Root Test to determine absolute convergence or divergence of a series. c) Describe a strategy for testing the convergence or divergence of a series.
In this lecture, we will review all what we have seen about infinite series so far in the course: sequences and their convergence, series and their convergence, the divergence and the integral test, the comparisson test, the alternating test and the ratio and the root tests.
At the core of this lecture, we will find a general discussion of how to approach problems regarding the convergence, absolute/conditional convergence and divergence of series.
A power series is a series of the form \[ \sum_{n=0}^\infty c_nx^n \] where $\{c_n\}$ is a sequence, $x$ a real variable and, by convention, $x^0=1$—even for $x=0$. The above series is said to be centered at $x=0$. However, we can consider also a power series centered at $x=a$ of the form \[ \sum_{n=0}^\infty c_n(x-a)^n. \]
Observe that $\sum_{n=0}^\infty c_n(x-a)^n$ always converges for $x=a$ (and to the value $c_0$). However, power series might converge to more values. In general, for a power series $\sum_{n=0}^\infty c_n(x-a)^n$, one of the following three exclusive possibilities happens:
In view of the above theorem, we call \[\left\{x\in\mathbb{R}\mid \sum_{n=0}^\infty c_n(x-a)^n\text{ converges}\right\}\] the interval of convergence of $\sum_{n=0}^\infty c_n(x-a)^n$, and \[ \sup\left\{R\geq 0\mid \text{for }|x-a|< R\text{, }\sum_{n=0}^\infty c_n(x-a)^n\text{ converges}\right\} \] the radius of convergence of $\sum_{n=0}^\infty c_n(x-a)^n$, which might be zero (case a) or $\infty$ (case c). Note that the interval of convergence can be of the form $(a-R,a+R]$ or $[a-R,a+R)$ as the power series \[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^n \] with interval of convergence $(-1,1]$ and radius of convergence $R=1$.
As of now, we have seen one power series, the geometric series $\sum_{n=0}^\infty x_n$. This power series has radious of convergence $1$ and interval of convergence $(-1,1)$, where this series satisfies \[\sum_{n=0}^\infty x_n=\frac{1}{1-x}.\] Note that combining this series and using the algebraic laws for series, we can write several rational functions where zero is not a root of the denominator as power series centered at $x=0$.
MathInContext: Maryna Viazovska, an Ukranian mathematician known for her work in sphere packing, where she solved the problem in eight and twenty-four dimensions. Because of these contributions, in 2022, she was awarded the Fields Medal which is the most prestigious prize that a mathematician under 40 years old can win.
Additional materials (all with Maryna Viazovska talking) to learn more: Maryna Viazovska is awarded the 2022 Fields Medal in EPFL in youtube. In conversation with Fields Medalist Maryna Viazovska in BMUCO in youtube. Maryna Viazovska: 2018 Breakthrough Prize Symposium in Breakthrough in youtube.
Learning Objectives: a) Identify a power series. b) Determine the interval of convergence and radius of convergence of a power series. c) Use a power series to represent certain functions.
How can generate new power series out of series that we now? We can add, multiply, differentiate and integrate existing power series in order to get new ones. In this way, given power series $\sum_{n=0}^\infty c_nx^n$ and $\sum_{n=0}^\infty \tilde{c}_nx^n$ and $x\in I$ where $I$ is an interval where both series converge absolutely, we have the following: \[\left(\sum_{n=0}^\infty c_nx^n\right)\pm \left(\sum_{n=0}^\infty \tilde{c}_nx^n\right)=\sum_{n=0}^\infty \left(c_n\pm \tilde{c}_n\right)x^n\] \[\left(\sum_{n=0}^\infty c_nx^n\right)\left(\sum_{n=0}^\infty \tilde{c}_nx^n\right)=\sum_{n=0}^\infty \left(\sum_{i=0}^n c_i \tilde{c}_{n-i}\right)x^n\] \[\left(\sum_{n=0}^\infty c_nx^n\right)'=\sum_{n=0}^\infty \left(c_n x^n\right)'=\sum_{n=0}^\infty n c_n x^{n-1}\] \[\int\left(\sum_{n=0}^\infty c_nx^n\right)\mathrm{d}x=\sum_{n=0}^\infty \int\left(c_n x^n\right)\mathrm{d}x=\sum_{n=0}^\infty \frac{c_n}{n+1} x^{n+1}+C\] \[\int_a^x\left(\sum_{n=0}^\infty c_nt^n\right)\mathrm{d}t=\sum_{n=0}^\infty \int_a^x\left(c_n t^n\right)\mathrm{d}t=\sum_{n=0}^\infty \frac{c_n}{n+1} x^{n+1}-\sum_{n=0}^\infty \frac{c_n}{n+1} a^{n+1}\] Hence, by doing arithmetic operations and differentiating and integrating term-by-term power series, we can power series for new functions in term of old functions. For example, by integrating \[\frac{1}{1+x^2}=\sum_{n=0}^{\infty}(-1)^nx^{2n},\] we obtain \[\arctan(x)=\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}x^{2n+1}\] with radius of convergence $1$. Now, for $x=1$, this series converges by the alternating series test, and so \[\frac{\pi}{4}=\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}.\]
MathInContext: Srinivasa Ramanujan, an Indian mathematician who made significant contributions mathematical analysis, number theory, infinite series and continued fractions, even though he lacked formal training as a mathematician.
After sharing his work with Indian mathematicians, they tried to get British mathematicians to pay attention to Ramanujan's work, because they were unable to fully grasp and understand his work. Because of this Ramanujan wrote letter to several leading mathematicians at University of Cambridge. Many mathematicians ignored the letter, but G.H. Hardy couldn't ignore the letter. In the owrds of Hardy himself:
The formulae... defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only have been written by a mathematician of the highest class. They must be true because, if they were not true, no one would have the imagination to invent them.
And this was the beginning of the mathematical career of Ramanujan in England. At some moment later on, Hardy said that Ramanujan was his greates mathematical discovery.
Additional materials to learn more: Srinivasa Aiyangar Ramanujan by J.J. O'Connor and E.F. Robertson in the MacTutor History of Mathematics Archive. The Man Who Loved Numbers - Srinivasa Ramanujan documentary (1988) in Muonium in youtube. La increíble historia de Srinivasa Ramanujan (Spanish) [The incredible story of Srinivasa Ramanujan] by E. Sáenz de Cabezón in Derivando in youtube. The letter that revealed Ramanujan's genius by T. Hendy in Tibees in youtube. Film to Watch: The Man Who Knew Infinity. Accuracy: The Man Who Knew Infinity: A Report on the Movie by G.E. Andrews in Notices of the AMS.
Learning Objectives: a) Combine power series by addition or subtraction. b) Multiply two power series together. c) Differentiate and integrate power series term-by-term. d) Use differentiation and integration of power series to represent certain functions as power series.
For now, we only now how to sum one power series: the geometric series. However, the Taylor series of a function $f:I\rightarrow \mathbb{R}$ at a point $a$, \[\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(a)(x-a)^n=f(a)+f'(a)(x-a)+\frac{f''(a)}{2}(x-a)^2+\frac{f'''(a)}{6}(x-a)^3+\cdots,\] allows us to obtain power series for the so-called analytic functions, which includes the exponential, the sine, the cosine, the tangent, the $n$th root (of a positive number) and all the functions obtained by adding, substacting, multiplying, dividing and composing them.
In general, it is difficult to tell if an arbitrary function can be written as a Taylor series. Even if the series converges, there is no guarantee of equality beyond the center. For example, the function \[f(x)=\begin{cases}\mathrm{e}^{-\frac{1}{x^2}}&\text{if }x\neq 0\\0&\text{if }x=0\end{cases}\] has the Taylor series $\sum_{n=0}^\infty 0x^n$ at $0$, but it is equal to zero only at $x=0$. However, to obtain obtaining the sum of different power series, it is enough to know the power series of the elementary functions mentioned in the previous paragrapgh and know how to combine them to obtain the power series we want to obtain the value of.
Nevertheless, the Taylor's theorem with Remainder states that for $x\in [a-R,a+R]$, \[\left|f(x)-\sum_{n=0}^N\frac{1}{n!}f^{(n)}(a)(x-a)\right|\leq \frac{1}{(N+1)!}\max_{t\in [a-R,a+R]}\left|f^{(N+1)}(t)\right|R^{N+1}.\] This result allow us to estimate any sufficiently good function $f$ by partial sums of its Taylor series around any point, even if the Taylor series does not converge to the function.
MathInContext: Joseph Fourier, a French mathematician and physicist most well known for the series and transform that bear his name: Fourier series and Fourier Transform. These notions led to the development of Harmonic analysis and of modern analysis as we know it today. Fourier series are an alternative to Taylor series, which in many areas—such quantum mechanics, partial differential equations and signal analysis—play a central role. The Fourier transform is very important in computation, where the so-called Fast Fourier Transform is at the center of many of today's efficient algorithms.
Additional materials to learn more: Jean Baptiste Joseph Fourier by J.J. O'Connor and E.F. Robertson in the MacTutor History of Mathematics Archive. But what is a Fourier series? From heat flow to drawing with circles | DE4 by G. Sanderson in 3Blue1Brown in youtube. But what is the Fourier Transform? A visual introduction by G. Sanderson in 3Blue1Brown in youtube.
Learning Objectives: a) Find a Taylor or Maclaurin series representation of a function. b) Find the radius of convergence of a Taylor Series or Maclaurin series. c) Finding a Taylor polynomial of a given order for a function. d) Use Taylor's Theorem to estimate the remainder for a Taylor series approximation of a given function.
At the end of each Fall and Spring Semester, the day prior to the beginning of the final examination period is designated as a Student Study Day. Classes do not meet during a Student Study Day. Furthermore, a Student Study Day is not to be used as a date on which papers are to be turned in, examinations are to be given, quizzes are to be scheduled, mandatory review sessions are to be held, or for any other class-related activities, other than office hours. Voluntary review sessions at which no new material is presented may be conducted by faculty on this day. There is no Student Study Day during the Summer Semester.
The final exam will take place on Wednesday, May 10, from 9.00 to 10.50 am. More details on the location will be given later.
The course will be based mainly in the following textbook:
The following supplementary material covers sometimes topics beyond the scope of our course, but the parts that correspond to our course can be useful to supplement the textbook, lectures and exercises of the course.
The grade in this course will depend on homework, three midterm exams and one (compulsory) final exam according to the following formula:
\[\frac{HW+ME_1+ME_2+ME_3+FE}{5}\in [0,100]\]where $HW\in [0,100]$ is the grade on the homework, $ME_1\in [0,100]$ is the grade on the 1st midterm exam, $ME_2\in [0,100]$ is the grade on the 2nd midterm exam, $ME_3\in [0,100]$ is the 3rd midterm exam and $FE\in [0,100]$ is the grade on the final exam. Note that each one of these grade is a numerical scale from $0$ to $100$.
Once the numerical grade is computed, this will be converted to a letter grade as follows:
A+ | 97.5-100 | A | 90.0-97.4 | A- | 87.5-89.9 |
B+ | 85.0-87.4 | B | 80.0-84.9 | B- | 77.5-79.9 |
C+ | 75.0-77.4 | C | 70.0-74.9 | C- | 67.5-69.9 |
D+ | 65.0-67.4 | D | 60.0-64.9 | D- | 57.5-59.9 |
F | 0-57.4 |
However, there are a couple of extra rules that will be taken into account when computing the final grade: the back up grade policy, and the extra credit for practice problems.
If the grade in the final exam is greater than the grade in any of the midterm exams, then the grade in the final exam will substitute the lowest grade in a midterm exam when computing the final grade. In other words, the actual formula to compute the final grade will be:
\[\frac{HW+\max\left\{ME_1+ME_2+ME_3+FE,ME_2+ME_3+2\,FE,ME_1+ME_3+2\,FE,ME_1+ME_2+2\,FE\right\}}{5}\in [0,100]\]where $HW\in [0,100]$ is the grade on the homework, $ME_1\in [0,100]$ is the grade on the 1st midterm exam, $ME_2\in [0,100]$ is the grade on the 2nd midterm exam, $ME_3\in [0,100]$ is the 3rd midterm exam and $FE\in [0,100]$ is the grade on the final exam.
Extreme example: Imagine that you were unable to take one of the midterm exams, because you fell asleep. Because of this, you got zero points in that midterm exam. However, not everything would be lost. The back up grade policy guarantees that your grade in the final exam will substitute the zero grade in the midterm exam when computing the final grade. However, note that if you miss two final exams, then the grade in the final exam would only substitute one of the zero grades, not both of them.
Before each of the midterm exams and the final exam, a set of practice problems will be available through WeBWork. The 10% of the grade (from 0 to 100 points) obtained in these practice set will be added to the corresponding exam. Hence a student can take an exam with as many as 10 extra bouns point if E completes successfully the practice problems before the exam.
All homework will be assigned through WeBWork. You can access WebWork through the following link using your UTSA credentials. In case you cannot access, please, let me know immediately.
To complete each of the assignments, you will have until the Sunday of the week after the opening date. In case you miss this first deadline, you will have an extra week to complete the assignments for 70% of the original credit.
There will be three midterm exams and one compulsory final exam, which must be taken to pass the course. In all the exams, the use of a scientific (non-graphing) calculator without internet capabilities is allowed. However, under no circumstances are students permitted to utilize an online resource, website or tutor to find solutions to exam problems.
There will be three midterm exams administered during the class. Each of these exams will consist of written problems similar to the homework problems and examples discussed in class. The exams will have a time limit of 50 minutes.
The midterm exams will take place in the following days (any change will be communicated in advance):
In case you cannot attend the class the day of the midterm exam, you should inform the instructor as soon as possible and always before the exam. Otherwise, no alternative arrangement will be possible.
In accordance with the 2023 Spring Final Exam Schedule, the final exam of this course will on Wednesday, May 10, from 9.00 am to 10:50 am. The final exam will consist of written problems similar to the homework problems and problems from the midterm exams. More details on the location will be given later.
There will be homework sessions by the teaching assistants of the course. Additionally, UTSA offers the possibilities to students to get free tutoring both in person and online.
The MAT1224 teaching assistants, Michael Brinkman and Sean Roberson, will offer homework help sessions and homework recitation sesions. In the homework help sessions, you can walk in person or online and work alone or with other fellow students on the homework problems with the assistance of the TA. In the homework recitations, the TA will review the material already covered and work through some selected examples. The recitations will only be online and will be recorded.
More details regarding the arrangements, schedule and location of the homework sessions can be found in the UTSA Blackboard platform of the course, which can be found in the Syllabus and Schedule sections of the course's blackboard.
UTSA students can access tutoring by dropping into the Tomás Rivera Center for Academic Excellence, located in MS 2.02.18, or utilizing the chat interface at their website. More information about these services and tutoring schedules can be found on their website.
UTSA continues its partnership with TutorMe, a free 24/7 online tutoring platform, to offer additional on-demand tutoring support to students this semester. UTSA students have access to six hours of free tutoring per week through this service. To access this, use the link available in the UTSA Blackboard platform of the course, which can be found in the Start Here and the Tools sections of the course's blackboard.
The following extra materials will be helpful.