stationary points. gradient of a curve copy this curve onto your whiteboard. mark on to the graph...
DESCRIPTION
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,TRANSCRIPT
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Stationary Points
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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?
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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
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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
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The Remainder Theorem
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Aims•To find the factors of cubic expressions•To explore remainders•To discover the remainder theorem
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The Remainder Theorem•Long division.
•Calculate 253626
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Remainders in algebraic division
This leads to the remainder theorem:
Raxxpxf
3451552
32 2
23
xxx
xxx
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Division of a polynomial with remainders Divide by
Method 1 (Equating Coefficients)
22 23 xxx 2x
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Division of a polynomial with remainders Divide by
Method 2 (Long Division)
22 23 xxx 2x
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Division of a polynomial with remainders Divide by
Method 3 (Synthetic Division)
22 23 xxx 2x
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Factor and Remainder Theorem 5127 23 xpxxxf
f(x) has a remainder of -5 when divided by (x + 2)Find the value of p