Trigonometry Self-study: Reading: Red Bostock and Chandler p , p , p

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1 Trigonometry Self-study: Reading: Red Bostock Chler p , p , p Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant, cotangent. be able to draw the graphs of the six trigonometric functions Inverse trigonometric functions be familiar with the six trigonometric functions be able to draw the graphs of the six inverse trigonometric functions underst the notations for inverse functions, e.g. the inverse function of could be written as or Note that. Be able to find all solutions of equations of the form, where is one of the six trigonometric functions is a specified range such as [ E.g. find the values of in the range [ for which. Trigonometric identities be familiar with the formulas on the formula sheet be able to use them to do the following find the possible values of given the value of, where are any of the six trigonometric functions e.g. given find the possible values of write expressions of the form in any one of the following four forms,, or,. hence be able to solve equations of the form solve trigonometric equations, prove trigonometric identities. Algebra 4 weeks, 8 lectures. Lecture 1: Quadratic functions. Reading: Red Bostock Chler p (ignore example 2), p be familiar with the shape of a quadratic curve (i.e a parabola), its symmetry about its minimum/maximum point, be able to sketch this curve. be familiar with the method of completing the square be able to use it to determine the coordinates of the minimum/maximum point of a quadratic, determine the range of a quadratic function, prove the formula for the roots of a quadratic function.

2 know the formula for the roots of a quadratic equation, underst the significance of the discriminant know what different values of the discriminant mean. be able to solve quadratic inequalities. Lecture 2: Long division factorisation Reading: Photocopied notes from Precalculus. Optional reading: Red Bostock Chler p32-34, yellow p Bostock Chler be able to use polynomial long division to find the quotient remainder when one polynomial is divided by another, underst that the remainder will always have a lower degree than the divisor. underst that all polynomials with real coefficients can be factorised uniquely as a product of irreducible polynomials, i.e. as a product of linear factors quadratics with negative discriminant. Lecture 3 4: Remainder factor theorem Reading: Red Bostock Chler p32-35, yellow Bostock Chler p (ignore the material on repeated roots). be able to both use prove the remainder factor theorem. be able to generalise this technique to find the remainder when a polynomial is divided by a quadratic know the rational root test i.e. that if (where p q are coprime) is a root of the polynomial with integer coefficients then is a factor of is a factor of. be able to factorise a given polynomial using the factor theorem, long division the quadratic formula as necessary be able to use the factorisation of a polynomial to determine the range of values of for which f(x) is positive or negative Lecture 5: Partial fractions Reading: Partial fractions of Paul s notes: Optional Reading: Red Bostock Chler p5-9 p be able to find the partial fraction decomposition of a rational function including examples where the numerator has a higher degree than the denominator. Note: Questions involving repeated quadratic factors will not be asked in exams. The cover-up rule may be used but if used then it must be justified somehow e.g. by saying by cover-up rule. Lecture 6: Proof by Induction Reading: Red B&C p Yellow B&C p be able to use proof by induction to prove given statements about integers. Lecture 7 8: Binomial Theorem Generalised Binomial theorem Reading: Binomial Theorem: Red B&C p p

3 Generalised Binomial Theorem: Red B&C p Optional Reading: Precalculus Mathematics: A problem solving approach p be able to find the expansion of an expression of the form find the coefficient of a particular term of the expansion without calculating the whole expansion know when it is appropriate to use the Binomial Theorem when it is appropriate to use the Generalised Binomial Theorem, know the range of validity of the expansion be able to use the Generalised Binomial Theorem to find approximations, underst how to improve approximations. Differentiation 4 weeks, 8 lectures Lecture 1: Limit of a function at a point Underst the concept of continuity as a curve you can draw without taking your pen off the page Underst left-limits, right-limits, limits at a point. Lecture 2: Definition of the derivative as a limit, derivative of Reading: Red Bostock Chler p Know the definition of derivative in terms of limits Given a specific function, for example e.g., students should be able to use the definition to find derivatives of it at a particular point, or at a general point Lecture 3: Derivatives of,, Reading: Red Bostock Chler p know the proofs that. The proofs of the results will not be assessed they may be used without proof. know the proof that where is defined as the number such that evaluated at 0 is one. Lecture 4: Rules of differentiation (chain, product quotient) Reading: Red Bostock Chler p know the proofs of the product rule the quotient rule (the proof of the chain rule will not be examined), be able to apply these rules appropriately to find the derivatives of a wide range of functions. Lecture 5 & 6: Implicit differentiation: derivatives of inverse functions, tangents normal Reading: Red Bostock Chler p & p

4 Be able to find when a curve that cannot be written in the form Be able to find the tangent normal of a curve at a specified point Be able to find derivatives of inverse functions such as, inverse trig functions by using implicit differentiation. Be able to differentiate functions of the form. Lecture 7: Finding classifying stationary points + finding global local minima/maxima Reading: Red Bostock Chler p be familiar with the 1 st 2 nd derivative test; they should be able to use their judgement about which might be more appropriate/easier to use in a given context, but they should also be able to use a specific test if told to do so. Underst the concept of concavity underst that global min/max can occur at end points, where or where is undefined. underst that a function may have multiple local min/max or none. Lecture 8: Optimisation Reading: Photocopied notes from Calculus. Be able to find use differentiation to solve practical problems involving optimisation. Note: Problems will only be asked about situations where the variables are defined on a closed interval. Curve sketching 2.5 weeks, 5 lectures. Lecture 1: Basics of graph sketching Reading: Scanned notes from Understing Pure Mathematics pages Know the main features that should be included on graphs of : -intercepts -intercepts (where can be reasonably solved, otherwise some note should be made of what range the root is in) stationary points places where is not defined vertical horizontal asymptotes (these two features to be covered in more detail later) Places where the function is not defined Underst what a point of inflection is be able to find points of inflection if asked (but non-stationary points of inflection do not need to be found put on graphs, unless that is specifically asked for). Lecture 2: Numerical methods for finding roots

5 Reading: Photocopied notes from Further Pure Mathematics 1" by Geoff Mannall Michael Kenwood Underst that if a function is continuous on [ have opposite signs, then must have a root in the range. Underst that that sometimes roots cannot be found exactly, that sometimes numerical methods are needed get estimates of roots Be able to use bisection method the Newton-Raphson Method. Lecture 3: Horizontal asymptotes, the power of functions/race to infinity Reading: See page 183 of Calculus, Possibly also: Know the following limits: for for for Be able to use those limits to calculate the limits, where are one of the following functions: a polynomial,, (or if the limits do not exist then work out if the function tends to or ) Be able to use this information about the limits to find the horizontal asymptotes of functions. Lecture 4: Vertical asymptotes Reading: Know that asymptotes can occur in graphs of functions of trig functions, rational functions functions involving logarithms. Using the information from this lecture, the previous lecture, the material covered in differentiation students should be able to determine if a function has a global minimum or maximum (or if the function is unbounded). Lecture 5: Transformations of curves Reading: Scanned notes from Understing Pure Mathematics page 280 & pages Know the relationships between:

6 When given a graph of be able to draw the graph for any of the related curves listed above ( simple combinations of them such as ). Integration 4 weeks, 8 lectures. Lecture 1: Indefinite integration stard integrals Reading: Red Bostock Chler p underst definite integration as the reverse of differentiation know (i.e. memorise) the integrals of following functions (where ): know that Lecture 2: Integrating products Reading: Red B&C p underst the methods of integration by substitution integration by-parts know to use substitution to solve integrals of the form be able to used judgement about when to use integration by-parts choose which function should be which should be including examples such as or Lecture 3: Definite integrals Reading: Red Bostock Chler p underst the Fundamental Theorem of Calculus underst that the definite integral can be positive, negative or 0 underst how to apply integration by substitution integration by-parts to definite integrals Lecture 4: Integrating quotients Reading: Red Bostock Chler p be able to solve integrals of the following forms

7 Lecture 5: Integrals involving trigonometric functions Reading: Red Bostock Chler p know (i.e. memorise ) the integrals: be able to use trigonometric identities to solve integrals to solve integrals involving trigonometric functions including (but not limited to) 1. where 2. where 3. where 4. where 5. where 6. where Lecture 6: Trigonometric substitution Reading: Red Bostock Chler p know ( be able to) use appropriate trigonometric substitutions to solve (indefinite or definite) integrals involving or Lecture 7: Finding areas Reading: Red Bostock Chler p p be able to use definite integration to calculate areas including areas such as: the area between a curve the -axis the area between a curve the -axis the lines (including cases where the curve cuts the -axis) the area between two curves the area between two curves the lines (including cases where the curve touch or cut each other multiple times) Be able to write an integral either in terms of or, use their judgement to decide which integral is easier to evaluate Lecture 8: Volumes of revolution Reading: Red Bostock Chler p be able to use definite integration to evaluate volumes generated when the area between, the -axis the lines is rotated about the axis the area between the two curves, the lines is rotated about the axis

8 Series 2.5 weeks, 5 lectures. the area between, the -axis the lines is rotated about the axis the area between the two curves, the lines is rotated about the axis Lecture 1&2: Arithmetic/Geometric progressions series Reading: Red Bostock Chler p be able to recognise an arithmetic or geometric progression write down a formula for term be able to derive the formulas for an arithmetic series or geometric series be able decide whether a given geometric series will converge or diverge, if it converges calculate the infinite sum be able to use the formulas for arithmetic progressions, geometric progressions, arithmetic series geometric series Lecture 3: Method of differences sum of squares cubes Reading: Red Bostock Chler p be able to use the method of differences / telescoping series to evaluate series where the terms can be written in the form, where (including, where it makes sense, infinite sums) in particular students should be able to use this method to derive the formulas be able to use the formulas for sum of square cubes to evaluate series involving squares or cubes Lecture 4&5: Maclaurin s series Taylor series Reading: Yellow Bostock Chler p Students: should be able to find the Maclaurin s series or Taylor series expansion of a given function up to a specified term, e.g. up to the or term. should know the Maclaurin s series for

9 are not expected to know or be able to find the valid range for a Maclaurin s series or Taylor series expansion, but they should underst that some expansions are not valid for all should be able to use Maclaurin s series or Taylor series to make approximations to numbers, underst that (roughly speaking) the accuracy of the approximations are improved by 1. increasing the number of terms used in the expansion 2. in the case of a Maclaurin s series using a smaller value of, or in the case of a Taylor series using a smaller value of. should be able to find derivatives integrals of power series. Complex numbers 3.5 weeks, 7 lectures. Lecture 1: Complex arithmetic Reading: Red Bostock Chler p underst to be the square root of to be the set { } know the functions be able to do basic arithmetic with complex numbers i.e. be able to familiar with perform the following operations: conjugation, taking the modulus of a complex number, addition, subtraction, multiplication division, know that if is a polynomial with real coefficients then but if is a polynomial where some of the coefficients are not real then does not imply that know that a polynomial of degree can be written uniquely as the product of linear factors (if we allow complex coefficients). Lecture 2: The Arg Diagram polar form Reading: Red Bostock Chler p be able to plot complex numbers on the Arg Diagram, whether they are in Cartesian form or polar form, be able to change a complex number from Cartesian form to polar form, or from polar form to Cartesian form (note that finding is not simply if is in the second or third quadrant), be able to multiply divide numbers that are in polar form, be familiar with able to use the following properties of complex numbers:

10 = [up to adding or taking away ] ( ) [up to adding or taking away ] Lecture 3: Exponential form De Moivre s Theorem Reading: Yellow Bostock Chler p , p be able to change a complex number from Cartesian or polar form into exponential form, or from exponential form into either polar or Cartesian form, know De Moivre s Theorem be able to prove it using either Maclaurin s Series, or (for positive integer values) by proof by induction, know De Moivre s Theorem the identity by using either of them be able to find formulas for in terms of powers of or formulas for as a sum of the form or. be able to write or in exponential form be able to use these forms to prove trigonometric identities Lecture 4: Simple Loci Reading: Yellow Bostock Chler p be able to draw loci such as: be able to draw loci based on inequalities such as Convert simple equations written in terms of modulus argument into equations written in terms of, where appropriate sketch this curve, e.g. write Lecture 5: Simple transformations as equations in terms of. Reading: Yellow Bostock Chler p

11 underst the geometric effect on the Arg diagram of: complex conjugation addition or subtraction by a complex number multiplication/division by a real number multiplication/division by where is real simple combinations of the above transformations be able to interpret what the geometric meaning of a given function is, or be able to write down the function which represents a given transformation be able to find the fixed points of a given transformation find the equation of a curve after a transformation has been applied to it, i.e. if then find the image of a curve in terms of. Lecture 6: Roots of unity Reading: Yellow Bostock Chler p be able to find all the complex roots of the equation for any positive integer value of, write them in Cartesian, polar or exponential form, be able to plot these points on an Arg diagram without first calculating their values, in particularly they should know 1 is one of the roots, the roots all lie on the unit circle, the -fold rotational symmetry, the symmetry in the -axis know be able to use the following properties of the roots the sum of the roots equals 0 if is a root of then Lecture 7: Roots of a general complex number Reading: Yellow Bostock Chler p be able to find all the complex roots of the equation for any complex number any positive integer value of write them in Cartesian, polar or exponential form, know the relationship between the roots of the roots of be able to plot these points on an Arg diagram without first calculating all of their values, in particularly they should know the roots all lie on the circle, the -fold rotational symmetry Vectors 3 weeks, 6 lectures. Lecture 1: Basic definitions & operations

12 Reading: Red Bostock Chler p & p Be able to perform the following operations on vectors underst the geometric meaning of these operations: addition, subtraction scalar multiplication. Be able to find the modulus of a vector, the unit vector in the same direction as a given vector. Know how to calculate the dot product of two vectors; know & use the properties of the dot product. Know the identity be able to use it to calculate the angle between two vectors. Lecture 2: Cross product (vector product) triple product Reading: Yellow Bostock Chler p64-71 Be able to calculate the determinant of a matrix Know how to calculate the cross product of two vectors; know & use the properties of the cross product. Know the identity be able to use it to calculate the area of a triangle or a parallelogram. Know how to calculate the triple product of three vectors; know & use the properties of the triple product. Be able to use the triple product to calculate the volume of a parallelepiped or a tetrahedron. Lecture 3: Equations of lines Reading: Red Bostock Chler p Be familiar with the 3 forms of describing lines in 3D i.e. the vector equation, parametric equations, Cartesian equations. Be able to convert one form of the description of a line into another. Be able to find the equation of a line when given enough geometric information to define the line, e.g. find the line that goes through 2 specified points, or that goes through a specified point in a specific direction. Lecture 4: Equations of planes Reading: Red Bostock Chler p Be familiar with the 3 forms of describing planes in 3D i.e. scalar product form, parametric form, the Cartesian equation. Be able to convert one form of the description of a plane into another.

13 Be able to find the equation of a plane when given enough geometric information to define the line, e.g. find the plane that goes through 3 specified points, or that goes through a specified point is parallel to 2 specified directions, etc. Lecture 5&6: 3d Geometry problems Reading: Red Bostock Chler p , p & p512-7 Be able to find the point(s) of intersection between lines & planes. Be able to solve geometric problems involving the distance between /or the angle between lines, planes points.