Calculus 1

✩ Language: English

✩ Place: Czech Technical University in Prague, Prague, Czech Republic

✩ Category: Theory + Labs

✩ Credits: 7 credits

✩ Classes: Tuesday 9.15 – 10.45 (Lecture) ★ Wednesday 9.15 – 10.45 (Lecture) ★ Wednesday 11.00 – 12.30 (Practice)

✩ When: Winter Term 2026 – 2027 (September 21 – January 10, 2027)

✩ Syllabus

• Lectures
  • The real line, elementary functions and their graphs, shifting and scaling.
  • Limits and continuity, tangent, velocity, rate of change. 
  • Derivative of functions, properties and applications.
  • Mean value theorem, L’Hospital’s rule. 
  • Higher derivatives, Taylor polynomial.
  • Local and global extrema, graphing of functions.
  • Indefinite integral, basic integration methods.
  • Integration of rational functions, more techniques of integration.
  • Definite integral, definition and properties, Fundamental Theorems of Calculus.
  • Improper integrals, tests for convergence. Mean value Theorem for integrals, applications.
  • Sequences of real numbers, numerical series, tests for convergence.
  • Power series, uniform convergence, the Weierstrass test.
  • Taylor and Maclaurin series.
• Tutorials
  • The real line, elementary functions and their graphs, shifting and scaling.
  • Limits and continuity, tangent, velocity, rate of change.
  • Derivative of functions, properties and applications.
  • Mean value theorem, L’Hospital’s rule. 
  • Higher derivatives, Taylor polynomial.
  • Local and global extrema, graphing of functions.
  • Indefinite integral, basic integration methods.
  • Integration of rational functions, more techniques of integration.
  • Definite integral, definition and properties, Fundamental Theorems of Calculus.
  • Improper integrals, tests for convergence. Mean value Theorem for integrals, applications.
  • Sequences of real numbers, numerical series, tests for convergence.
  • Power series, uniform convergence, the Weierstrass test. 
  • Taylor and Maclaurin series.

✩ Evaluation system

★Lectures: attendance is not mandatory, but highly recommended. 

★Practice: attendance is mandatory, 2 absences are tolerated.

★Midterms: During the labs of Week 6 and Week 11 – 45 minutes and 3 problems

★Homeworks: 10 accepted homeworks is one of the requirements for zápočet (assessment).

★Zápočet (assesment): In order to obtain zápočet (needed for the final exam), students are required to actively participate in the labs, hand in the assigned homeworks and obtain a sufficient score (at least 9 points out of 20) in the midterms. 

★Final exam: it is composed of written and oral part. The written final exam will be in January-February (the dates will be announced later), it will consist of six problems (one about theory) to be solved in 90 minutes. The oral final exam is optional, it is used to improve the grade up to ten points. Questions about theory will be asked (definitions, theorems, proofs). In order to pass the exam a minimum of 50 points in the written test is required, students with more then 60 points in the written part of the exam will be allowed to improve their grade with the oral part of the exam.

★Grading: the grade is calculated as follows (this is the grade from the final exam only!):

➤0 – 49 points: F

➤ 50 – 59 points: E

➤ 60 – 69 points: D

➤ 70 – 79 points: C 

➤ 80 – 89 points: B

➤ 90 – 100 points: A

❖ You have at most 3 attempts to pass the final exam, each is entirely independent.

❖ Repeating the exam is obvious in case you fail, but you can also refuse a grade when you are not happy with it, then you are assigned the grade F and you can try again (if you still have attempts left).

❖ The midterm and the final test are subject to the following rules: No textbooks, no notes, no calculators, no cellphones are allowed. 

★ (week 1) The real line, elementary functions and their graphs, shifting and scaling.

★ (week 2) Limits and continuity, tangent, velocity, rate of change.

★ (week 3) Derivative of functions, properties and applications.

★ (week 4) Mean value theorem, L’Hospital’s rule.

★ (week 5) Higher derivatives, Taylor polynomial.

★ (week 6) Local and global extrema, graphing of functions.

★ (week 7) Indefinite integral, basic integration methods.

★ (week 8) Integration of rational functions, more techniques of integration.

★ (week 9) Definite integral, definition and properties, Fundamental Theorems of Calculus.

★ (week 10) Improper integrals, tests for convergence. Mean value Theorem for integrals, applications.

★ (week 11) Sequences of real numbers, numerical series, tests for convergence.

★ (week 12) Power series, uniform convergence, the Weierstrass test.

★ (week 13) Taylor and Maclaurin series.

★ Week 1 (September 22nd and September 23rd)

✩  Presentation of the course

✩ The real line and inequalities

✩  Functions of one real variable

☛ Notes:

#1 September 22nd

#2 September 23rd

☛ Slides: 

#0 Presentation of the course

#1 September 22nd

#2 September 23rd

***

★ Week 2 (September 29th and September 30th)

✩  Limits

✩ Continuity

 ☛ Notes:

#3 September 29th

#4 September 30th

☛ Slides: 

#3 September 29th

#4 September 30th

***

★ Week 3 (October 6th and October 7th)

✩  Derivatives

✩ First definitions and rules

 ☛ Notes:

#5 October 6th

#6 October 7th

☛ Slides: 

#5 October 6th

#6 October 7th

***

★ Week 4 (October 13th and October 14th)

✩  Mean value theorems

✩ L’Hospital’s Rule

 ☛ Notes:

#7 October 13th

#8 October 14th

☛ Slides: 

#7 October 13th

#8 October 14th

***

★ Week 5 (October 20th and October 21st)

✩  Extrema

✩ Drawing a graph of a function

 ☛ Notes:

#9 October 20th

#10 October 21st

☛ Slides: 

#9 October 20th

#10 October 21st

***

★ Week 6 (October 27th and October 28th)

✩ October 27th: Midterm 1 (up to Week 5, Week 5 included)

✩ October 28th: public holiday

***

★ Week 1 – September 23rd – Slides – Elementary functions

★ Week 2 – September 30th – Slides – Limits and Continuity (Group activity)

★ Week 3 – October 7th – Slides – Derivatives (Group activity – Solutions of the group activity – Scores)

★ Week 4 – October 14th – Slides – Mean value theorems and L’Hopital Rules (Group activity)

★ Week 5 – October 21st – Slides – Extrema and drawing graphs

★ Week 6 – October 28th – public holiday

Every week there will be a different homework. Ten of them should be accepted for započet.

★ Homework #1 (deadline: Sep 29th at 23.59) – Elementary functions

★ Homework #2 (deadline: Oct 6th at 23.59) – Limits and Continuity

★ Homework #3 (deadline: Oct 13th at 23.59) – Derivatives

★ Homework #4 (deadline: Oct 19th at 23.59) – L’Hopital

★ Homework #5 (no yet! Wait till next week!) – Drawing graphs

The exercises are separated by topic. Please take into account that in the exercises might appear some terminology that we haven’t seen in class (for instance, removable discontinuity). The student, therefore, should look it up and learn these concepts by themselves.

★ Background

★ Limits and Continuity

★ Differentiability

The exercises are separated by week. Please take into account that in the exercises might appear some terminology that we haven’t seen in class (for instance, removable discontinuity). The student, therefore, should look it up and learn these concepts by themselves. For the videos, you should select the quality you want but it should be HD.

★ Week 1: Inequalities and functions (video)

★ Week 2: Limits and continuity (video)

★ Week 3: Continuity and derivatives (video)

★ Week 4: Mean value theorems and L’Hopital (video)

★ Week 5: Extrema and drawing graphs (video)

Midterm 1: Tuesday, October 27th, 2026 – 9.15am – 10.00am – Dejvice (up to and including Week 5)
Midterm 2: Wednesday, December 2nd, 2026 – 11.00am – 11.45am – Dejvice (up to and including Week 10)

★ Midterm 1: 10 points (template)
★ Midterm 2: 10 points
★ Minimum required for započet : any combination that gives ≥ 9 points.
★ The midterms will consist of 3 problems that should verify the knowledge of basics Calculus theory.
★ Take a look at the sections “Exercises” and “Solved Exercises”.

The final exams will have the following dates.

Final 1: Month Day

Final 2: Month Day

Final 3: Month Day

***

Here is a sample.

Homeworks and extra points (updated on Oct 13th at 22.34)

Midterm 1 – Day, Month Day at XX.XX (Dejvice)

Midterm 2 – Day, Month Day at XX.XX (Dejvice)

Final exam #1 (Early bird exam) – Day, Month Day at X.XX a.m. (Dejvice)

Final exam #2

Final exam #3

Final exam #4

Final exam #5

In each midterm there will be a true or false question. Practice here!

WEEK 1

1. If A is a subset of ℝ that is bounded above and sup(A) ∈ A, then max(A) = sup(A).
2. If f: A → B is injective, then its inverse function exists.
3. Since the sine function is odd, the function f(x) = sin(|x|) is also odd.
4. The domain of f ∘ g is always Dom(f) ∩ Dom(g).
5. The graph of y = f(x + 3) is obtained by shifting the graph of y = f(x) three units to the right.

WEEK 2

1. Suppose that f(x) → 0 as x → a and that g is bounded near a. Then f(x)g(x) → 0 as x → a.
2. If the limit of f(x) as x → a is L, then f(a) must be equal to L.
3. If the left- and right-hand limits of f at a are both L, then the limit of f(x) as x → a is L, even if f(a) is not equal to L.
4. Suppose that f is continuous on [a,b] and f(a) and f(b) have the same sign. Then f has no zeros in (a,b).
5. Every continuous function defined on the open interval (0,1) attains both a maximum and a minimum.

WEEK 3

1. If f is differentiable at a and f′(a) = 0, then f has a local maximum or minimum at a.
2. If f and g are differentiable at a and (fg)′(a) = 0, then f′(a) = g′(a) = 0.
3. If f is differentiable and one-to-one, then its inverse f⁻¹ must be differentiable everywhere on its domain.
4. Suppose that f(a) = 0. If |f| is differentiable at a, then f is differentiable at a and f′(a) = 0.
5. The formula y′ = −x/y is not defined at (1,0). Therefore, the circle x² + y² = 1 has no tangent line at that point.

WEEK 4

1. Suppose that the equation f(x) = 0 has at least three distinct solutions. Then the equation f′(x) = 0 has at least two distinct solutions.
2. Suppose that f is differentiable on [0,1], f(0) = f(1) = 0 and the equation f′(x) = 0 has exactly one solution c in (0,1). Then c must be a local maximum.
3. Suppose that f(x) → 0 and g(x) → 0 as x → a. If f′(x)/g′(x) has no limit, then f(x)/g(x) also has no limit.
4. Suppose that f(x) → 0 and g(x) → 0 as x → 0. If lim as x → 0 of f(x)/g(x) = L, then lim as x → 0 of f′(x)/g′(x) = L.
5. Suppose that f and g are differentiable on an interval and that f′(x) > g′(x) at every point. Then f(x) > g(x) at every point of the interval.

WEEK 5

1. Let f be differentiable on ℝ. If f(x) → +∞ as x → ±∞, then f has an absolute minimum.
2. If f has a local maximum at x = c and f″(c) exists, then f″(c) < 0.
3. Suppose that f′(x) > 0 to the left of c and f′(x) < 0 to the right of c. Then f must have a local maximum at c.
4. If the line y = mx + n is an asymptote of a function, then the graph cannot cross this line.
5. If f has a local minimum at x = a and another local minimum at x = b, where a < b, then there is at least one critical point between a and b.

CRACK IT!

★ CRACK IT – WEEK 1 ★

Let f,g: ℝ → ℝ. Suppose that f ∘ g is injective and g ∘ f is surjective. Are both f and g bijective?

★ CRACK IT – WEEK 2 ★

Let f: [0,1] → ℝ be continuous and suppose that f(x) ∈ ℚ for every x ∈ [0,1]. Must f be constant?

★ CRACK IT – WEEK 3 ★

Consider the function f(x) = x² sin(1/x) if x ≠ 0 and f(0) = 0. Is f differentiable at 0? Is f′ continuous at 0?

★ CRACK IT – WEEK 4 ★

Let f: ℝ → ℝ be differentiable and suppose that f(x + 1) = f(x) + 1 for every real number x. Must the equation f′(x) = 1 have infinitely many solutions?

★ CRACK IT – WEEK 5 ★

Let f be differentiable on ℝ and suppose that f has exactly two critical numbers. Then at least one of them must be a local maximum or a local minimum.