Introduction to Calculus

The University of Sydney

The Introduction to Calculus course at The University of Sydney offers a comprehensive exploration of foundational concepts and applications in calculus. This course is designed to provide students with a strong understanding of fundamental mathematical principles and their real-world applications in science, engineering, and commerce.

Throughout this course, students will delve into precalculus, mastering the manipulation of equations and elementary functions. They will then progress to the preliminary methodology of tangents and limits, and the definition of a derivative, followed by methods of differential and integral calculus with practical applications. The curriculum emphasizes a balance between theory and application, ensuring students gain fluency with key threshold concepts in foundational mathematics.

By the end of the course, students will have developed a strong foundation in calculus, preparing them for advanced studies and practical implementation in their respective fields. Enroll now to gain a deep understanding of calculus and its significance in various disciplines.

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Introduction to Calculus
Course Modules

The Introduction to Calculus course is comprised of five modules, covering precalculus, differential calculus, integral calculus, and foundational mathematical concepts, providing a comprehensive understanding of calculus and its applications.

Precalculus (Setting the scene)

The Precalculus module sets the scene by familiarizing students with key ideas and methodology, including the manipulation of equations, elementary functions, and the real number line. Students will also explore algebraic expressions, surds, and approximations, enabling a strong foundational understanding of precalculus.

Functions (Useful and important repertoire)

The Functions module further builds on the precalculus foundation, introducing students to functions as rules, composite functions, inverse functions, and various types of functions, such as polynomial, power, exponential, logarithmic, and trigonometric functions. This module provides a comprehensive repertoire of functions and their applications.

Introducing the differential calculus

The Introducing the Differential Calculus module delves into the methodology of slopes, average rates of change, tangents, secants, and different kinds of limits, leading to the understanding of derivatives and their applications. Students will also explore Leibniz notation, differentials, and practical applications of differentials.

Properties and applications of the derivative

The Properties and Applications of the Derivative module focuses on increasing and decreasing functions, sign diagrams, maxima, minima, concavity, inflections, curve sketching, and various differentiation rules and their applications. This module equips students with the essential tools for analyzing functions using differentiation.

Introducing the integral calculus

The Introducing the Integral Calculus module covers inferring displacement from velocity, areas bounded by curves, Riemann sums, definite integrals, the Fundamental Theorem of Calculus, and applications of definite integrals. This module also explores the connection between areas and derivatives, integration by substitution, odd and even functions, and practical applications of the integral calculus.

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