## Quick Maclaurin’s series expansion using GC

In Maclaurin’s series expansion, where up to f”(0) need to be evaluated, can be done quickly using the GC.

Example

Expand  up to powers of 2.

Using TI 84 Plus (OS 2.55)

From G.C,

f(0)= 3/2

f'(0) = -3/4

f”(0)= 5/4

Therefore using Maclaurin’s series expansion,

If powers higher than 2 such as 3 are required, then do differentiation to find the first derivative. Then enter into Y1. Y2 and Y3 will then give the second and third derivative.

## Quick binomial expansion using GC

Binomial expansion can be done quickly using the GC.

Consider binomial expansion of , where a is a real number, and q is a fraction or negative integer.

The recurrence formula for the coef works out to be

Example 1

Expand   in ascending powers up to and including the term in

Using TI 84 Plus (OS 2.55)

Therefore, the expansion is

The recurrence formula can also be used to expand in descending powers of x.

Example 2

Expand  in descending powers of x up to the 7th term.

Using TI 84 Plus (OS 2.55)

Therefore, the expansion is

## Using GC to determine the nature of stationary points

Suppose we are required to determine the nature of stationary points for the following:

Using TI-84 Plus (OS 2.55)

Observation of stationary points:

Left stationary point: First derivative changes from +ve to -ve. Therefore, it is a maximum point

Middle stationary point: No change in sign of deriative. Therefore, it is a point of inflexion.

Right stationary point. First derivative changes from -ve to +ve. Therefore, it is a minimum point

## How to quickly find the intersection between planes

Finding the intersection between planes is typically 4 marks in A level exam, and can be solved using this method in 1 min 🙂

For example, find the line of intersection of r. ( 1 1 0 )=3 and r.(-1 -1 2 )= 7

Basically, we are finding the solution of these two equations:

x+y = 3

-x-y+2z = 7

Use TI 84 plus calculator.

so we have

x= 3-y

y= y

z=5

So the line of intersection is r = (3 0 5) + a(- 1 1 0),  where a is a real number.