Maximum and minimum problems, and stationary points with integration
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 7 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- The best shape
- Optimisation steps
- A simple warm-up
Read the transcript of the teaching part
Last time
Before we start: last time, in Connected rates of change and small increments, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
The best shape
A tank with a fixed amount of metal. A field with a fixed amount of fence. Which shape holds the most? Write what you want to maximise as a function of one variable, differentiate, and set it to zero. And often the question hands you the derivative and asks you to work backwards: a stationary point tells you the constant, and integrating gives the whole curve.
Optimisation steps
First write the quantity to maximise, V say, using the constraint to remove all but one variable. Differentiate and set to zero. Solve for the variable. Use the second derivative, or a sign check, to confirm maximum or minimum. Then find the value asked.
A simple warm-up
A rectangle has perimeter 40. If one side is x, the other is 20 minus x. The area is x times 20 minus x. Differentiate: 20 minus 2x. Zero when x is 10. So the biggest rectangle is a square, area 100.
Chapter 1.7: Differentiation
- Differentiation: standard functions, chain, product and quotient rules 5 min
- Tangents and normals 5 min
- Stationary points and the derivative tests 6 min
- Connected rates of change and small increments 6 min
- Maximum and minimum problems, and stationary points with integration 6 min
Also for A Level Maths 9709 Paper 1
Which topics come up most in A Level Maths 9709 Paper 1 past papers, counted from every paper since 2004.