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: Newton and Leibniz: Crash Course History of Science #17 by CrashCourse. Gottfried Wilhem Leibniz by Strayer. Philosophy of rationalism—Descartes and Leibniz by Philosophy Animal.
Media to Watch: Integration and the fundamental theorem of calculus by 3Blue1Brown. What does area have to do with slope? by 3Blue1Brown.
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, 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.
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 Johnson: An American Hero by NASA. The Woman Behind Project Mercury (Outlier) by Timeline. Film to Watch: Hidden Figures.
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: Cavalieri's principle in 3D by Khan Academy. 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 \[\int_a^b\,2\pi xf(x)\,\mathrm{d}x.\] 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: How taking a bath led to Archimedes' principle by Mark Salata. The real story behind Archimedes' Eureka! by Armand D'Angour.
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 3Blue3Brown.
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: DicKtionary — M is for Mathematics — Newton and Hooke by TimeGhost History. Newton and the equations of Nature by OpenMind. Micrographia: turning the pages of Robert Hooke's masterpiece by The Royal Society.
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 \[\mathrm{vol}(S)=\mathrm{Area}(K)\cdot\mathrm{distance}(L,\mathrm{centroid}(K))\] where $\mathrm{centroid}(K)$ is the center of mass of $K$ assuming that the density of $K$ is uniform.
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.
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.
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.
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 \[ P(x)+\sum_{i=1}^r\sum_{j=1}^{m_i}\frac{a_{i,j}}{(x-\zeta_i)^j}+\sum_{k=1}^s\sum_{l=1}^{n_k}\frac{u_{k,l}x+b_{k,l}}{(x^2+b_k x+c_k)^l} \] 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 $x^2+b_1 x+c_1,\ldots,x^2+b_s x+c_s$ 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.
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.
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.
Learning Objectives: a) Recognize separable differential equations. b) Use separation of variables to solve a differential equation. c) Develop and analyze elementary mathematical models.
Learning Objectives: a) Find an integrating factor and use it to solve a first-order linear differential equation. b) Solve applied problems involving first-order linear differential equations.
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.
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.
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) Use the p-Series Test to determine whether a series converges or diverges. d) Estimate the sum of a series by finding bounds on its remainder term.
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.
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.
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.
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.
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.
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.
Learning Objectives:
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 2022 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.