DIFFERENTIATION IT LESSON 4 - STATIONARY POINTS
(SEC 4 ADDITIONAL MATHEMATICS)
Software: LiveMath
(previously known as MathView)
Thinking Skills: Induction
and Deduction
At the end of the lesson, the students should be able to:
(1) explain how the first derivative or gradient changes over a stationary
point,
(2) apply the first derivative test to determine the nature of the
stationary point.
A. MINIMUM POINT
If you cannot view the LiveMath
document below, click here to
troubleshoot.
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Question 1: | Write down the coordinates of the point where the gradient
is 0.
This point is called a stationary point. |
Question 2: | As x increases from –2 to 4, what do you notice about the sign of the
gradient of the curve?
This stationary point is called a minimum point. |
B. ANIMATION
If you cannot view the LiveMath
document below, click here to
troubleshoot.
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C. MAXIMUM POINT
1. | Using the first LiveMath document above, change the values of a, b, c and d so that it shows the graph of y = 2 –x– x2. Change the value of x1 to those given in the table below and record the gradient of the curve at various points of P in the same table. |
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Question 3: | Write down the coordinates of the stationary point.
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Question 4: | As x increases from –3 to 2, what do you notice about the sign of the
gradient of the curve?
This stationary point is called a ______________ point. |
D. FIRST DERIVATIVE TEST
1. | How do you determine the nature of the stationary point? By this, we mean how you decide whether the stationary point is a maximum or a minimum or neither. |
Question 5: | Define a stationary point: A stationary point is a point
on the curve where the gradient is equal to __________.
Can you explain why they are called stationary points?
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Question 6: | A stationary point can be a maximum point, a minimum point or a point
of inflexion (see Enrichment below). Maximum points and minimum points
are also called turning points. Can
you explain why they are called turning points?
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Question 7: | Define a minimum turning point: A minimum turning point is a point on the curve where the gradient changes sign from _____________ to _____________ as x increases through the point. |
Question 8: | Define a maximum turning point: A maximum turning point is a point on the curve where the gradient changes sign from _____________ to _____________ as x increases through the point. |
Question 9: | Define a turning point: A turning point is a point on the curve where the gradient changes ________ as x increases through the point. |
Question 10: | In conclusion, explain how you would test whether a stationary point
is a maximum or a minimum point.
This is called the first derivative test (as it makes use of the first derivative). |
E. POINT OF INFLEXION (ENRICHMENT)
1. | Using the first LiveMath document above, change the values of a, b, c and d so that it shows the graph of y = x3. Change the value of x1 to those given in the table below and record the gradient of the curve at various points of P in the same table. |
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Question 11: | Write down the coordinates of the stationary point.
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Question 12: | Is this stationary point a maximum or a minimum point?
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Question 13: | As x increases from –3 to 3, what do you notice about the sign of the
gradient of the curve?
This stationary point is called a point of inflexion. |