Presentation on theme: "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."— Presentation transcript:
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
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
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%