Mathematics MA1131
Textbook
A textbook you could refer to (one that contains many more details than we will cover) is Thomas' calculus (by Maurice D. Weir, Joel Hass and Frank R. Giodano), published by Pearson/Addison Wesley, [Hamilton 515.1 K82*10;35 ]. The one in the library is a version from 2005.
Here is a guide to the relevant sections of the book, although we certainly have not covered all of the material in the sections listed.
- Lecture 1, October 1
- Introductory remarks, some notation, functions, beginning of limits.
(Thomas: 1.1 Real numbers and the real line; 1.3 Functions and their graphs; 1.5 Combining functions; shifting and scaling graphs; 2.1 Rates of change and limits) - Lecture 2, October 2
- Limits of polynomials and rational functions. Definition of derivative.
(2.1 Rates of change and limits; 2.2 Calculating limits using the lmit laws) - Lecture 3, October 8
- Interpretations of the derivative; rules for differentiation.
(2.7 The derivative as a function; 3.8 Linearization adn differentials; 3.1 Differentiation rules; 3.5 The chain rule [and parametric equations]) - Lecture 4, October 9
- Implicit differentiation; rational powers; trigonometric functions;
sign of the derivative and monotone functions.
(3.6 Implicit differentiation; 3.4 Differentiation of trigonometric functions; 4.3 Monotonic functions and the first derivative test) - Lecture 5, October 15
- Exponential function;
inverse functions.
(7.3 The exponential function; 7.7 Exponential growth and decay; 7.1 Inverse functions and their derivatives) - Lecture 6, October 16
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The natural logarithm;
arbitrary powers of positive numbers;
Inverse trigonometric functions;
hyperbolic functions.
(7.2 Natural logarithms; 7.4 a^x and log_a x; 7.7 Inverse trigonometric functions; 7.8 Hyperbolic functions) - Lecture 7, October 22
-
Inverse hyperbolic functions.
(7.8 Hyperbolic functions) - Lecture 8, October 23
-
Starting towards functions of several variables. (First some recap on
coordinates and vectors in space; planes and lines; dot products;
cross products. Assumed known from linear algebra!) Vector valued
functions of one variable (parametric curves).
(Chapter 12 Vectors and geometry of space [12.1 - 12.5]; 13.1 Vector functions; 14.1 Functions of several variables) - Lecture 9, October 29
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Graphs of functions of two variables, partial derivatives.
(14.1 Functions of several variables; 14.3 Partial derivatives) - Lecture 10, October 30
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Directional derivatives, tangent plane to a graph (of a function
of two variables),
linear approximation for functions of two variables,
(total) derivatives.
(14.5 Directional derivatives and gradient vectors; 14.6 Tangent planes and differentials)
- Lecture 11, November 19
-
Gradient vector, its relation to directional derivatives,
a chain rule for partial derivatives, level curves and the fact
that the gradient is perpendicular to them.
(14.5 Directional derivatives and gradient vectors; 14.4 The chain rule) - Lecture 12, November 20
-
Statement of the implicit function theorem and relation to implicit
differentiation. Functions of 3 variables, level surfaces, partials,
directional derivatives, gradient vector, gradient perpendicular
to level surface, gradient and direction of fastest inrease.
(14.4 The chain rule; 14.5 Directional derivatives and gradient vectors; 14.6 Tangent planes and differentials) - Lecture 13, November 26
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Integrals in one variable, both as antiderivatives and as definite
integrals (defined by a limit of sums).
(5.5 Indefinite integrals and the substitution rule; 5.3 The definite integral) - Lecture 14, November 27
-
The fundamental theorem of integral calculus connects the two concepts
of `integral'. Continuous functions do have antiderivatives.
Start of techniques of integration (substitution).
(5.4 The fundamental theorem of calculus; 5.5 Indefinite integrals and the substitution rule; 8.1 Basic integration formulas) - Lecture 15, December 3
-
More on techniques of integration (integration by parts, trigonometric
integrals, more examples of substitutions, and partial fractions).
(8.2 Integration by parts; 8.4 Trigonometric integrals; 8.5 Trigonometric substitutions; 8.3 Integration of rational functions by partial fractions) - Lecture 16, December 4, 11 & 17
-
Double and triple integrals. How to compute them (Fubini theorem) via
iterated single integrals. Double integrals in polar coordinates.
(15.1 Double integrals; 15.4 Triple integrals in rectangular coordinates; 15.3 Double integrals in polar form; 15.7 Substututions in multiple integrals)