Connected rates of change and small increments
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 2 worked examples, 5 practice questions, 1 real past-paper question with the answer explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- A balloon being inflated
- The chain rule for rates
Read the transcript of the teaching part
Last time
Before we start: last time, in Practical maxima and minima, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
A balloon being inflated
Air goes into a balloon at a steady rate. The volume rises evenly, but the radius grows more slowly as the balloon gets bigger. Connecting these rates, and estimating a small change, are two applications of the chain rule.
The chain rule for rates
If two quantities depend on a third, such as time, their rates are linked by the chain rule. dV by dt equals dV by dr, times dr by dt. You need the formula connecting the quantities, differentiated with respect to the first variable.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Connected rates of change and small increments.
The whole chapter: Applications of Differentiation.
Chapter 14: Differentiation and integration
- Differentiation: standard functions, chain, product and quotient rules 5 min
- Tangents and normals 5 min
- Stationary points and the derivative tests 5 min
- Practical maxima and minima 5 min
- Connected rates of change and small increments 5 min
- Integration: powers of x and the reverse of differentiation 5 min
- Integrating (ax + b)ⁿ, exponentials and trigonometric functions 5 min
- Definite integrals and plane areas 5 min
- Kinematics: displacement, velocity, acceleration and graphs 5 min
Also for O Level Additional Maths 4037
Which topics come up most in O Level Additional Maths 4037 past papers, counted from every paper since 2004.