This document discusses how calculus can be used to optimize real-world problems. It provides an example of using calculus to find the dimensions that maximize the volume of a box with a quadratic relationship between its width, length, and height. The key steps are to write an equation for the volume of the box in terms of x, take the derivative of the volume equation to find where the gradient is zero, and solve the derivative equation to find the critical values that give the maximum volume.

1. Calculus
How can we use algebra to model real world problems?
How can we use calculus to help us optimise real world problems?
Key words: gradient, derivative, function, differentiate
14 February 2023
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2. Revision

3. Quadratic models: Applications

4. Quadratic models: Applications
1) Using GDC

5. Quadratic models: Applications
2) Algebraically

6. Calculus: optimisation
3) Using calculus

7. Calculus: Optimisation
Find the x coordinate of the local maximum

8. Calculus: Optimisation
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9. Calculus: Optimisation
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10. Calculus: Optimisation

11. Calculus: Optimisation

12. Calculus: Optimisation
𝑤𝑖𝑑𝑡ℎ = 12 − 2𝑥
𝑙𝑒𝑛𝑔𝑡ℎ = 12 − 2𝑥
Area = (12 − 2𝑥)(12 − 2𝑥)
ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑥
= 144 − 48𝑥 + 4𝑥2
𝑉𝑜𝑙𝑢𝑚𝑒 = 144 − 48𝑥 + 4𝑥2 𝑥 = 144𝑥 − 48𝑥2 + 4𝑥3

13. Calculus: Optimisation
𝑣𝑜𝑙𝑢𝑚𝑒 = 144𝑥 − 48𝑥2 + 4𝑥3 𝑑𝑦
𝑑𝑥
= 144 − 96𝑥 + 12𝑥2
0 = 12(12 − 8𝑥 + 𝑥2)
0 = 12(𝑥 − 6)(𝑥 − 2)
𝑥 = 2 𝑥 = 6

14. Calculus: Optimisation

15. Calculus: Optimisation

16. Calculus: optimisation

17. Calculus: optimisation

18. Calculus: optimisation

19. Calculus: optimisation

20. Calculus: Optimisation

21. Calculus: Optimisation

22. Calculus: Optimisation

23. Quadratic models: Applications

24. Calculus: optimisation
3) Using calculus

26. Starter - Extension
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27. Revision