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NSW Y11 Maths Standard NSW Y11 Maths Advanced NSW Y11 Maths Extension 1

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Functions Polynomials Inverse Trig Functions and Identities Rates of Change Permutations and Combinations

NSW Y11 Maths - Extension 1 Rates of Change Related Rates

Resources for Related Rates

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Related Rates Theory

NSW Syllabus Reference

NSW Syllabus Reference: ME-C1.3: Related rates of change. This will require student to 

  • solve problems involving related rates of change as instances of the chain rule (ACMSM129)
  • develop models of contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, and to which the chain rule can be applied

Ref: https://educationstandards.nsw.edu.au/

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  • Related Rates - Video - Related Rates - Applications of the Chain Rule

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Theory

If \(y = f(x)\) and the rate of change of \(x\) with respect to \(t\) is \(\dfrac{{dx}}{{dt}}\), then the rate of change of \(y\) with respect to \(t\) is given by \(\dfrac{{dy}}{{dt}} = \dfrac{{dy}}{{dx}} \times \dfrac{{dx}}{{dt}}\)

For example: Given the surface area of a sphere is \(A = 4\pi {r^2}\), the radius \(r = 2\,cm\) and the rate of change of the radius with respect to time \(t\) is \(\dfrac{{dr}}{{dt}} = 0.2\,cm{s^{ - 1}}\), find the rate of change of the area with respect to time \(\dfrac{{dA}}{{dt}}\).

\(\dfrac{{dA}}{{dt}} = \dfrac{{dA}}{{dr}} \times \dfrac{{dr}}{{dt}}\), \(\dfrac{{dA}}{{dr}} = 8\pi r\)

\(\therefore \,\,\dfrac{{dA}}{{dt}} = 8\pi \times 2 \times 0.2 = 3.2\pi \,\,c{m^2}{s^{ - 1}}\)