Calculus: JAMB Mathematics Notes

The mathematics of rates of change (differentiation) and accumulation (integration) -- JAMB mostly tests polynomial functions.

Differentiation

To differentiate $x^{n}$ , multiply by the power and reduce the power by 1: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ . A constant differentiates to 0.

Example 1: Differentiate $y = 3 x^{4} - 2 x^{2} + 5 x - 7$ .

Apply the rule to each term: $\frac{d y}{d x} = 12 x^{3} - 4 x + 5$ .

Finding the gradient at a point

Once you have the derivative, substitute the x-value to get the gradient there.

Example 2: Find the gradient of $y = x^{2} - 3 x$ at $x = 2$ .

$\frac{d y}{d x} = 2 x - 3$ . At $x = 2$ : gradient $= 2 (2) - 3 = 1$ .

Turning points (maxima and minima)

A curve is momentarily flat at a turning point, so its gradient is zero there.

Example 3: Find the turning point of $y = x^{2} - 4 x + 1$ .

Differentiate: $\frac{d y}{d x} = 2 x - 4$ . Set to zero: $2 x - 4 = 0$ , so $x = 2$ . Substitute back: $y = 4 - 8 + 1 = - 3$ . The turning point is $(2, - 3)$ .

To tell whether it's a maximum or minimum, check the second derivative: if $\frac{d ^{2} y}{d x ^{2}} > 0$ it's a minimum; if negative, a maximum. Here $\frac{d ^{2} y}{d x ^{2}} = 2$ , which is positive, so $(2, - 3)$ is a minimum.

Integration

Integration reverses differentiation: raise the power by 1 and divide by the new power, then add a constant $C$ : $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ .

Example 4: Integrate $6 x^{2} - 4 x + 3$ .

$\int (6 x^{2} - 4 x + 3) d x = 2 x^{3} - 2 x^{2} + 3 x + C$ .

Example 5: Evaluate $\int_{1}^{3} (2 x + 1) d x$ .

Integrate first: $x^{2} + x$ . Substitute the limits: $(9 + 3) - (1 + 1) = 12 - 2 = 10$ .

Calculus

Differentiation

Finding the gradient at a point

Turning points (maxima and minima)

Integration

Key terms

Practise Calculus questions

Related topics

More JAMB Mathematics notes