Applications of differentiation

Learning objectives

Find the equation of the tangent and of the normal to a curve at a given point, using the fact that the normal gradient is the negative reciprocal of the tangent gradient

Locate stationary points by solving dy/dx = 0, and determine whether each is a maximum or a minimum using the sign of the second derivative or a sign change of the first derivative (points of inflexion are not included in 0606)

Find the intervals on which a function is increasing or decreasing from the sign of its derivative, and use this to describe the shape of the curve

Solve practical maximum and minimum problems by forming an expression for the quantity to be optimised, eliminating any second variable with a constraint, and justifying that the stationary value is the required maximum or minimum

Solve connected rates of change problems using the chain rule in the form dy/dt = (dy/dx)(dx/dt), stating the units of the answer

Use the approximation δy ≈ (dy/dx)δx to estimate the small change in one quantity produced by a small change in another, and express it as a percentage change