This document provides an overview of the calculus concepts covered in school and in various university courses at the Electrotechnical University “LETI” in Saint Petersburg, Russia. It outlines the key competencies developed in functions, sequences, series, logarithmic/exponential functions, rates of change, differentiation, integration, and other topics. The levels of mastery increase across the core courses in Calculus, Computing Mathematics, and some additional advanced topics covered in only two specialized groups.

1. Calculus. Saint Petersburg
Electrotechnical University “LETI”
Andrey Kolpakov
December, 2014

2. Core 0
Functions and their inverses
In School:
• define a function, its domain and its range
• use the notation f(x)
• determine the domain and range of simple functions
• relate a pictorial representation of a function to its graph and to its algebraic definition
• understand how a graphical translation can alter a functional description
• understand how a reflection in either axis can alter a functional description
• understand how a scaling transformation can alter a functional description
• determine the domain and range of simple composite functions
• recognize the properties of the function 1/x
In Course of Calculus (1st semester):
• determine whether a function is injective, surjective, bijective
• obtain the inverse of a function by a pictorial representation, graphically or algebraically
• determine the domain and range of the inverse of a function
• determine any restrictions on f(x) for the inverse to be a function
• obtain the inverse of a composite function
• understand the concept of the limit of a function
In Course of Computing Mathematics (4th semester):
• use appropriate software to plot the graph of a function

3. Core 0
Sequences, series, binomial expansions
In School:
• recognize an arithmetic progression and its component parts
• find the general term of an arithmetic progression
• find the sum of an arithmetic series
• recognize a geometric progression and its component parts
• find the general term of a geometric progression
• find the sum of a finite geometric series
• interpret the term 'sum' in relation to an infinite geometric series
• find the sum of an infinite geometric series when it exists
• find the arithmetic mean of two numbers
• find the geometric mean of two numbers
In Course of Calculus (1st semester):
• define a sequence
• obtain the binomial expansions of (a+b)^s, (1+x)^s for s a rational number
In Course of Calculus (3rd semester):
• define a series and distinguish between a sequence and a series

4. Core 0
Logarithmic and exponential functions
In School:
• recognize the graphs of the power law function
• define the exponential function and sketch its graph
• define the logarithmic function as the inverse of the exponential function
• use the laws of logarithms to simplify expressions
• solve equations involving exponential and logarithmic functions
• In course of calculus (1st semester):
• use the binomial expansion to obtain approximations to simple rational
functions

5. Core 0
Rates of change and differentiation
In School:
• solve problems using growth and decay models
• define average and instantaneous rates of change of a function
Both in School and in Course of Calculus (1st semester):
• understand how the derivative of a function at a point is defined
• recognize the derivative of a function as the instantaneous rate of change
• interpret the derivative as the gradient at a point on a graph
• use the notations dy/dx, f'(x), y'(x) etc.
• use a table of the derived functions of simple functions
• recall the derived function of each of the standard functions
• use the multiple, sum, product and quotient rules
• use the chain rule
• relate the derivative of a function to the gradient of a tangent to its graph
• obtain the equation of the tangent and normal to the graph of a function
In Course of Calculus (3rd semester):
• distinguish between 'derivative' and 'derived function'

6. Core 0
Stationary points, maximum and minimum values
Both in School and in Course of Calculus (1st semester):
• use the derived function to find where a function is increasing
or decreasing
• define a stationary point of a function
• distinguish between a turning point and a stationary point
• locate a turning point using the first derivative of a function
• classify turning points using first derivatives
• obtain the second derived function of simple functions
• classify stationary points using second derivatives

7. Core 0
Indefinite integration
Both in School and in Course of Calculus (2nd semester):
• reverse the process of differentiation to obtain an indefinite
integral for simple functions
• understand the role of the arbitrary constant
• use a table of indefinite integrals of simple functions
• understand and use the notation for indefinite integrals
• use the constant multiple rule and the sum rule
• use indefinite integration to solve practical problems such as
obtaining velocity from a formula for acceleration or
displacement from a formula for velocity

8. Core 0
Definite integration, applications to areas and
volumes
Both in School and in Course of Calculus (2nd semester):
• understand the idea of a definite integral as the limit of a sum
• realize the importance of the Fundamental Theorem of the Calculus
• obtain definite integrals of simple functions
• use the main properties of definite integrals
• calculate the area under a graph and recognize the meaning of a negative
value
• calculate the area between to curves
In course of calculus (2nd semester):
• calculate the volume of a solid of revolution
In course of Computing Math (4th semester):
• use trapezium and Simpson's rules to approximate the value of a definite
integral

9. Core 0
Complex numbers
In course of Algebra (1st semester):
• define a complex number and indentify its component parts
• represent a complex number on an Argand diagram
• carry out the operations of addition and subtraction
• write down the conjugate of a complex number and represent
it graphically
• identify the modulus and argument of a complex number
• carry out the operations of multiplication and division in both
Cartesian and polar form
• solve equations of the form z^n = a, where a is a real number

10. Core 0
Proof
In School:
• understand how a theorem is deduced from a
set of assumptions
• appreciate how a corollary is developed from
a theorem
• follow a proof of Pythagoras' theorem
• follow proofs of theorems for example, the
concurrency of lines related to triangles
and/or the equality of angles related to circles

11. Level 1
Almost all competences of all contents are studied in LETI’s
courses (not in School) :
- Algebra, 1st semester (Complex numbers, Rational functions)
- Calculus, 1st semester (Hyperbolic functions, Convex ,
Differentiation)
- Calculus, 2nd semester (Methods of integration, Applications
of integration)
- Calculus, 3rd semester (Series, Functions of 2 variables)
- Computing Mathematics, 4th semester (Solution of non-linear
equations )

12. Level 2
Almost all competences of contents:
• Ordinary differential equations
• First order ordinary differential equations
• Laplace transforms
• Second order equations - complementary function and particular integral
are studied in Course of Calculus (2nd semester)
(another competences are not studied)
Almost all competences of contents:
• Functions of several variables
• Fourier series
• Double integrals
• Further multiple integrals
• Vector calculus
• Line integrals
are studied in Course of Calculus (3rd semester)
(another competences are not studied)

13. Level 2
Such contents as
- Surface integrals, integral theorems
- Linear optimization
- The simplex method
- z transforms
are not studied in LETI
Only 2 groups study
- Complex functions
- Complex series and contour integration
- Introduction to partial differential equations
- Solving partial differntial equations

14. Level 3
In Course of Computing Math (4th semester):
- Numerical solutions of ordinary differential equations
Another contents of level 3 are not studied in LETI
(only 2 groups study)

15. ありがとう
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