|DESCRIPTION||Calculus can be regarded as the mathematical study of change. This calculus course is intended to supplement parallel study-units within the technology course. Concepts from this course are essential for adopting a STEM approach toward the learning of the main domains of electrical/electronics knowledge, materials/mechanical knowledge and graphical communication/engineering drawing knowledge since sound functional design is almost always supported by mathematics. Within this course students will learn how to describe, quantify, interpret and communicate change through calculus notation, relationships and diverse methods for calculation. Topics covered within this unit include the following: finding the derivative of a function using different methods, finding the integral of a function using different methods, link these to practical applications such as area and volume.
The aims of this study-unit are:
1. To present the mathematical descriptors and representations of the concept of "change";
2. To enable technology students to acquire essential concepts and techniques of differential and integral calculus and its application in relation to other topics in technology.
1. Knowledge & Understanding:
By the end of the study-unit the student will be able to interpret concepts involved in differential and integral calculus and their application to series expansions and the solution of differential equations. In addition, the students will understand how to make approximations through the use of or in relation to calculus.
By the end of the study-unit the student will be able to:
- find a derivative from first principles, and learn how to derive standard functions (like x^n, e^x, ln x, sin x, cos x, tan x);
- find the derivative of non-standard functions including composite functions, products and quotients;
- link differentiation to the gradient of a tangent to a curve at a point;
- use their knowledge of the nature of the second derivative for the determination of stationary points;
- link differentiation to practical situations of rates of change;
- find the integral of a function as the inverse of differentiation and interpret it as a process of summation (idea of limit);
- work with standard integrals, integrate rational functions, products of sines and cosines of multiple angles, powers of sin x and cos x;
- demonstrate knowledge on how to integrate by parts and by substitution;
- apply definite integrals for mensuration in the Cartesian coordinate system (area between graph and x-axis, mean value of a function, volume of solids of revolution, and curved surface area of solids of revolution);
- perform a binomial expansion with non-integer powers;
- work with Maclaurin’s series expansions;
- determine the Fourier series that converges to a function f defined by f(x) = f(x+2L);
- reduce a relationship to linear form for practical purposes;
- approximate definite integrals (Trapezium and Simpson’s rule);
- locate and approximate a root by the Newton-Raphson method;
- find general and particular solutions of ordinary differential equations (ODEs) of the first order with variables separable, and ODEs of the exact type (use of integrating factor);
solve second order ODEs with constant coefficients, homogeneous or non-homogeneous.