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Calculus I

• Course name: Calculus I
• Course number: XYZ

Course Characteristics

Key concepts of the class

• Calculus for the functions of one variable: differentiation
• Calculus for the functions of one variable: integration
• Basics of series
• Multivariate calculus: derivatives, differentials, maxima and minima
• Multivariate integration
• Functional series. Fourier series
• Integrals with parameters

What is the purpose of this course?

The course is designed to provide Software Engineers the knowledge of basic (core) concepts, definitions, theoretical results and techniques of calculus for the functions of one and several variables. The goal of the course is to study basic mathematical concepts that will be required in further studies. This calculus course will provide an opportunity for participants to understand key principles involved in differentiation and integration of functions: solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities, get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation. All definitions and theorem statements (that will be given in lectures and that are needed to explain the keywords listed above) will be formal, but just few of these theorems will be proven formally. Instead (in the tutorial and practice classes) we will try these definitions and theorems on work with routine exercises and applied problems.

Course objectives based on Bloom’s taxonomy

- What should a student remember at the end of the course?

By the end of the course, the students should be able to

• what the partial and directional derivatives of functions of several variables are
• basic techniques of integration of functions of one variables
• how to calculate line and path integrals
• distinguish between point wise and uniform convergence of series and improper integrals
• decompose a function into Fourier series
• calculate Fourier transform of a function

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to

• how to find minima and maxima of a function of various orders
• how to represent double integrals as iterated integrals and vice versa
• what the length of a curve and the area of a surface is
• properties of uniformly convergent series and improper integrals
• how to find Fourier transform of a function

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to

• take derivatives of various type functions and of various orders
• integrate the functions of one and several variables
• apply definite integration
• expand functions into Taylor series
• find multiple, path, surface integrals
• find the range of a function in a given domain
• decompose a function into Fourier series

Course evaluation

Course grade breakdown

Type	Points
Labs/seminar classes	12
Interim performance assessment	48
Exams	140

Grades range

Course grading range

Grade	Points
A	[180, 200]
B	[150, 179]
C	[120, 149]
D	[0, 119]

Resources and reference material

• Claudio Canuto, Anita Tabacco Mathematical Analysis I (Second Edition), Springler
• Claudio Canuto, Anita Tabacco Mathematical Analysis II (Second Edition), Springler

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Section 1

Section title

Sequences and Limits

Topics covered in this section

• Sequences. Limits of sequences
• Limits of sequences. Limits of functions
• Limits of functions. Continuity. Hyperbolic functions

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. A sequence, limiting value
2. Limit of a sequence, convergent and divergent sequences
3. Increasing and decreasing sequences, monotonic sequences
4. Bounded sequences. Properties of limits
5. Theorem about bounded and monotonic sequences.
6. Cauchy sequence. The Cauchy Theorem (criterion).
7. Limit of a function. Properties of limits.
8. The first remarkable limit.
9. The Cauchy criterion for the existence of a limit of a function.
10. Second remarkable limit.

Typical questions for seminar classes (labs) within this section

1. Find a limit of a sequence
2. Find a limit of a function

Test questions for final assessment in this section

1. Find limits of the following sequences or prove that they do not exist:
2. $a_{n}=n-{\sqrt {n^{2}-70n+1400}}$;
3. $d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}$;
4. $x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}$.

Section 2

Section title

Differentiation

Topics covered in this section

• Derivatives. Differentials
• Mean-Value Theorems
• l'Hopital’s rule
• Taylor Formula with Lagrange and Peano remainders
• Taylor formula and limits
• Increasing / decreasing functions. Concave / convex functions

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. A plane curve is given by $x(t)=-{\frac {t^{2}+4t+8}{t+2}}$, $y(t)={\frac {t^{2}+9t+22}{t+6}}$. Find \begin{enumerate}
2. the asymptotes of this curve;
3. the derivative $y'_{x}$.

Typical questions for seminar classes (labs) within this section

1. Differentiation techniques: inverse, implicit, parametric etc.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function

Test questions for final assessment in this section

1. Find a derivative of a (implicit/inverse) function
2. Apply Leibniz formula Find $y^{(n)}(x)$ if $y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x$.
3. Draw graphs of functions
4. Find asymptotes
5. Apply l'Hopital’s rule
6. Find the derivatives of the following functions: \begin{enumerate}
7. $f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}$;
8. $y(x)$ that is given implicitly by $x^{3}+5xy+y^{3}=0$.

Section 3

Section title

Integration and Series

Topics covered in this section

• Antiderivative. Indefinite integral
• Definite integral
• The Fundamental Theorem of Calculus
• Improper Integrals
• Convergence tests. Dirichlet's test
• Series. Convergence tests
• Absolute / Conditional convergence
• Power Series. Radius of convergence
• Functional series. Uniform convergence

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Find the indefinite integral $\displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx$.
2. Find the length of a curve given by $y=\ln \sin x$, ${\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}$.
3. Find all values of parameter $\alpha$ such that series $\displaystyle \sum \limits _{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}$ converges.

Typical questions for seminar classes (labs) within this section

1. Integration techniques
2. Integration by parts
3. Calculation of areas, lengths, volumes
4. Application of convergence tests
5. Calculation of Radius of convergence

Test questions for final assessment in this section

1. Find the following integrals:
2. $\int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx$;
3. $\int 2^{2x}e^{x}\,dx$;
4. $\int {\frac {dx}{3x^{2}-x^{4}}}$.
5. Use comparison test to determine if the following series converge. $\sum \limits _{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}$;
6. Use Cauchy criterion to prove that the series $\sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}$ is divergent.
7. Find the sums of the following series:
8. $\sum \limits _{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}$;
9. $\sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}$.

Section 4

Section title

Differential Analysis of Functions of Several Variables

Topics covered in this section

• Limits of functions of several variables
• Partial and directional derivatives of functions of several variables. Gradient
• Differentials of functions of several variables. Taylor formula
• Maxima and minima for functions of several variables
• Maxima and minima for functions of several variables subject to a constraint

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	0

Typical questions for ongoing performance evaluation within this section

1. Find $\lim \limits _{x\to 0}\lim \limits _{y\to 0}u(x;y)$, $\lim \limits _{y\to 0}\lim \limits _{x\to 0}u(x;y)$ and $\lim \limits _{(x;y)\to (0;0)}u(x;y)$ if $u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}$.
2. Find the differential of a function: (a)~$u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)$; \; (b)~$u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}$.
3. Find the differential of $u(x;y)$ given implicitly by an equation $x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0$ at points $M(1;1;1)$ and $N(1;1;-2)$.
4. Find maxima and minima of a function subject to a constraint (or several constraints): \begin{enumerate}
5. $u=x^{2}y^{3}z^{4}$, \quad $2x+3y+4z=18$, $x>0$, $y>0$, $z>0$;
6. $u=x-y+2z$, \quad $x^{2}+y^{2}+2z^{2}=16$;
7. $u=\sum \limits _{i=1}^{k}a_{i}x_{i}^{2}$, \quad $\sum \limits _{i=1}^{k}x_{i}=1$, $a_{i}>0$;

Typical questions for seminar classes (labs) within this section

1. Let us consider $u(x;y)={\begin{cases}1,&x=y^{2},\\0,&x\neq y^{2}.\end{cases}}$ Show that this function has a limit at the origin along any straight line that passes through it (and all these limits are equal to each other), yet this function does not have limit as $(x;y)\to (0;0)$.
2. Find the largest possible value of directional derivative at point $M(1;-2;-3)$ of function $f=\ln xyz$.
3. Find maxima and minima of functions $u(x,y)$ given implicitly by the equations: \begin{enumerate}
4. $x^{2}+y^{2}+u^{2}-4x-6y-4u+8=0$, $u>2$;
5. $x^{3}-y^{2}+u^{2}-3x+4y+u-8=0$.

Test questions for final assessment in this section

1. Find all points where the differential of a function $f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}$ is equal to zero.
2. Show that function $\varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)$ satisfies the equation $2xz\varphi _{x}+2yz\varphi _{y}+\left(2x^{2}+y\right)\varphi _{z}=0$.
3. Find maxima and minima of function $u=2x^{2}+12xy+y^{2}$ under condition that $x^{2}+4y^{2}=25$. Find the maximum and minimum value of a function
4. $u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}$ on a domain given by inequality \quad $x^{2}+y^{2}\leq 4$;

Section 5

Section title

Integration of Functions of Several Variables

Topics covered in this section

• Z-test
• Double integrals. Fubini's theorem and iterated integrals
• Substituting variables in double integrals. Polar coordinates
• Triple integrals. Use of Fubini's theorem
• Spherical and cylindrical coordinates
• Path integrals
• Area of a surface
• Surface integrals

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	0

Typical questions for ongoing performance evaluation within this section

1. Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: $\iint \limits _{D}f(x;y)\,dx\,dy$ where $D=\left\{(x;y)\left|x^{2}+y^{2}\leq 9,\,x^{2}+(y+4)^{2}\geq 25\right.\right\}$.
2. Represent integral $I=\displaystyle \iiint \limits _{D}f(x;y;z)\,dx\,dy\,dz$ as iterated integrals with all possible (i.e. 6) orders of integration; $D$ is bounded by $x=0$, $x=a$, $y=0$, $y={\sqrt {ax}}$, $z=0$, $z=x+y$.
3. Find line integrals of a scalar fields $\displaystyle \int \limits _{\Gamma }(x+y)\,ds$ where $\Gamma$ is boundary of a triangle with vertices $(0;0)$, $(1;0)$ and $(0;1)$.

Typical questions for seminar classes (labs) within this section

1. Change order of integration in the iterated integral $\int \limits _{0}^{\sqrt {2}}dy\int \limits _{y}^{\sqrt {4-y^{2}}}f(x;y)\,dx$.
2. Find the volume of a solid given by $0\leq z\leq x^{2}$, $x+y\leq 5$, $x-2y\geq 2$, $y\geq 0$.
3. Change into polar coordinates and rewrite the integral as a single integral: $\displaystyle \iint \limits _{G}f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx\,dy$, $G=\left\{(x;y)\left|x^{2}+y^{2}\leq x;\,x^{2}+y^{2}\leq y\right.\right\}$.
4. Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at $A$ and finishes at $B$: $\displaystyle \int \limits _{\Gamma }\left(x^{4}+4xy^{3}\right)\,dx+\left(6x^{2}y^{2}-5y^{4}\right)\,dy$, $A(-2;-1)$, $B(0;3)$;

Test questions for final assessment in this section

1. Domain $G$ is bounded by lines $y=2x$, $y=x$ and $y=2$. Rewrite integral $\iint \limits _{G}f(x)\,dx\,dy$ as a single integral.
2. Represent the integral $\displaystyle \iint \limits _{G}f(x;y)\,dx\,dy$ as iterated integrals with different order of integration in polar coordinates if $G=\left\{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};\,|x|-y\geq 0\right.\right\}$.
3. Find the integral making an appropriate substitution: $\displaystyle \iiint \limits _{G}\left(x^{2}-y^{2}\right)\left(z+x^{2}-y^{2}\right)\,dx\,dy\,dz$, $G=\left\{(x;y;z)\left|x-1<y<x;\,1-x<y<2-x;\,1-x^{2}+y^{2}<z<y^{2}-x^{2}+2x\right.\right\}$.
4. Use divergence theorem to find the following integrals $\displaystyle \iint \limits _{S}(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy$ where $S$ is the outer surface of a tetrahedron ${\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1$, $x\geq 0$, $y\geq 0$, $z\geq 0$;

Section 6

Section title

Uniform Convergence of Functional Series. Fourier Series

Topics covered in this section

• Uniform and point wise convergence of functional series
• Properties of uniformly convergent series
• Fourier series. Sufficient conditions of convergence and uniform convergence
• Bessel's inequality and Parseval's identity.

What forms of evaluation were used to test students’ performance in this section?

Form	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	0
Testing (written or computer based)	1
Reports	0
Essays	0
Oral polls	0
Discussions	0

Typical questions for ongoing performance evaluation within this section

1. Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. $\sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}$, $\Delta _{1}=(0;1]$, $\Delta _{2}=[1;+\infty )$;
2. $\sum \limits _{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}$, $\Delta _{1}=(0;1)$, $\Delta _{2}=(1;+\infty )$
3. Show that sequence $f_{n}(x)=nx\left(1-x\right)^{n}$ converges non-uniformly on $[0;1]$ to a continuous function $f(x)$, but $\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx=\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx$.
4. Decompose the following function determined on $[-\pi ;\pi ]$ into Fourier series using the standard trigonometric system $\left.\left\{1;\cos kx;\sin kx\right\}\right|_{k=1}^{\infty }$. Draw the graph of the sum of Fourier series obtained. $f(x)={\begin{cases}1,\;0\leq x\leq \pi ,\\0,\;-\pi \leq x<0.\end{cases}}$
5. Prove that if for an absolutely integrable function $f(x)$ on $[-\pi ;\pi ]$ \begin{enumerate}
6. $f(x+\pi )=f(x)$ then $a_{2k-1}=b_{2k-1}=0$, $k\in \mathbb {N}$;
7. $f(x+\pi )=-f(x)$ then $a_{0}=0$, $a_{2k}=b_{2k}=0$, $k\in \mathbb {N}$.

Typical questions for seminar classes (labs) within this section

1. Show that sequence $f_{n}(x)=nx\left(1-x^{2}\right)^{n}$ converges on $[0;1]$ to a continuous function $f(x)$, and at that $\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx\neq \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx$.
2. Show that sequence $f_{n}(x)=x^{3}+{\frac {1}{n}}\sin \left(nx+{\frac {n\pi }{2}}\right)$ converges uniformly on $\mathbb {R}$, but $\left(\lim \limits _{n\rightarrow +\infty }f_{n}(x)\right)'\neq \lim \limits _{n\rightarrow +\infty }f'_{n}(x)$.
3. Decompose $\cos \alpha x$, $\alpha \notin \mathbb {Z}$ into Fourier series on $[-\pi ;\pi ]$. Using this decomposition prove that $\cot y={\frac {1}{y}}+\sum \limits _{k=1}^{\infty }{\frac {2y}{y^{2}-\pi ^{2}k^{2}}}$.
4. Function $f(x)$ is absolutely integrable on $[0;\pi ]$, and $f(\pi -x)=f(x)$. Prove that \begin{enumerate}
5. if it is decomposed into Fourier series of sines then $b_{2k}=0$, $k\in \mathbb {N}$;
6. if it is decomposed into Fourier series of cosines then $a_{2k-1}=0$, $k\in \mathbb {N}$.

Test questions for final assessment in this section

1. Find out whether the following functional series converge uniformly on the indicated intervals. Justify your answer. $\sum \limits _{n=1}^{\infty }{\frac {xn+{\sqrt {n}}}{n+x}}\ln \left(1+{\frac {x}{n{\sqrt {n}}}}\right)$, $\Delta _{1}=(0;1)$, $\Delta _{2}=(1;+\infty )$;
2. Show that sequence $f_{n}(x)={\frac {\sin nx}{\sqrt {n}}}$ converges uniformly on $\mathbb {R}$ to a differentiable function $f(x)$, and at that $\lim \limits _{n\rightarrow +\infty }f'_{n}(0)\neq f'(0)$.