All straight lines are found using y – b = m(x – a)

The gradient, m, can be found using the derivative of the curve

The gradient is the derivative evaluated at the point of tangency

If you’re only given the x co-ordinate, substitute into the function to find y

The values of a function are the f(x) or y values

Increasing function means the values are increasing

Increasing effectively means the graph of the function is going up to the right

The slope is measured by the gradient which is just the derivative

So, we use the functions derivative to show that it is always increasing

Equations with logarithms are solved by removing the logarithm

The properties, or rules, of logarithms are a key skill here

Logarithms can be removed by ‘dropping’ the logarithm in some cases

Logarithms can be removed by rewriting as an exponential in some cases

Collinear refers to points laying in a straight line

Collinearity can be shown using vectors or gradients

Using gradients we show the vectors are parallel and share a point

Vectors are parallel if one is a multiple of the other

Points can be collinear but there are no collinear vectors

Definite integrals can be used to find the area under a curve

If the area is above the x-axis then the definite integral is exactly the area

If the area is below the x-axis the definite integral will come out negative but the negative is dropped when stating the area

Areas can be used to evaluate definite integrals but you should keep in mind whether the area is above or below the x-axis

Integration by Parts is for evaluating integrals written as a product of functions

It is similar to the product rule for derivatives

The key is assigning the correct function as ‘u’ and ‘v’

Sometimes you will have to repeat the integration by parts as the integral in the first round will also need to be integrated by parts

Integration by parts works for both indefinite and definite integrals

Proof by Induction is a method of demonstrating that a Mathematical result is true

It is particualrly useful when you have a large number of cases, or infinitely many cases, to prove the result for

Proof by Induction is a two step process – the ‘basic’ step and the ‘inductive’ step

The basic step proves the result for the first case

The inductive step says that if the result is true for n = k it has to be true for n = k+1

How you perform the inductive step will depend on the type of result you are trying to prove

The area ‘under a curve’ is the area beneath the curve and between the x-axis and limits

The area under is a curve is found by setting up and evaluating a definite integral

Practice definite integrals to be successful in finding the area under a curve

If the area is underneath the x-axis the definite integral will evaluate to a negative number. Ignore the negative when writing the area since an area cannot be negative

Definite integrals give a numerical solution not another function

The numerical solution can be positive or negative

Definite integrals are evaluated using anti-derivatives and The Fundamental Theorem of Calculus

The anti-derivative technique used will depend on the type of function being integrated

To solve a trigonometric equation without a calculator use exact values

Learn the exact values for Sin, Cos & Tan for 0, 30, 45, 60 & 90 degrees

The ‘trig triangles’ can help you work out the exact values

Trigonometric equations also require the use of the CAST diagram