The focus and themes of the Introduction to Calculus course address the most important foundations for applications of mathematics in science, engineering and commerce. The course emphasises the key ideas and historical motivation for calculus, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics.
Students taking Introduction to Calculus will:
• gain familiarity with key ideas of precalculus, including the manipulation of equations and elementary functions (first two weeks),
• develop fluency with the preliminary methodology of tangents and limits, and the definition of a derivative (third week),
• develop and practice methods of differential calculus with applications (fourth week),
• develop and practice methods of the integral calculus (fifth week).
Introduction to Calculus
proposé par
Université de Sydney
Programme du cours : ce que vous apprendrez dans ce cours
Semaine
1Precalculus (Setting the scene)
This module begins by looking at the different kinds of numbers that fall on the real number line, decimal expansions and approximations, then continues with an exploration of manipulation of equations and inequalities, of sign diagrams and the use of the Cartesian plane.
10 vidéos (Total 109 min), 8 lectures, 9 quiz
10 vidéos
Real line, decimals and significant figures15 min
The Theorem of Pythagoras and properties of the square root of 211 min
Algebraic expressions, surds and approximations10 min
Equations and inequalities17 min
Sign diagrams, solution sets and intervals (Part 1)10 min
Sign diagrams, solution sets and intervals (Part 2)10 min
Coordinate systems8 min
Distance and absolute value5 min
Lines and circles in the plane14 min
8 lectures
Notes: Real line, decimals and significant figures20 min
Notes: The Theorem of Pythagoras and properties of the square root of 220 min
Notes: Algebraic expressions, surds and approximations20 min
Notes: Equations and inequalities20 min
Notes: Sign diagrams, solution sets and intervals20 min
Notes: Coordinate systems20 min
Notes: Distance and absolute value20 min
Notes: Lines and circles in the plane20 min
9 exercices pour s'entraîner
Real line, decimals and significant figures20 min
The Theorem of Pythagoras and properties of the square root of 220 min
Algebraic expressions, surds and approximations20 min
Equations and inequalities20 min
Sign diagrams, solution sets and intervals20 min
Coordinate systems20 min
Distance and absolute value20 min
Lines and circles in the plane20 min
Module 1 quizs
Semaine
2Functions (Useful and important repertoire)
This module introduces the notion of a function which captures precisely ways in which different quantities or measurements are linked together. The module covers quadratic, cubic and general power and polynomial functions; exponential and logarithmic functions; and trigonometric functions related to the mathematics of periodic behaviour. We create new functions using composition and inversion and look at how to move backwards and forwards between quantities algebraically, as well as visually, with transformations in the xy-plane.
13 vidéos (Total 142 min), 12 lectures, 13 quiz
13 vidéos
Parabolas and quadratics11 min
The quadratic formula10 min
Functions as rules, with domain, range and graph11 min
Polynomial and power functions13 min
Composite functions7 min
Inverse functions12 min
The exponential function13 min
The logarithmic function8 min
Exponential growth and decay13 min
Sine, cosine and tangent9 min
The unit circle and trigonometry16 min
Inverse circular functions11 min
12 lectures
Notes: Parabolas and quadratics20 min
Notes: The quadratic formula20 min
Notes: Functions as rules, with domain, range and graph20 min
Notes: Polynomial and power functions20 min
Notes: Composite functions20 min
Notes: Inverse functions20 min
Notes: The exponential function20 min
Notes: The logarithmic function15 min
Notes: Exponential growth and decay20 min
Notes: Sine, cosine and tangent20 min
Notes: The unit circle and trigonometry20 min
Notes: Inverse circular functions20 min
13 exercices pour s'entraîner
Parabolas and quadratics20 min
The quadratic formula20 min
Functions as rules, with domain, range and graph20 min
Polynomial and power functions20 min
Composite functions20 min
Inverse functions20 min
The exponential function20 min
The logarithmic function20 min
Exponential growth and decay20 min
Sine, cosine and tangent20 min
The unit circle and trigonometry20 min
Inverse circular functions20 min
Module 2 quizs
Semaine
3Introducing the differential calculus
This module introduces techniques of differential calculus. We look at average rates of change which become instantaneous, as time intervals become vanishingly small, leading to the notion of a derivative. We then explore techniques involving differentials that exploit tangent lines. The module introduces Leibniz notation and shows how to use it to get information easily about the derivative of a function and how to apply it.
12 vidéos (Total 132 min), 10 lectures, 11 quiz
12 vidéos
Slopes and average rates of change10 min
Displacement, velocity and acceleration11 min
Tangent lines and secants10 min
Different kinds of limits12 min
Limit laws15 min
Limits and continuity9 min
The derivative as a limit10 min
Finding derivatives from first principles14 min
Leibniz notation14 min
Differentials and applications (Part 1)13 min
Differentials and applications (Part 2)7 min
10 lectures
Notes: Slopes and average rates of change20 min
Notes: Displacement, velocity and acceleration20 min
Notes: Tangent lines and secants20 min
Notes: Different kinds of limits20 min
Notes: Limit laws20 min
Notes: Limits and continuity20 min
Notes: The derivative as a limit20 min
Notes: Finding derivatives from first principles20 min
Notes: Leibniz notation20 min
Notes: Differentials and applications20 min
11 exercices pour s'entraîner
Slopes and average rates of change20 min
Displacement, velocity and acceleration20 min
Tangent lines and secants20 min
Different kinds of limits20 min
Limit laws20 min
Limits and continuity20 min
The derivative as a limit20 min
Finding derivatives from first principles20 min
Leibniz notation20 min
Differentials and applications20 min
Module 3 quizs
Semaine
4Properties and applications of the derivative
This module continues the development of differential calculus by introducing the first and second derivatives of a function. We use sign diagrams of the first and second derivatives and from this, develop a systematic protocol for curve sketching. The module also introduces rules for finding derivatives of complicated functions built from simpler functions, using the Chain Rule, the Product Rule, and the Quotient Rule, and how to exploit information about the derivative to solve difficult optimisation problems.
14 vidéos (Total 155 min), 13 lectures, 14 quiz
14 vidéos
Increasing and decreasing functions11 min
Sign diagrams12 min
Maxima and minima12 min
Concavity and inflections10 min
Curve sketching16 min
The Chain Rule9 min
Applications of the Chain Rule14 min
The Product Rule8 min
Applications of the Product Rule9 min
The Quotient Rule8 min
Application of the Quotient Rule10 min
Optimisation12 min
The Second Derivative Test16 min
13 lectures
Notes: Increasing and decreasing funtions20 min
Notes: Sign diagrams20 min
Notes: Maxima and minima20 min
Notes: Concavity and inflections20 min
Notes: Curve sketching20 min
Notes: The Chain Rule20 min
Notes: Applications of the Chain Rule20 min
Notes: The Product Rule20 min
Notes: Applications of the Product Rule20 min
Notes: The Quotient Rule20 min
Notes: Application of the Quotient Rule20 min
Notes: Optimisation20 min
Notes: The Second Derivative Test20 min
14 exercices pour s'entraîner
Increasing and decreasing functions20 min
Sign diagrams20 min
Maxima and minima20 min
Concavity and inflections20 min
Curve sketching20 min
The Chain Rule20 min
Applications of the Chain Rule20 min
The Product Rule20 min
Applications of the Product Rule20 min
The Quotient Rule20 min
Application of the Quotient Rule20 min
Optimisation20 min
The Second Derivative Test20 min
Module 4 quizs
4.9
6 avisMeilleurs avis
par SG•Jan 11th 2019
The teacher is great and if you are beginner this course is really for you
Enseignant
David Easdown
Associate ProfessorDepartment of Mathematics and Statistics
À propos de Université de Sydney
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