Stationary Points

Gradient of a Curvedxdy

Copy this curve onto your whiteboard.

Mark on to the graph where it has a positive gradient.

Mark where it has a negative gradient

If y stands for the distance travelled by a car and x stands for time, when is the car stationary?

The terms for a peak and a trough of a curve are the maximum and minimum points. They are examples of turning points.

Examples of problems with stationary points are:•Finding the maximum profit for a business•Finding the time at which chemicals are reacting fastest•Find the point at which a missile reaches its peak height•Finding the peak of a sound wave•Finding the mode of a statistical distribution•Minimising the cost of restocking a supermarket

At a turning point,

This is an equation that you must solve to find the values of x

At a turning point, the tangent is parallel to the x-axis

0dxdy i.e, 0' xf

Summary of Finding a Stationary Point1. D

2. F

3. S

If you need to determine the nature (type) of the stationary point(s)4. Differentiate again to obtain the formula for

5. Substitute the x value(s) you found into and look at its sign

If then the turning point is a Minimum pointIf then the turning point is a Maximum pointIf then

The Remainder Theorem

Aims•To find the factors of cubic expressions•To explore remainders•To discover the remainder theorem

The Remainder Theorem•Long division.

•Calculate 253626

Remainders in algebraic division

This leads to the remainder theorem:

Raxxpxf

3451552

32 2

23

xxx

xxx

Division of a polynomial with remainders Divide by

Method 1 (Equating Coefficients)

22 23 xxx 2x

Division of a polynomial with remainders Divide by

Method 2 (Long Division)

22 23 xxx 2x

Division of a polynomial with remainders Divide by

Method 3 (Synthetic Division)

22 23 xxx 2x

Factor and Remainder Theorem 5127 23 xpxxxf

f(x) has a remainder of -5 when divided by (x + 2)Find the value of p