Rates of Change

DIFFERENTIATION IT LESSON 3 - RATES OF CHANGE
(SEC 4 ADDITIONAL MATHS)

Software: LiveMath (previously known as MathView)
Thinking Skills: Induction and Deduction
At the end of the lesson, the students should be able to:
(1) find the rate of change of x with respect to t using the derivative dx/dt,
(2) identify whether the change is increasing or decreasing,
(3) identify whether the rate of change is constant or not.

A. INTRODUCTION TO RATES OF CHANGE

1. The radius, r cm, of a circle changes with time, t seconds, and they are related by the equation r = 3t³ + 2. But what does this equation mean? Is r increasing or decreasing at a constant rate?

2. Drawing a table of values of r for different values of t will help. Calculate and record the values of r in the table below.

t / s	0	1	2	3
r / cm


Question 1:	Is the radius r increasing or decreasing with time t?

Question 2:	Is the radius r changing at a constant rate? How do you know?

B. ANIMATION

If you cannot view the LiveMath document below, click here to troubleshoot.

1. The first document (IT4AMDiffnRatesOfChange.thp) shows a circle with radius r = 3t³ + 2. The equations of the circle in parametric form are x = r cos p and y = r sin p. What you need to know about these equations is where the radius r is.

Note: To plot a circle on a computer using its Cartesian equation will take a long time. That is why the parametric form is used. You will get to study the parametric form in a junior college later.

2. Click on the graph to animate the circle for t = 0 to 3 in steps of 1. Notice that the circle "suddenly becomes very big", i.e. the rate of change is not constant.

Note: Although the box below the graph says t = 0 to 4, it does not include this last value.

Question 3: Differentiate r = 3t³ + 2 with respect to t to get dr/dt which is the rate of change of r with respect to t. Does the value of dr/dt depend on the time t? So does it tell you whether dr/dt is constant or not?

C. CONSTANT (OR STEADY) RATES OF CHANGE

1. Change r = 3t³ + 2 in the LiveMath document above to r = 3t¹ + 2 by highlighting the power 3 and typing in the number 1. Animate this graph for t = 0 ... 26 in steps of 2 by highlighting the relevant values and changing them accordingly. What does this equation mean?

2. Calculate and record the values of r in the table below.

t / s	0	1	2	3
r / cm


Question 4:	Is the radius r increasing or decreasing with time t?

Question 5:	Is the radius r changing at a constant rate? How do you know?

Question 6:	Differentiate r = 3t + 2 with respect to t to get the rate of change dr/dt. Does the value of dr/dt depend on the time t? So does it tell you whether dr/dt is constant or not?


3.	Click on the graph above to animate the circle for r= 3t + 2 again. Do you observe that the radius of the circle is increasing at a constant rate? A constant rate of change is also called a steady rate.

D. INCREASING OR DECREASING?

1. In the above example, r = 3t + 2 implies dr/dt = 3 and you can observe from the animation of the graph that the radius of the circle increases with time. What happens if the radius decreases with time?

2. Change the equation in the LiveMath document above to r = –3t + 70 and animate this graph for t = 0 ... 26 in steps of 2.

Question 7: Is the radius r increasing or decreasing with time t?

Question 8: Differentiate r = –3t + 70 with respect to t to get the rate of change dr/dt. What does the negative value of dr/dt tell you about the type of change of r with respect to t?

E. EXERCISE

1. In general, if the rate of change of x with respect to t is constant, i.e. dx/dt = m where m is a constant, then you can express x in terms of t linearly, i.e. x = mt + c (the equation of a straight line where the gradient m is a constant). The constant c can be found if you know the initial value of x when t = 0.

2. In the example in Sect C, the rate of change of the radius r of the circle with respect to t is dr/dt = 3 cm/s, which is a constant. So r= 3t + c. If we know that the initial radius of the circle is r = 2 cm, i.e. r = 2 when t = 0, then c = 2. Therefore r =3t + 2.

Question 9: The side of the square, x cm, changes with time, t s, such that dx/dt = 4 cm/s. If the square has a side of 1 cm initially, find an expression for x in terms of t.

3. It is important to take note of this difference. If x = mt + c, then x is linearly related to t, but the rate of change of x w.r.t. t is constant (not linear).

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1.	The radius, r cm, of a circle changes with time, t seconds, and they are related by the equation r = 3t³ + 2. But what does this equation mean? Is r increasing or decreasing at a constant rate?

2.	Drawing a table of values of r for different values of t will help. Calculate and record the values of r in the table below.

1.	Change r = 3t³ + 2 in the LiveMath document above to r = 3t¹ + 2 by highlighting the power 3 and typing in the number 1. Animate this graph for t = 0 ... 26 in steps of 2 by highlighting the relevant values and changing them accordingly. What does this equation mean?

2.	Calculate and record the values of r in the table below.

1.	In the above example, r = 3t + 2 implies dr/dt = 3 and you can observe from the animation of the graph that the radius of the circle increases with time. What happens if the radius decreases with time?

2.	Change the equation in the LiveMath document above to r = –3t + 70 and animate this graph for t = 0 ... 26 in steps of 2.


Question 7:	Is the radius r increasing or decreasing with time t?

Question 8:	Differentiate r = –3t + 70 with respect to t to get the rate of change dr/dt. What does the negative value of dr/dt tell you about the type of change of r with respect to t?

1.	In general, if the rate of change of x with respect to t is constant, i.e. dx/dt = m where m is a constant, then you can express x in terms of t linearly, i.e. x = mt + c (the equation of a straight line where the gradient m is a constant). The constant c can be found if you know the initial value of x when t = 0.

2.	In the example in Sect C, the rate of change of the radius r of the circle with respect to t is dr/dt = 3 cm/s, which is a constant. So r= 3t + c. If we know that the initial radius of the circle is r = 2 cm, i.e. r = 2 when t = 0, then c = 2. Therefore r =3t + 2.


3.	It is important to take note of this difference. If x = mt + c, then x is linearly related to t, but the rate of change of x w.r.t. t is constant (not linear).