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
oferecido por
The University of Sydney
Programa - O que você aprenderá com este curso
Semana
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 videos (Total 109 min), 8 leituras, 9 testes
10 videos
Real line, decimals and significant figures15min
The Theorem of Pythagoras and properties of the square root of 211min
Algebraic expressions, surds and approximations10min
Equations and inequalities17min
Sign diagrams, solution sets and intervals (Part 1)10min
Sign diagrams, solution sets and intervals (Part 2)10min
Coordinate systems8min
Distance and absolute value5min
Lines and circles in the plane14min
8 leituras
Notes: Real line, decimals and significant figures20min
Notes: The Theorem of Pythagoras and properties of the square root of 220min
Notes: Algebraic expressions, surds and approximations20min
Notes: Equations and inequalities20min
Notes: Sign diagrams, solution sets and intervals20min
Notes: Coordinate systems20min
Notes: Distance and absolute value20min
Notes: Lines and circles in the plane20min
9 exercícios práticos
Real line, decimals and significant figures20min
The Theorem of Pythagoras and properties of the square root of 220min
Algebraic expressions, surds and approximations20min
Equations and inequalities20min
Sign diagrams, solution sets and intervals20min
Coordinate systems20min
Distance and absolute value20min
Lines and circles in the plane20min
Module 1 quizmin
Semana
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 videos (Total 142 min), 12 leituras, 13 testes
13 videos
Parabolas and quadratics11min
The quadratic formula10min
Functions as rules, with domain, range and graph11min
Polynomial and power functions13min
Composite functions7min
Inverse functions12min
The exponential function13min
The logarithmic function8min
Exponential growth and decay13min
Sine, cosine and tangent9min
The unit circle and trigonometry16min
Inverse circular functions11min
12 leituras
Notes: Parabolas and quadratics20min
Notes: The quadratic formula20min
Notes: Functions as rules, with domain, range and graph20min
Notes: Polynomial and power functions20min
Notes: Composite functions20min
Notes: Inverse functions20min
Notes: The exponential function20min
Notes: The logarithmic function15min
Notes: Exponential growth and decay20min
Notes: Sine, cosine and tangent20min
Notes: The unit circle and trigonometry20min
Notes: Inverse circular functions20min
13 exercícios práticos
Parabolas and quadratics20min
The quadratic formula20min
Functions as rules, with domain, range and graph20min
Polynomial and power functions20min
Composite functions20min
Inverse functions20min
The exponential function20min
The logarithmic function20min
Exponential growth and decay20min
Sine, cosine and tangent20min
The unit circle and trigonometry20min
Inverse circular functions20min
Module 2 quizmin
Semana
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 videos (Total 132 min), 10 leituras, 11 testes
12 videos
Slopes and average rates of change10min
Displacement, velocity and acceleration11min
Tangent lines and secants10min
Different kinds of limits12min
Limit laws15min
Limits and continuity9min
The derivative as a limit10min
Finding derivatives from first principles14min
Leibniz notation14min
Differentials and applications (Part 1)13min
Differentials and applications (Part 2)7min
10 leituras
Notes: Slopes and average rates of change20min
Notes: Displacement, velocity and acceleration20min
Notes: Tangent lines and secants20min
Notes: Different kinds of limits20min
Notes: Limit laws20min
Notes: Limits and continuity20min
Notes: The derivative as a limit20min
Notes: Finding derivatives from first principles20min
Notes: Leibniz notation20min
Notes: Differentials and applications20min
11 exercícios práticos
Slopes and average rates of change20min
Displacement, velocity and acceleration20min
Tangent lines and secants20min
Different kinds of limits20min
Limit laws20min
Limits and continuity20min
The derivative as a limit20min
Finding derivatives from first principles20min
Leibniz notation20min
Differentials and applications20min
Module 3 quizmin
Semana
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 videos (Total 155 min), 13 leituras, 14 testes
14 videos
Increasing and decreasing functions11min
Sign diagrams12min
Maxima and minima12min
Concavity and inflections10min
Curve sketching16min
The Chain Rule9min
Applications of the Chain Rule14min
The Product Rule8min
Applications of the Product Rule9min
The Quotient Rule8min
Application of the Quotient Rule10min
Optimisation12min
The Second Derivative Test16min
13 leituras
Notes: Increasing and decreasing funtions20min
Notes: Sign diagrams20min
Notes: Maxima and minima20min
Notes: Concavity and inflections20min
Notes: Curve sketching20min
Notes: The Chain Rule20min
Notes: Applications of the Chain Rule20min
Notes: The Product Rule20min
Notes: Applications of the Product Rule20min
Notes: The Quotient Rule20min
Notes: Application of the Quotient Rule20min
Notes: Optimisation20min
Notes: The Second Derivative Test20min
14 exercícios práticos
Increasing and decreasing functions20min
Sign diagrams20min
Maxima and minima20min
Concavity and inflections20min
Curve sketching20min
The Chain Rule20min
Applications of the Chain Rule20min
The Product Rule20min
Applications of the Product Rule20min
The Quotient Rule20min
Application of the Quotient Rule20min
Optimisation20min
The Second Derivative Test20min
Module 4 quizmin
Instrutores
David Easdown
Associate ProfessorDepartment of Mathematics and Statistics
Sobre The University of Sydney
The University of Sydney is one of the world’s leading comprehensive research and teaching universities, consistently ranked in the top 1 percent of universities in the world. In 2015, we were ranked 45 in the QS World University Rankings, and 100 percent of our research was rated at above, or well above, world standard in the Excellence in Research for Australia report.
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