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 Differentiation 1

The derivative of f(x) as the gradient of the tangent to the graph y = f(x) at a point

The notations f’(x) or dy will be used

                                   dx

By first principles is not required.

RISP 36 - First steps into differentiation 

Autograph Introducing Differentiation

Autograph Pre-Calculus Activity

The gradient of the tangent as a limit

Interpretation as  rate of change

A general appreciation only of the derivative when interpreting it is required. Differentiation from first principles will not be tested

Standards Unit C3 - Functions and Derivatives

RISP 21  - Advanced Arithmogons

Autograph Differentiation of Quadratic

Autograph Differentiation of Cubic

Differentiation of polynomials

Differentiation of xⁿ, where n is a rational number, and related sums and differences

Eg x ^3/2 + 3x²          

including terms which can be expressed as a single power such as xÖ x

Standards Unit C2 - Functions with non-integer Indices 

Applications of differentiation to gradients, tangents and normals

The equation of the tangent and normal at a given point to the circle

Implicit differentiation is not required. Candidates will be expected to use the coordinates of the centre and a point on the circle or of other appropriate points to find relevant gradients

Autograph Tangent to a Curve

Applications of differentiation to:

         maxima and minima

         stationary points

         increasing and decreasing functions

Questions will not be set requiring the determination of or knowledge of points of inflection

Questions may be set in the form of a practical problems where a function of a single variable has to be optimised

Standards Unit C5 - Stationary Points on cubics 

Second order derivatives

Application to determining maxima and minima

Differentiation of ex, sin x, cos x, tan x and linear combinations of these functions. To include simple composite functions.

Differentiation using the product rule, the quotient rule, the chain rule and by the use of dy/dx = 1/(dx/dy)

E.g.

x²lnx

e^3xsinx

(e^2x-1)/(e^2x+1)

(2x+1)/(3x-2)

 Algebra and Functions 1

Laws of indices for all rational exponents

Standards Unit N12 - Using Indices

Use and manipulation of surds

To include simplification and rationalization of the denominator of  a fraction

√12 + 2√27 = 8√3

   1        = √2 + 1

√2 -1

Standards Unit N11 -  Manipulating Surds

Completing the square

Autograph Completing the Square

Quadratic functions and their graphs

Reference to the vertex and line of symmetry of the graph

Standards Unit C1 - Properties of Quadratic Graphs 

The discriminant

Conditions for real roots, distinct real roots and for no real roots

Factorization of quadratic poynomials

Eg 2x² + x – 6

RISP 3 

Solution of quadratic equations

Any method

Autograph Roots of Quadratics

Algebraic manipulation of polynomials, including expanding brackets and collecting like terms

Knowledge of the effect of simple transformations on the graph y = f(x) as represented by

y = af(x)         y = f(x) +a

y = f(x + a)      y = f(ax)

Expected to use the terms reflection, translation and stretch in the x or y direction in their descriptions of these transformation

Eg y = sin 2x       y= cos(x + 30°)

     Y = 2 ^x+3        y = 2 ^–x

Descriptions involving combinations of more than one transformation will not be tested.

Knowledge of the effect of simple transformations on the graph y = f(x) as represented by

y = af(x)         y = f(x) +a

y = f(x + a)      y = f(ax)

Expected to use the terms reflection, translation and stretch in the x or y direction in their descriptions of these transformation

Eg y = sin 2x       y= cos(x + 30°)

     Y = 2 ^x+3        y = 2 ^–x

Descriptions involving combinations of more than one transformation will not be tested.

Simplification of rational expressions including factorising and cancelling, and algebraic division.

Use of the Remainder and Factor Theorems.

Expressions of the type

    (3+2x)       
(x-3(2x² + 1)

RISP 11 - Remainder & Factor Theorem

Partial fractions (denominators not more complicated than repeated linear terms). 

Greatest level of difficulty  and

Solution of linear & quadratic inequalities

2x² + x ³ 6

Simple algebraic division

Quadratic or cubic polynomial divided by a linear term of the form (x+ a) ( x – a)  where a is a small whole number

Standards Unit A11 - Factorising Cubics 

RISP 21  - Advanced Arithmogons

RISP 22

Use of the remainder theorem

When a quadratic or cubic polynomial f(x) is divided by ( x – a) the remainder is f(a) and, that when f(a) = 0, then ( x – a) is a factor and vice versa

Use of the factor theorem

Greatest level of difficulty

 x³ – 5x² + 7x – 3

ie always a factor ( x + a) ( x – a ) including cases of three distinct linear factors, repeated linear factors or a quadratic factor which cannot be factorised

 Coordinate Geometry

Equation of a straight line, including the forms y – y1 = m( x – x1 ) , ax + by  c = 0 and y = mx + c

Problems using gradients, mid-points and the distance between two points

Identifying Points on Straight Lines Jigsaw 

Autograph Defining Gradient

Autograph Distance, Gradient & Midpoint

Conditions for two straight lines to be parallel or perpendicular to each other

Product of two perpendicular lines is -1

Standards Unit A10 - Perpendicular Lines

Straight Line Facts Jigsaw

Autograph Perpendicular Lines

Autograph Perpendicular Bisector

The equation of a circle in the form

( x – a) ² + ( y – b ) ² = r²

Completing the square to find the centre and radius of a circle

Eg x² + 4x + y² -6y – 12 = 0

Coordinate geometry of the circle

Use of the following circle properties is required

i) the angle in a semicircle is a right angle

ii) the perpendicular from the centre to a chord bisects the chord

iii) the tangent to a circle is perpendicular to the radius at its point of contact

Cartesian and parametric equations of curves and conversion between the two forms.  Knowledge of the curves given by the following parametric equations will be expected:  

x = aat² , y = 2at for a parabola,  

x = asint, y = bcost for an ellipse or circle (a=b),

x = ct, y = c/t for a rectangular hyperbola.

e.g.

x=t² , y=2t;

x = acost, y = bsint;

x = 1/t, y = 3t;

x = t+1/t, y = t-1/t  Þ (x+y)(x-y)=4

Standards Unit A14 - Exploring Parametric Equations shown Graphically

RISP 37 - Parabolic Clues
RISP 27 - Parametric Equations

 Integration 1

Indefinite integration as the reverse of differentiation

Standards Unit C4 - Non Integer Indices 

Autograph Integration as Reverse of Diff

Integration of polynomials

Autograph Integration of Quadratics

Evaluation of definite integrals

Interpretation of the definite integral as the area under the curve

Area between a curve and the x axis

Areas wholly below the x axis, knowledge that the integral will give a negative value.

Areas partially above and below the x axis will not be set

RISP 25 - The Area is 1 

Autograph Calculus Areas

see Numerical Methods

Integration of xⁿ , n is a rational number and n ≠ -1

also related sums and differences

Expressions such as  x ^3/2 + 2x ^-1/2

Or x + 2 = x ^½ + 2x ^{– 1//2}

       Öx

Integration of sinx, cosx.  To include ∫cos²xdx etc

Simple cases of integration by substitution and integration by parts.

Inspection

∫e^-3xdx;    ∫sin4xdx;    ∫x(1+x²)^0.5dx

Substitution

∫x(2x+6)⁶dx;   ∫x(2x-3)^0.5dx

By parts

∫xe^2xdx;   ∫xsin3xdx;    ∫xlnxdx

These methods as the reverse processes of the chain and product rules respectively.

∫ f'(x)/f(x)dx by inspection or substitution

Evaluation of standard integrals.  To include

∫1/(a²+x²)dx and ∫1/(a²-x²)^0.5dx

Standard integrals given in the formula booklet

Evaluation of volume of revolution.  The axes of revolution will be restricted to x and y axes.

∫∏y²dx about the x-axis   or   

∫∏x²dy about the y-axis

 Proof

Construction and presentation of rigorous mathematical arguments through appropriate use of precise statements and logical deduction.

Not counter examples

 RISP 1 - Triangle Number Differences

Correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, implies’, ‘is implied by’, ‘necessary’, ‘sufficient’, and notation for Û, Ü, or Þ therefore

|X+3|<3|X| * X>3

Replace * by Û, Ü, or Þ

 Algebra and Functions 2 

Solving simultaneous equations

               Two linear

               A linear and a quadratic

Analytical solution by substitution

RISP 8

RISP 21  - Advanced Arithmogons

Phone Bills Problem

Excel Phone Bills

Intersection of a straight line and a curve

Solutions from intersection points

Applications will be to either circles or graphs of quadratic functions

Interpretation of geometrical implication of equal roots, distinct real toots or no real roots

RISP 5 

Autograph Simultaneous Equations

Graphs of functions: sketching,

f(x) notation, but only a general idea of the concept of a function is required. Domain and range are not included

linear, quadratic, cubic

graphs of circles are included

Knowledge of the effect of translations on graphs and their equations

Applied to quadratic graphs and circles ie
y = ( x – a )² + b as a translation of
y = x² and  (x – a)² + ( y –b)² = r² as a      translation of x² + y² = r²

RISP 17 

Domain and range of functions. Composition of functions. Inverse functions.

Notation such as f(x)=x2-4

Domain and range may be expressed as x>1 for example

fg(x)=f(g(x))

RISP 18 - When does fg = gf?
RISP 16 - Combining Negative functions

Autograph Composite Functions

Graphs of functions and their inverses; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations.

Inverse of f(x) written as f-1(x)

To include reflection in y = x.

The modulus function.

To include related graphs and the solution of inequalities such as |x+2|<3|x| using solutions of |x+2|=3|x|

 Trigonometry

Sine, cosine and tangent functions

Their graphs, symmetries and periodicity

Concepts of odd and even functions are not required

Use the sine and cosine rules

The area of a triangle in the form

                                               ½ab sin C

Degree and radian measure

RISP 24 - 3 Fact Triangles 

RISP 23 - Radians and Degrees

Converting between degrees & radians

Sector arc length and area

Knowledge of the formulae l = rθ

                                    Area = ½ r² θ

Knowledge and use of

Tan θ = sin θ/cosθ

sin² θ + cos² θ = 1

Solution of simple trigonometric equations in a given interval of degrees or radians

Max level of difficulty ;

Sin 2θ = - 0.4

Sin (θ - 20°) = 0.2

2Sin θ – cos θ = 0

2 sin² θ + 5 cos θ = 4

Knowledge of secant, cosecant and cotangent and arcsin, arccos and arctan. Their relationships to sine, cosine and tangent.  Understanding of their graphs and appropriate restricted domains.

Knowledge that, -∏/2 <arcsinx<∏

0 <arccosx<∏,    -∏/2 <arctanx<∏

The graphs of these functions as reflections of the relevant parts of the trigonometric graphs in x y plane included. The addition formulae for inverse functions are not required.

Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a , y = f(x + a), y = f(ax) and combinations of these transformations.

lnx leading to 2ln(x-1);

secx leading to 3sec2x

Autograph asin(x) & cos(x)+d

Autograph asin(bx) & tan(x+d)

Knowledge and use of,

1 + tan²x = sec²x and 1 + cot²x = cosec²x

Use in simple identities. Solution of trigonometric equations and inequalities in a given interval. 

Knowledge and use of double angle formulae. Use of formulae for sin(A±B) & cos(A±B) and tan(A±B) and of expressions for acosx+bsinx in the equivalent forms of rcos(x±a) or rsin(x±a)

Knowledge that sin2x = 2sinxcosx

cos2x = cos²x-sin²x = 2cos²x-1 = 1 - 2sin²x

and tan 2x = 2tan2x

                      1-tan²x

Solution of trigonometric equations in a given interval

e.g. 2sinx + 3cosx = 1.5

       3sin2x = cosx

Use in simple identities

Sin3x = sin(2x + x) = sinx(3 – 4sin²x)

Use in integration e.g. ∫cos²xdx

RISP 26 - Generating Compound Angle Formula 

Laying Out Trig Identities

GSP File - asinx+bcosx=rsin(x+θ)

Sequences and Series

Sequences , including those given by a formula for the nth term

Position to term formulae

Standards Unit N13 Arithmetic & Geometric

Sequences generated by a simple relation of the form x _n+1 = f(x_n)

Iterative formulae

To include their use in finding of a limit L as n à ¥ by putting L = f(L)

RISP 2 - Sequence Tiles 

Arithmetic series, including the formula for the sum of the first n natural numbers

To include the å notation for sums of series

RISP 20 - When does Sn = Un

The sum of a finite geometric series

RISP 20 - When does Sn = Un

Sigma and Series - Jigsaw

The sum to infinity of a convergent

( -1<r<1 ) geometric series

Should be familiar with the notation êr ê < 1 in this context

The binomial expansion of (a + b)ⁿ for positive integer n

Should be familiar with the notation n ! and nCr and [n]

                r

Use of Pascal’s triangle or formulae to expand ( a + b)ⁿ will be accepted

RISP 32 - Exploring Pascal's Triangle

The binomial expansion ( 1 + x ) n for positive integer n

Binomial series for any rational n. To include.

(a+x)ⁿ, |x|<a

Series expansion of rational functions including the use of partial fractions.

Greatest level of difficulty

(2+3x)^-2 = 1/4(1+3x/2)^-2

Greatest level of difficulty .

      9+2x²        

(2x+1)(x-3)²

RISP 19 - Extending the Binomial Theorem 

 Exponentials and logarithms

y = a^x and its graph

Using the laws of indices where appropriate

Simple Applet looking at fish growth Sliders in the applet control panel are used to change parameters included in the definition of the exponential function which has the form

The values of the coefficients a, b, c, d, and the base B may be changed continuously (small increments).

Logarithms and the laws of logarithms

log_a x + log_ay = log_a (xy)

log_ax – log_ay = log_a (x/y)

klog _ax = log_a (x^k)

The equivalence of

y = a^x and x = log_ay

RISP 31 Building Log Equations

Standards Unit A13 Exploring Log Expressions

Rules of logs Jigsaw

Rules of logs - scroll to end

The solution of equations of the form

 a^x = b

Use of a calculator logarithm function to base 10 ( or base e) to solve for example 3 ^2x =2

Wilkpedia - e

RISP 13 - Introducing e
Generates single problems and gives solution

The function e^x and its graph

Autograph: Describe a series of transformations which will map

Y=e^x onto y=e^2(x-1)

The function lnx and its graph; lnx as the inverse function of ex.

Exponential growth and decay. The use of exponential functions as models.

 Numerical Methods

Location of roots of f(x)=0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous. To include interval bisection and linear interpolation.

Approximate solutions of equations using simple iterative methods, including Newton - Raphson.  Rearrangement of equations to the form x = g(x).

f(x)=6-x-lnx

Show that f(x) has a root between 4 and 5 and determine whether the root is closer to 4 or 5.

Identifying Roots of Equations Domino

Autograph Iteration 1
Excel Iterations 1
Excel Iterations 2
Iterations Spreadsheet

Staircase and cobweb diagrams.

Autograph Iteration

Approximation of the area under a curve using the trapezium rule

The term ‘ordinate’ will be used.

To include a graphical determination of whether the rule over- or under- estimates the area and improvement of an estimate by increasing the number of steps.

RISP 25 - The Area is 1 

Autograph Trapezium Rule

Numerical integration of functions using the mid-ordinate rule and Simpson’s rule

To include a geometrical determination of whether the rule over- or under-estimates the area and improvement of an estimate by increasing the number of steps.

Autograph Area under Straight Line

Autograph Area under a Curve 1

Autograph Area under a Curve 2

Autograph Calculus Areas

Autograph Mid-Ordinate Rule

 Differentiation and Integration 2

Formation of simple differential equations.  To include the context of growth and decay.

Autograph Differential Equations

Simple cases of integration using partial fractions.

Maximum level of difficulty

∫         (x+1)        dx     ∫        x²        dx

     (3x-4)(x+3)²               (x+5)(x-3)

Analytical solution of simple first order differential equations with separable variables.

RISP 28 - Modelling with DE

RISP 30 - Differential Equations

Autograph Terminal Velocity

Differentiation of simple functions defined implicitly or parametrically. To include the examples in the co-ordinate geometry section above.

The second derivative of curves defined implicitly or parametrically is not required.

Differentiating using different variables - Jigsaw 

Equations of tangents and normals for curves specified implicitly or in parametric form.

 Vectors

Vectors in two and three dimensions. Position vectors.

Distinguish between the terms Vector and position vector.

Expressing vectors from a diagram 

Magnitude of a vector.

The distance between two points.

Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations.

Vector equations of lines. Equations of lines in the form .  To include the intersection of two straight lines in two and three dimensions and skew lines in three dimensions.

Call some point in the room the origin. Use a piece of string to represent the line. Get two students to hold the string it's more visually impressive if you get someone to stand on a chair) and refer to them as position vectors (ie
vector Alice and vector Bob). Another student represent a general point on the line, so the vector Caroline is somewhere along the Alice->Bob Vector. She can be moved to various fractions of the Alice-Bob Vector.
Students can they say what her position is based on vectors of the two original people, and the origin. This gets you to Caroline=Alice+t*(Alice->Bob) which can then be made more algebraic.

Autograph Vector Equation of a line 1

Autograph Vector Equation of a line 2

The scalar product. Its use for calculating the angle between two lines.

The co-ordinates of the foot of the perpendicular from a point to a line. The perpendicular distance from a point to a line.