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Maths Aim Higher Calculus of Small increments

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A first principles approach In general, the derivative f ’ (x) evaluated at x = a can be defined as Click here to see how this works with quadratics

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Practice makes perfect ! Click here to practice first principles on quadratics Click here for first principles applied to rational expressions Click here to apply first principles to surdal expressions Click here for a check test on first principles

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Setting up a formula Another way to express this is …. If P ( x, y ) is a point on the curve y = f (x) and Q (x +, y + ) is close to P. Then is a small increase in x, and is the corresponding increase in y.

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The formula is established Now, by definition. So when is small, Hence, We can now investigate changes affecting other variables

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Example: y = ln x We know that Hence When x = 1, y = ln 1 = 0 Taking (small) and Then ln 1.1 = = 0 + 0.1 = 0.1 So ln 1.1 0.1

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Percentage increments If x increases by r% then Hence, corresponding % increase in y is given by Let’s see an example in action ……..

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The pendulum problem Given Find % change in T when l is increased by 2% Well, So Given And so …………

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Pendulum problem (ctd) Since Then So % change in T = So period T increases by 1%

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