Mathematics I, 2024-2025, Winter semester
Link to the course in teams (currently as of October 2, 2024, it is accessible only for students enrolled in hybrid form)
Midterm: 7 November 2024
Exams:
written: 15.1.2025, 22.1.2025, 29.1.2025,
oral: 16.1.2025, 23.1.2025, 30.1.2025.
Lectures Week 01 (Wednesday 2.10.2024, 9:30-10:50, 11:00-12:20)
Presentation, we did pages 1-19 out of 50, or 1-58 out of 157. We proved one of de Morgan’s law. We considered truth tables for negation, conjunction, disjunction, implication. We stoped just before “Methods of proof” (was not considered).
Superseminar Week 01 (Thursday 3.10.2024, 9:30-10:50)
We did 1, 2, 3, one of properties in 5, 6, partially 7.
Lectures Week 02 (Wednesday 9.10.2024, 9:30-10:50, 11:00-12:20)
Same Presentation as previous week 1, pages 20-36 out of 50. We considered methods of proof. We proved Cauchy-Schwartz inequality (as an example of direct proof), also the fact that \(\sqrt{2}\) is irrational (as an example of proof by contradiction). We introduced real numbers by stating axioms that the set of real numbers must satisfy. We then introduced natural, integer and rational numbers. We deduced from the axioms some basic properties of real numbers.
We formulated some further theorems about real numbers:
- existence of integer partially
- Archimedean property
- existence of n-th root for a positive numbers
- density of rational and irrational numbers in \(\mathbb{R}\)
Link to recording in Teams part 1
Link to recording in Teams part 2
Superseminar Week 02 (Thursday 10.10.2024, 9:30-10:50)
Link to problems and solutions in moodle.
Lectures Week 03 (Wednesday 16.10.2024, 9:30-10:50, 11:00-12:20)
Presentation part 1: updated version of presentation from previous weeks 1, 2,
pages 34-37 out of 53, or 102-114 out of 114 (proof of existence of integer part. Other theorems there without proof).
Presentation part 2: sequences. We went through pages 1-25 out of 44, or 1-74 out of 144. We looked at the trick of adding and subtraction, in the proof of arithmetics of limits for the product of sequences. We finished before sandwich theorem (not including).
We skipped exercise on page 9/44.
Superseminar Week 03 (Thursday 17.10.2024, 9:30-10:50)
We did problems 1-6. We considered scale of growth of some sequences, solved problems on the use of the scale and use of arithmetics of limits, and also problems involving difference of roots.
Lectures Week 04 (Wednesday 23.10.2024, 9:30-10:50, 11:00-12:20)
Presentation part 1: Sequences: sequences. We went through pages 26-44 out of 44, or 75-144 out of 144.
We started a new topic Functions: presentation part 2. We went through pages 1-23 out of 47, or 1-14 out of 27.
Link to recording in Teams part 1
Link to recording in Teams part 2
Link to recording in Teams part 3
Superseminar Week 04 (Thursday 24.10.2024, 9:30-10:50)
We did problems 1, 2, 3. We also looked at a sample for the midterm test.
Lectures Week 05 (Wednesday 30.10.2024, 9:30-10:50, 11:00-12:20)
Previous file Functions, first part. We went through pages 15-27 out of 27.
Second part: Functions: second part. We went through pages 1-30 out of 43.
Superseminar Week 05 (Thursday 31.10.2024, 9:30-10:50)
Same file as last week: Problems
We did problems 4, 5, 14. We also looked at a sample for the midterm test.
We started the topic “limits of functions”, we computed the following limits:
- \[\lim\limits_{x\to1}\frac{x^3 + x^2}{x+10}\]
- \[\lim\limits_{x\to1}\frac{x^2-1}{x-1}\]
- \[\lim\limits_{x\to1}\frac{x^2-1}{x^3-1}\]
- \[\lim\limits_{x\to1}\frac{x^2-1}{2x^2-x-1}\]
Lectures Week 06 (Wednesday 6.11.2024, 9:30-10:50, 11:00-12:20)
Updated file Functions 2. We went through pages 30, 35, 36-47, 49 out of 49. We proved Bolzano intermediate value theorem.
Superseminar Week 06 (Thursday 7.11.2024, 9:30-10:50)
Midterm
Lectures Week 07 (Wednesday 13.11.2024, 9:30-10:50, 11:00-12:20)
File Derivatives. We went through pages 1-25 out of 154. We started with the inequality between geometric and arithmetic means; we finished before derivative of an inverse function.
Superseminar Week 07 (Thursday 14.11.2024)
- \[\lim\limits_{x\to0}\dfrac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{x^3}\]
- \[\lim\limits_{x\to 0}\dfrac{\log(\cos x)}{x^2}\]
- \[\lim\limits_{x\to\infty}\left(\dfrac{x+2}{x-1}\right)^x\]
- \[\lim\limits_{x\to \frac{\pi}{4}}\left(\tan x\right)^{\tan2x}\]
- \[\lim\limits_{x\to 1}\left(1 + \sin \pi x\right)^{\cot \pi x}\]
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\[\lim\limits_{x\to \infty}x\left(\dfrac{\pi}{4} - \arctan\dfrac{x}{x+1}\right)\]
Hint: substitution \(y = \dfrac{1}{x},\) then \(u = \dfrac{\pi}{4} - \arctan\dfrac{1}{1+y},\) then use trigonometric manipulations for difference of cotangents. Answer: \(\dfrac12.\)
Lectures Week 08 (Wednesday 20.11.2024, 9:30-10:50, 11:00-12:20)
Substituted by Tomáš Bárta.
File Derivatives. Till Theorem 9 (page 34 out of 83).
Superseminar Week 08 (Thursday 21.11.2024)
Find derivatives:
- \[\log\left(\cos\left(\arctan\left(\sin\,2x\right)\right)\right)\]
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\[\dfrac{1+x^2}{\sqrt[3]{x^4}\,\sin^7x}\]
through fraction and product rule or via logarithmic derivative
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\[(2+\cos x)^x\]
rewrite as exponent
-
\[x^{2^x}\]
is \(x^{\left(2^x\right)}\), not to confuse with \(\left(x^{2}\right)^x = x^{2x}\)
- \[\sqrt[13]{9+7\sqrt[5]{2x}}\]
- find \(f'(-3)\) if
Superseminar Week 09 (Thursday 28.11.2024)
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Find intervals of monotonicity, local and global extrema:
- \[y = 4x^3-21x^2 + 18x + 7\]
- \[y = 8x^3 - x^4\]
- \[y = x^5 - 5x^4 + 5x^3 - 1\]
- \[y = (x-1)^3(2x+3)^2\]
- \[y = xe^{-3x}\]
- \[y = x^2 - 10\log x\]
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Find intervals where the function is convex, concave, and inflexion points:
- \[y = 2x^4 - 3x^2 + x - 1\]
- \[y = x^5 - 10x^2 + 3x\]
- \[y = \frac{x}{1-x^2}\]
- \[y = \frac{x}{\sqrt[3]{x^2-1}}\]
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Find asymptotes (vertical and slant):
- \[y = \frac{x^3}{4(2-x)^2}\]
- \[y = \frac{x^3}{x^2 - 1}\]
- \[y = \frac{4x^2 + 2x}{3x-1}\]
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Plot the graph of a function:
- \[y = \frac{x^3}{x^2-1}\]
Lectures Week 10 (Wednesday 4.12.2024, 9:30-10:50, 11:00-12:20)
File Derivatives (updated version).
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p. 35/84. Computation of one-sided derivative, proof.
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pp. 36-39. L’Hopital rule, proof in the case of \(a\in\mathbb{R},\) \(\lim\limits_{x\to a}f(x) = \lim\limits_{x\to a}g(x) = 0.\)
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pp. 39-41: L’Hopital rule, proof in the case of \(\lim\limits_{x\to a}g(x) =\pm\infty:\) a question for extra points in exam.
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pp.42-57: convex, concave functions, points of inflection. Till Sufficient condition for inflection.
Superseminar Week 10 (Thursday 5.12.2024)
Investigate behaviour of a function and plot its graph:
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\[y = \sqrt{\dfrac{x^3-2x^2}{x-3}}.\]
Hint: \(y'(x) = \sqrt{\dfrac{x-2}{x-3}}\cdot\dfrac{(x-4)(2x-3)}{2(x-2)(x-3)}\cdot \mathrm{sgn}(x),\) \(y''(x) = \sqrt{\dfrac{x-2}{x-3}}\cdot\dfrac{11x-24}{4(x-2)^2(x-3)^2}\cdot \mathrm{sgn}(x).\)
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\[y = x\sqrt[3]{(x+1)^2}.\]
Hint: \(y'(x) = \dfrac{\frac53\left(x+\frac35\right)}{\sqrt[3]{x+1}},\) \(y''(x) = \dfrac{\frac{10}{9}\left(x+\frac65\right)}{\sqrt[3]{(x+1)^4}}.\)
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\[y = \dfrac{x^2-4}{x}\exp\left({\frac{-5}{3x}}\right).\]
Hint: \(y'(x) = \dfrac{(x-1)(3x^2+8x+20)}{3x^3}\exp\left({\frac{-5}{3x}}\right),\) \(y''(x) = \dfrac{-\left(47x^2-240x+100\right)}{9x^5}\exp\left({\frac{-5}{3x}}\right).\)
We did the first task.