Stationary Points

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.

1. The 1st LiveMath document (IT4AMDiffnStatPts1.thp) shows the graph of y = ax³ + bx² + cx + d (blue curve) where a = 0, b = 1, c = –2 and d = 0. The red line is the tangent to the curve at P(x₁, y₁) where x₁ = –2.

2. The gradient of the curve (or the tangent) at P(x₁, y₁) is denoted by y₁' under the subheading "Derivative of y". Please note that LiveMath uses a different symbol for dy/dx. [Actually, the symbol is for partial derivatives.]

3. Change the value of x₁ to those given in the table below and record the gradient of the curve y = x² – 2x at various points of P in the same table.

x-coord of P –2 –1 0 1 2 3 4

Gradient of Curve at P

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.

1. The 2nd document (IT4AMDiffnStatPts2.thp) shows the graph of the same curve y = x² – 2x. The red line is the tangent to the blue curve at the point P where the x-coordinate is h.

2. Click on the graph and it will show the animation of the tangent at different points on the curve. The x-coordinates of these points, h, will increase from –2 to 4 in steps of 0.5. The main purpose is to observe the gradient of the tangent (and so the gradient of the curve) and see how it changes its sign as it goes past the stationary point from left to right (or as x increases).

Note: The animation will stop at 4. It will not include the last value 4.5.

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.

x-coord of P	–3	–2	–1	–0.5	0	1	2
Gradient of Curve at P


Question 3:	Write down the coordinates of the stationary point.

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?

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?

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.

x-coord of P	–3	–2	–1	0	1	2	3
Gradient of Curve at P


Question 11:	Write down the coordinates of the stationary point.

Question 12:	Is this stationary point a maximum or a minimum point?

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.

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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.

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.


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?

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?

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).