Introduction to Calculus

About this Course

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).

Learner Career Outcomes

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English

Subtitles: Arabic, French, Portuguese (European), Italian, Vietnamese, German, Russian, English, Spanish

Skills you will gain

• logic
• Mathematics
• Calculus

Learner Career Outcomes

11%

22%

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

English

Subtitles: Arabic, French, Portuguese (European), Italian, Vietnamese, German, Russian, English, Spanish

Instructor

Offered by

Our excellence in research and teaching makes the University of Sydney one of the top universities in Australia and highly ranked among the best universities in the world. In 2020, we were ranked second in the Times Higher Education (THE) University Impact Rankings, and first in Australia in the QS Graduate Employability Rankings.

Week 1

9 hours to complete

Precalculus (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.

9 hours to complete

10 videos (Total 109 min), 8 readings, 9 quizzes

Welcome and introduction to Module 14m

Real line, decimals and significant figures15m

The Theorem of Pythagoras and properties of the square root of 211m

Algebraic expressions, surds and approximations10m

Equations and inequalities17m

Sign diagrams, solution sets and intervals (Part 1)10m

Sign diagrams, solution sets and intervals (Part 2)10m

Coordinate systems8m

Distance and absolute value5m

Lines and circles in the plane14m

Notes: Real line, decimals and significant figures20m

Notes: The Theorem of Pythagoras and properties of the square root of 220m

Notes: Algebraic expressions, surds and approximations20m

Notes: Equations and inequalities20m

Notes: Sign diagrams, solution sets and intervals20m

Notes: Coordinate systems20m

Notes: Distance and absolute value20m

Notes: Lines and circles in the plane20m

Real line, decimals and significant figures20m

The Theorem of Pythagoras and properties of the square root of 220m

Algebraic expressions, surds and approximations20m

Equations and inequalities30m

Sign diagrams, solution sets and intervals30m

Coordinate systems30m

Distance and absolute value30m

Lines and circles in the plane30m

Module 1 quiz1h

Week 2

13 hours to complete

Functions (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 hours to complete

13 videos (Total 142 min), 12 readings, 13 quizzes

Introduction to Module 21m

Parabolas and quadratics11m

The quadratic formula10m

Functions as rules, with domain, range and graph11m

Polynomial and power functions13m

Composite functions7m

Inverse functions12m

The exponential function13m

The logarithmic function8m

Exponential growth and decay13m

Sine, cosine and tangent9m

The unit circle and trigonometry16m

Inverse circular functions11m

Notes: Parabolas and quadratics20m

Notes: The quadratic formula20m

Notes: Functions as rules, with domain, range and graph20m

Notes: Polynomial and power functions20m

Notes: Composite functions20m

Notes: Inverse functions20m

Notes: The exponential function20m

Notes: The logarithmic function15m

Notes: Exponential growth and decay20m

Notes: Sine, cosine and tangent20m

Notes: The unit circle and trigonometry20m

Notes: Inverse circular functions20m

Parabolas and quadratics30m

The quadratic formula30m

Functions as rules, with domain, range and graph30m

Polynomial and power functions30m

Composite functions30m

Inverse functions30m

The exponential function30m

The logarithmic function30m

Exponential growth and decay30m

Sine, cosine and tangent30m

The unit circle and trigonometry30m

Inverse circular functions30m

Module 2 quiz1h

Week 3

12 hours to complete

Introducing 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 hours to complete

12 videos (Total 132 min), 10 readings, 11 quizzes

Introduction to Module 31m

Slopes and average rates of change10m

Displacement, velocity and acceleration11m

Tangent lines and secants10m

Different kinds of limits12m

Limit laws15m

Limits and continuity9m

The derivative as a limit10m

Finding derivatives from first principles14m

Leibniz notation14m

Differentials and applications (Part 1)13m

Differentials and applications (Part 2)7m

Notes: Slopes and average rates of change20m

Notes: Displacement, velocity and acceleration20m

Notes: Tangent lines and secants20m

Notes: Different kinds of limits20m

Notes: Limit laws20m

Notes: Limits and continuity20m

Notes: The derivative as a limit20m

Notes: Finding derivatives from first principles20m

Notes: Leibniz notation20m

Notes: Differentials and applications20m

Slopes and average rates of change30m

Displacement, velocity and acceleration30m

Tangent lines and secants30m

Different kinds of limits30m

Limit laws30m

Limits and continuity30m

The derivative as a limit30m

Finding derivatives from first principles30m

Leibniz notation30m

Differentials and applications30m

Module 3 quiz1h

Week 4

14 hours to complete

Properties 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 hours to complete

14 videos (Total 155 min), 13 readings, 14 quizzes

Introduction to Module 41m

Increasing and decreasing functions11m

Sign diagrams12m

Maxima and minima12m

Concavity and inflections10m

Curve sketching16m

The Chain Rule9m

Applications of the Chain Rule14m

The Product Rule8m

Applications of the Product Rule9m

The Quotient Rule8m

Application of the Quotient Rule10m

Optimisation12m

The Second Derivative Test16m

Notes: Increasing and decreasing functions20m

Notes: Sign diagrams20m

Notes: Maxima and minima20m

Notes: Concavity and inflections20m

Notes: Curve sketching20m

Notes: The Chain Rule20m

Notes: Applications of the Chain Rule20m

Notes: The Product Rule20m

Notes: Applications of the Product Rule20m

Notes: The Quotient Rule20m

Notes: Application of the Quotient Rule20m

Notes: Optimisation20m

Notes: The Second Derivative Test20m

Increasing and decreasing functions30m

Sign diagrams20m

Maxima and minima30m

Concavity and inflections30m

Curve sketching30m

The Chain Rule30m

Applications of the Chain Rule30m

The Product Rule30m

Applications of the Product Rule30m

The Quotient Rule30m

Application of the Quotient Rule30m

Optimisation30m

The Second Derivative Test30m

Module 4 quiz1h

Week 5

11 hours to complete

Introducing the integral calculus

This fifth and final module introduces integral calculus, looking at the slopes of tangent lines and areas under curves. This leads to the Fundamental Theorem of Calculus. We explore the use of areas under velocity curves to estimate displacement, using averages of lower and upper rectangular approximations. We then look at limits of approximations, to discover the formula for the area of a circle and the area under a parabola. We then develop methods for capturing precisely areas under curves, using Riemann sums and the definite integral. The module then introduces indefinite integrals and the method of integration by substitution. Finally, we discuss properties of odd and even functions, related to rotational and reflectional symmetry, and the logistic function, which modifies exponential growth.

11 hours to complete

14 videos (Total 162 min), 10 readings, 9 quizzes

Introduction to Module 52m

Inferring displacement from velocity15m

Areas bounded by curves17m

Riemann sums and definite integrals17m

The Fundamental Theorem of Calculus and indefinite integrals16m

Connection between areas and derivatives (Part 1)9m

Connection between areas and derivatives (Part 2)10m

Integration by substitution (Part 1)11m

Integration by substitution (Part 2)8m

Odd and even functions (Part 1)10m

Odd and even functions (Part 2)9m

The logistic function (Part 1)12m

The logistic function (Part 2)6m

The escape velocity of a rocket15m

Notes: Inferring displacement from velocity20m

Notes: Areas bounded by curves20m

Notes: Riemann sums and definite integrals20m

Notes: The Fundamental Theorem of Calculus and indefinite integrals20m

Notes: Connection between areas and derivatives20m

Notes: Integration by substitution20m

Notes: Odd and even functions20m

Notes: The logistic function20m

Notes: The escape velocity of a rocket20m

Formula Sheet10m

Inferring displacement from velocity30m

Areas bounded by curves30m

Riemann sums and definite integrals30m

The Fundamental Theorem of Calculus and indefinite integrals30m

Connection between areas and derivatives30m

Integration by substitution30m

Odd and even functions30m

The logistic function30m

Module 5 quiz1h