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. |
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Question 1: | Is the radius r increasing or decreasing with time t?
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Question 2: | Is the radius r changing at a constant rate? How do you know?
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B. ANIMATION
If you cannot view the LiveMath
document below, click here to
troubleshoot.
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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. |
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Question 4: | Is the radius r increasing or decreasing with time t?
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Question 5: | Is the radius r changing at a constant rate? How do you know?
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Question 6: | Differentiate r
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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?
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Question 8: | Differentiate r
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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 |
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.
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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). |