N6 Mathematics

Mathematics Downloaded from hlayiso.com N6

Module 1: Differentiation INTRODUCTION Differentiation is one of the most important fundamental concepts to understand in mathematics. It allows us to find the rate of change of one variable with respect to another variable. Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Differentiation (continued) PARTIAL DIFFERENTIATION The partial derivative of a function of several variables is the derivative with respect to one of the variables, while the other variables are held constant. Partial derivatives are themselves functions of the variables concerned and as such, they can also be differentiated. Downloaded from hlayiso.com www.futuremanagers.com

Module 1: Differentiation (continued) DIFFERENTIATION OF PARAMETRIC EQUATIONS Parametric equations are a number of equations that take into account several different independent variables or parameters in order to express coordinate points. This is done to define complex geometric objects such as surfaces and curves. Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Integration techniques INTEGRATION BY PARTS Every integration rule corresponds to a differentiation rule. Integration by parts is no different. This rule corresponds to the product rule of differentiation: 𝑑𝑑 𝑓𝑓 𝑔𝑔 = 𝑓𝑓 ′ 𝑔𝑔 + 𝑓𝑓 𝑔𝑔′ 𝑑𝑑𝑑𝑑 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 � 𝑓𝑓 ′ 𝑥𝑥 𝑔𝑔 𝑥𝑥 + 𝑓𝑓 𝑥𝑥 𝑔𝑔′ 𝑥𝑥 = 𝑓𝑓 𝑥𝑥 𝑔𝑔( ) 𝑥𝑥 Downloaded from hlayiso.com www.futuremanagers.com

Module 2: Integration techniques (continued) INTEGRATION BY MEANS OF COMPLETING THE SQUARE The idea of completing the square is to find values for a, h and k so that you can write any polynomial in the form 𝑎𝑎 2 + 𝑏𝑏 + 𝑥𝑥 𝑏𝑏 𝑥𝑥 as 𝑎𝑎 𝑥𝑥 − ℎ 2 + 𝑘𝑘. This is done by first focussing on the first two terms, 𝑎𝑎 2 + 𝑏𝑏 and then forming a 𝑥𝑥 𝑏𝑏 𝑏𝑏 2 2 + 𝑏𝑏 + 𝑏𝑏 square with just these two terms. 𝑎𝑎 + = 𝑎𝑎( ) 2𝑎𝑎 𝑎𝑎 2𝑎𝑎 𝑥𝑥 𝑥𝑥 Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Partial fractions INTRODUCTION Partial fractions is a method used to break apart fractions containing polynomials. Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart or decomposing it into its initial polynomial fractions. Downloaded from hlayiso.com www.futuremanagers.com

Module 3: Partial fractions (continued) THE RULES FOR DECOMPOSING INTO PARTIAL FRACTIONS 1. The numerator of the given functions must be of a lower degree than that of the denominator. 2. Factorise the denominator into its irreducible factors. 𝐴𝐴 3. A linear factor 𝑎𝑎 + 𝑏𝑏 results in a partial fraction of the form 𝑎𝑎𝑎𝑎+𝑏𝑏 𝑎𝑎 𝐴𝐴 𝐵𝐵 𝐺𝐺 4. A linear factor repeated has partial fractions + + ⋯ + 𝑎𝑎𝑎𝑎+𝑏𝑏 𝑎𝑎𝑎𝑎+𝑏𝑏 2 𝑎𝑎𝑎𝑎+𝑏𝑏 𝑛𝑛 2 + 𝑏𝑏 𝐵𝐵 5. An irreducible quadratic factor 𝑎𝑎 + 𝑐𝑐 gives fraction 𝐴𝐴 + 2 Downloaded from hlayiso.com 𝑎𝑎 +𝑏𝑏 +𝑐𝑐 𝑥𝑥 𝑏𝑏 𝐴𝐴 𝑏𝑏 𝑥𝑥 www.futuremanagers.com

Module 4: Differential equations INTRODUCTION A differential equation is an equation that involves the derivatives of one or more functions. These equations can be used to model very complex systems and they are an important aspect of calculus. Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Differential equations (continued) FIRST ORDER DIFFERENTIAL EQUATIONS First order differential equations, where the dependent variable appears in the first degree, are said to be linear. The general form of such a linear 𝑑𝑑𝑑𝑑 differential equation is + 𝑃𝑃𝑃𝑃 = 𝑄𝑄 where 𝑃𝑃 and 𝑄𝑄 are functions of . 𝑑𝑑 𝑥𝑥 𝑑𝑑 Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Differential equations (continued) SECOND ORDER DIFFERENTIAL EQUATIONS Second order differential equations are useful for modelling problems of mechanical and electrical systems. The general form of a second order 𝑑𝑑 2 𝑑𝑑𝑑𝑑 differential equation is: a 2 + 𝑏𝑏 + 𝑐𝑐 = 𝑓𝑓( ) 𝑦𝑦 𝑑𝑑 𝑑𝑑 𝑥𝑥 𝑐𝑐 𝑑𝑑 𝑥𝑥 Downloaded from hlayiso.com www.futuremanagers.com

Module 4: Differential equations (continued) NON-HOMOGENOUS DIFFERENTIAL EQUATIONS Non-homogeneous equations are the same as homogeneous differential equations, except they can have terms involving only x (and constants) on the right side. Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Applications of the definite integral VOLUMES The volume of the representative element can be approximated as a cylinder giving the volume equation: Δ𝑉𝑉 = 𝜋𝜋 2 Δ 𝑦𝑦 𝑥𝑥 Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Applications of the definite integral (continued) CENTROIDS The geometric centre of an object; the arithmetic mean position of all points in a shape. The centroid of any shape can be found using x- and y coordinates on a cartesian plane. Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Applications of the definite integral (continued) CENTRES OF GRAVITY The centre of gravity is the average location of the weight of an object. Also known as the centre of mass (when the gravitational field is uniform across the object). Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Applications of the definite integral (continued) SECOND MOMENT OF AREA The second moment of area is a geometrical property of a specific area which describes how all the points of the area are distributed with regard to a particular axis. This concept helps to determine the resistance of an object to bending along a certain axis. This is extremely important in structural engineering and helps to identify how and where beams will bend when subjected to a load. Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Applications of the definite integral (continued) MOMENT OF INERTIA Inertia represents a tendency to remain unchanged. The basic equation for the second moment of area in physics is: 𝐼𝐼 = 𝑚𝑚𝑟𝑟 2 Downloaded from hlayiso.com www.futuremanagers.com

Module 5: Applications of the definite integral (continued) CENTRES OF FLUID PRESSURE Fluid pressure is the pressure exerted by a static fluid on a body or object. Pressure is the force applied perpendicularly to the surface of an object per unit area. It can be explained as: 𝐹𝐹 𝑃𝑃 = or 𝑃𝑃 = 𝑝𝑝 𝐴𝐴 𝑝𝑝𝑝 Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Applications where differentiation and integration techniques are combined LENGTH OF NON-PARAMETRIC CURVES The arc length can be determined for continuous and derivative functions over a given range. The non-parametric functions are functions of one another. For non parametric functions, the formula used to determine the length is the same as the general formula: 𝑎𝑎 2 𝑑𝑑 𝐿𝐿 = � 1+ 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 Downloaded from hlayiso.com 𝑏𝑏 www.futuremanagers.com

Module 6: Applications where differentiation and integration techniques are combined (continued) LENGTH OF PARAMETRIC CURVES The parametric curve is defined as the path of a point moving along the plane. This path is effected through the coordinates of 𝑥𝑥 ; 𝑦𝑦 = 𝑓𝑓 𝑡𝑡 ; 𝑔𝑔 𝑡𝑡 . The arc is thus loosely defined as a straight line between two points and the length is derived through the Pythagorean theorem. Downloaded from hlayiso.com www.futuremanagers.com

Module 6: Applications where differentiation and integration techniques are combined (continued) SURFACES OF REVOLUTION A surface of revolution is formed when a curve is rotated about a line. Using integration and differentiation techniques, it is possible to determine the area of such curve. Downloaded from hlayiso.com www.futuremanagers.com