Maclaurin Series are Taylor Series centered at zero

You have to learn the formula for a Maclaurin Series expansion, or know the formula for a Taylor Series expansion and use the special case centered at zero

It’s good practice to create a table of required derivatives before plugging into the formula – doing this on the fly can get messy and introduce errors

It is useful to memorise common Maclaurin Series expansions

The key to Trig Equations is good fundamental Trig skills. These include the CAST diagram, trig exact values, and working with radians

Some solutions to trig equations are best taken off the graph of the trig function

Trig equations often have to be factorised so be sure to equate to zero

Keep an eye on the allowed range of values for solutions to be sure you include all solutions but don’t go outside of the range

The Limit Comparison Test requires a series with a known result to compare to

The comparison series can be deduced from the series to be tested

The Limit Comparison Test is not a convergence or divergence test, it simply tells you whether the two series have the same convergence / divergence result

If the result of the Limit Comparison Test is positive and finite the two series share the same convergence / divergence result

A derived function graph helps visualise the derivative of a function

The key to sketching a derived function graph is using the stationary points

No other key points on the original graph are needed to sketch the derived graph

Where the function is increasing the derived graph is above the x-axis

Where the function is decreasing the derived graph is below the x-axis

Where the function has stationary points the derived graph has roots

The derived graph should be of a type one power less then the function

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