Calculus

• Higher tier only.
• A curve has a variable rate of change: its gradient is different at different points.
• Differentiate integer powers: if y = axⁿ then dy/dx = anxⁿ⁻¹. This rule is not on the formulae sheet.
• Stationary (turning) points: solve dy/dx = 0, then find y.
• Decide maximum or minimum from the general shape of the graph.
• Kinematics: v = ds/dt and a = dv/dt.

Why curves need calculus

A straight line has the same gradient everywhere. A curve does not: on y = x² the gradient is 2 at x = 1 and 6 at x = 3. The derivative dy/dx is a formula that gives the gradient (the rate of change of y with respect to x) at any x. Drawing a tangent gives an estimate; differentiating gives the exact value.

Multiply by the power, reduce the power by one

Differentiate each term separately. A term in x becomes a constant; a constant on its own differentiates to 0. Rewrite fractions with x on the bottom as negative powers before differentiating.

dy/dx = 0 at a turning point

Substitute an x-value into dy/dx to get the gradient there. At a stationary point the gradient is zero; for 4MA1 these are the turning points (maxima and minima). Solve dy/dx = 0 for x, then substitute each x into the original equation to find y.

Use the shape of the graph

The 4MA1 specification asks you to decide by the general shape of the graph:

You can also compare the y-values: on a positive cubic the maximum has the larger y-value. Checking the sign of the gradient just either side of the point also works.