Stationary Points. Gradient of a Curve Copy this curve onto your whiteboard. Mark on to the graph where it has a positive gradient. Mark where it has.

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Presentation transcript:

Stationary Points

Gradient of a Curve 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 i.e,

Summary of Finding a Stationary Point 1.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 point If then the turning point is a Maximum point If 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

Remainders in algebraic division This leads to the remainder theorem:

Division of a polynomial with remainders Divide by Method 1 (Equating Coefficients)

Division of a polynomial with remainders Divide by Method 2 (Long Division)

Division of a polynomial with remainders Divide by Method 3 (Synthetic Division)

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