Calculus: Advanced Techniques and Applications

In this module of Maths Extension 1, you will expand your understanding of both differential and integral calculus by exploring advanced techniques to simplify complex problems into familiar forms. You will learn how to apply integral calculus to problems involving area and volume, and will be introduced to differential equations, a broad field of study in mathematics.

Integration Techniques involving Variable Substitution

In this section you will learn how to apply the Change of Variables Theorem—a concept closely related to the chain rule—to perform u-substitution and rearrange complicated integrals into more manageable expressions. You will also learn about how trigonometric substitution can be used handle integrals involving radicals that we have not been able to solve thus far.

• The Change of Variables Theorem
• Integration Using u-Substitution
• Rearranging Integrals for u-Substitution
• Trigonometric Substitution

Integration Techniques involving Trigonometric and Inverse Functions

This section explores how identities—such as the double angle formula—can simplify integrals involving powers of trigonometric functions and how to integrate squared trigonometric expressions. You will also learn to differentiate inverse functions and apply inverse trigonometric functions to new integration problems.

• Integrating powers of trigonometric functions
• Finding the derivative of an inverse function
• The derivative of the inverse trigonometric functions
• Integrating fractions involving irreducible quadratics

Applications of Integral Calculus: Areas and Volumes

In this section, you will see how integration is used to determine the area between curves and compute the volume of solids formed by rotating a curve around an axis.

• Calculating the area between two curves
• Calculating the volume of a solid of revolution

Introduction to Differential Equations

Differential equations are a cornerstone of mathematical modeling in science, engineering, and economics. This section introduces you to differential equations and demonstrates how they can be used to describe dynamic systems and change over time. You will learn to construct and interpret slope fields—a graphical tool that helps visualize solutions—and solve various forms of first-order differential equations. You will also explore how to apply initial conditions to select particular solutions and see the real-world impact of these equations.

• Constructing a slope field for a differential equation
• Graphing solutions using a slope field
• Solving differential equations of the form $\frac{dy}{dx} = f(x)$
• Solving differential equations of the form $\frac{dy}{dx} = g(y)$
• Solving differential equations of the form $\frac{dy}{dx} = f(x)g(y)$
• Applying an initial condition to differential equations
• Recognising and applying differential equations to real world problems

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