Functions, Graphs, Equations and Inequalities

Transcription

1 CAEM DPP Learning Outcomes per Module Module Functions, Graphs, Equations and Inequalities Learning Outcomes 1. Functions, inverse functions and composite functions 1.1. concepts of function, domain and range 1.. use of notations such as and 1.3. finding inverse functions and composite functions 1.4. conditions for the existence of inverse functions and composite functions 1.5. domain restriction to obtain an inverse function 1.6. relationship between a function and its inverse as reflection in the line. Graphs and transformations.1. use of graphing tool to graph a given function.. important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following: ; ; ;.3 determining the equations of asymptotes, axes of symmetry, and restrictions on the possible values of x and/or y.4 effect of transformations on the graph of as represented by and, and combinations of these transformations.5 graphs relating the graphs of and to the graph of..6 Simple parametric equations and their graphs

2 3. Equations and inequalities 3.1. formulating an equation, a system of linear equations, or inequalities from a problem situation 3.. solving an equation exactly or approximately using a graphing tool 3.3. solving a system of linear equations using a graphing tool 3.4. solving inequalities of the form where and are linear expressions or quadratic expressions that are either factorisable or always positive 3.5. concept of, and use of relations 3.6. solving inequalities by graphical methods and, in the course of solving inequalities Further Calculus 1. Differentiation 1.1. recall on Rules of Differentiation (Chain Rule, Product Rule and Quotient Rule) 1.. technique of Implicit Differentiation ' " ' find and evaluate f x, f x and interpret values associated with f x, f ' x 0, f " x 0and f " x 0 Note: also include x and x ) 1.4. find rate of change, related rate of change by finding the derivative of functions parametrically or implicitly 1.5. formulate and solve related rate of change problems using chain rule or implicit differentiation appropriately. Integration.1. perform integration for the following forms of expressions: f x f x ; n ' sin mx or cos sin mx ; and mx cos nx or sin mx sin nx or cos mx cos nx. perform integration for an expression when given a substitution.3 evaluate volume of solid of revolution (disk method) bounded by one or more graphs and axes (exclude shell method)

3 3. Differential Equation 3.1 identify the order and degree of an ordinary differential equation 3. solve first and second order differential equations of the following forms: dy f x dx ; dy g y dx ; d y f x ; dx (reducing forms (ii) and (iii) to form (i) by means of a given substitution) d y dy a by 0 dx dx ; and d y dy a by f x dx dx where kx f x pe or f x acos kx bsin kx 3.3 formulate, solve and interpret a differential equation from a given problem situation Vectors 1. Basic Properties of vectors in two- and three dimensions 1.1. use of notations such as 1.. position vectors, displacement vectors and direction vectors 1.3. magnitude of a vector 1.4. unit vectors 1.5. addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations 1.6. distance between two points 1.7. concept of direction cosines 1.8. collinearity 1.9. use of the ratio theorem in geometrical applications

4 . Scalar and vector products of vectors.1. concepts of scalar product and vector product of vectors and their properties.. calculation of the magnitude of a vector and the angle between two vectors.3. geometrical meanings of and, where is a unit vector 3. Three-dimensional geometry 3.1. vector and Cartesian equations of lines and planes 3.. finding the foot of the perpendicular and distance from a point to a line or to a plane 3.3. finding the angle between two lines, between a line and a plane, or between two planes 3.4. relationships between two lines (coplanar or skew) a line and a plane two planes Sequences and Series 1. Sequences and Series 1.1. concepts of sequence and series for finite and infinite cases; y f n where n is a positive integer; 1.. sequence as a function, 1.3. relationship between nth term, un and sum of n terms, S n 1.4. sequence given by a formula for the nth term 1.5. use of sigma notation, 1.6. sum and difference of two series 1.7. summation of series by the method of differences 1.8. formula for the nth term and the sum of a finite arithmetic series 1.9. formula for the nth term and the sum of a finite geometric series 1.10 convergence of a series and the sum to infinity 1.11 condition for convergence of an infinite geometric series 1.1 formula for the sum to infinity of a convergent geometric series

5 . Maclaurin Series.1. standard series expansion of 1 x n for any rational n, x e, sin x, cos x and ln 1 x.. derivation of the first few terms of the Maclaurin series by..1. repeated differentiation (example: sec x ); 3... repeated implicit differentiation (example: y y y x x ); x 1 x..3. using standard series (example: e cosx, ln 1 x ;..4. range of values of x for which a standard series converges and application for concept of approximation ; and small angle approximations (example:sin x x, cos x 1 x, tan x x) 3. Recurrence Relations and Mathematical Induction 3.1. sequence generated by a recurrence relation 3.. behavior of a sequence, such as the limiting behavior of a sequence 3.3. use of method of mathematical induction to establish a given result involving series and recurrence relations. Probability & Statistics I 1. Probability 1.1. addition and multiplication principles for counting 1.. concepts of Permutation ( n P r) and Combination ( n C r) 1.3. arrangements of objects in a line or in a circle, including cases involving repetition and restriction 1.4. addition and multiplication of probabilities 1.5. mutually exclusive events and independent events 1.6. use of tables of outcomes, Venn diagrams, tree diagrams, and permutation and combination techniques to calculate probabilities 1.7. calculation of conditional probabilities. Discrete Random Variables.1. concept of discrete random variables, probability distributions, expectations and variances.. concept of binomial distribution B(n, p).3. use of mean and variance of binomial distribution

6 3. Normal Distribution 3.1. concept of a normal distribution 3.. standard normal distribution 3.3. finding the value of P(X < x) or a related probability 3.4. symmetry of the normal curve and its properties Probability & Statistics II 4. Sampling 4.1. concepts of population, random and non-random samples 4.. concept of the sample mean as a random variable 4.3. distribution of sample means from a normal population 1. Sampling 1.1. use of the Central Limit Theorem to treat sample means as having normal distribution when the sample size is sufficiently large 1.. calculation and use of unbiased estimates of the population mean and variance from a sample. Hypothesis Testing.1 concepts of null hypothesis (H 0) and alternative hypotheses (H 1), test statistic, critical region, critical value, level of significance and p-value. formulation of hypotheses and testing for a population mean based on:..1 sample from a normal population of known variance.. large sample from any population.3 1-tail and -tail tests.4 interpretation of the results in a hypothesis test in the context of the problem 3. Correlation and Linear Regression 3.1. use of scatter diagram to determine if there is a plausible linear relationship between the two variables 3.. correlation coefficient as a measure of the fit of a linear model to the scatter diagram 3.3. finding and interpreting the product moment correlation coefficient 3.4. concepts of linear regression 3.5. concepts of interpolation and extrapolation 3.6. use of the appropriate regression line to make prediction or estimate a value

7 Complex Numbers, Matrices and Linear Spaces 1. Complex Numbers 1.1. complex roots of quadratic equations 1.. conjugate roots of a polynomial equation with real coefficients 1.3. representation of complex numbers in Argand diagram 1.4. use of Euler s formula 1.5. complex numbers expressed in the form, or where and 1.6. geometrical effects of conjugating, addition, subtraction, multiplication and division of complex numbers 1.7. loci of simple equations and inequalities such as and arg 1.8. use of de Moivre s theorem to find the powers and th roots of a complex number, and to derive trigonometric identities.. Matrices and Linear Spaces.1. use of matrices to represent a set of linear equations.. operations on 3 3 matrices.3. determinant of a square matrix and inverse of a non-singular matrix ( and 3 3 matrices only).4. use of matrices to solve a set of linear equations.5. linear spaces and subspaces, and the axioms Polar Coordinates and Real World Mathematics 1. Polar Curves and Conic Sections 1.1. simple polar curves 1.. definitions and defining geometric properties of conic sections, including their general equations: Circle: Ellipse: Parabola: Hyperbola: ; 1.3. conic sections in polar form given by or where e > 0 is the eccentricity and is the distance between the focus (pole) and the directrix

8 . Equation Formulation.1. unary, binary, and multi-parameter functions.. equation formulation within a closed system (limited parameters).3. equation formulation within an open system (multiple parameters).4. making assumptions and their effects on solution precision. 3. Parameter Acquisition 3.1. obtaining parameters through data gathering or mensuration 3.. precision of gathered data and its effect on solution accuracy 4. Creating original Real-world questions 4.1. presentation of original question concept and the impact of the solution on a relevant community 4.. class discourse on parameters used, assumed, approximated, or measured/found 4.3. presentation of answers to original question and class discourse on accuracy of answer, efficiency of method, and beneficial impact of the answer to the community