# Maths Aim Higher Calculus of Small increments. A first principles approach In general, the derivative f ’ (x) evaluated at x = a can be defined as Click.

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Maths Aim Higher Calculus of Small increments

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

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.

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

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

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

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

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

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