AQA Core 3

 

14.1 Functions

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))

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|

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.

ex leading to e2x - 1

lnx leading to 2ln(x-1);

secx leading to 3sec2x

Transformations of functions included in modules Core 1 and Core 2

14.2 Trigonometry

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.

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 and use of 1 + tan2x = sec2x and 1 + cot2x = cosec2x

Use in simple identities.

 

Solution of trigonometric equations and inequalities in a given interval. 

14.3 Exponentials and Logarithms

The function ex and its graph

Autograph: Describe a series of transformations which will map

Y=ex onto y=e2(x-1)

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

14.4 Differentiation

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

e3xsinx

 

(e2x-1)/(e2x+1)

 

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

14.5 Integration

 

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

∫xe2xdx;   ∫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 volume of revolution.  The axes of revolution will be restricted to x and y axes.

14.6 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.

Staircase and cobweb diagrams.

Numerical integration of functions using the mid-ordinate rule and Simpsonís 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.