Introductory Calculus
Differentiation and integration
Introductory Calculus tests the derivative as a limit and as a tool for gradients, stationary points and rates of change, together with integration as the reverse process and the definite integral as area.
Core rules
- Derivative from first principles: f'(x) = lim(h -> 0) [f(x + h) - f(x)]/h.
- Power rule: if y = x^n, dy/dx = n x^(n-1). Constant multiples and sums are differentiated term by term.
- Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v - uv')/v^2. Chain rule: dy/dx = (dy/du)(du/dx).
- At a stationary point, dy/dx = 0. Use the second derivative or a sign table to classify: f'' < 0 gives a maximum, f'' > 0 gives a minimum.
- Integral of x^n dx = x^(n+1)/(n+1) + C for n not equal to -1. For a definite integral, evaluate the antiderivative at the upper limit minus the lower limit.
- The area between a curve and the x-axis is found by integration; regions below the axis give a negative integral, so split the interval or take the absolute value.
Worked example
Find and classify the stationary points of y = x^3 - 3x^2 + 4.
Differentiate: dy/dx = 3x^2 - 6x = 3x(x - 2). Set it to zero: x = 0 or x = 2. Substitute into the original: when x = 0, y = 4; when x = 2, y = 8 - 12 + 4 = 0. So the stationary points are (0, 4) and (2, 0). The second derivative is d^2y/dx^2 = 6x - 6. At x = 0, it is -6 < 0, so (0, 4) is a maximum. At x = 2, it is 6 > 0, so (2, 0) is a minimum.
Traps
Do not forget + C in an indefinite integral. In the chain rule, multiply by the derivative of the inside function; omitting it is the most common error. In the quotient rule, keep the order u'v - uv', not the reverse. A stationary point is not automatically a maximum or minimum - always test it. When finding an area, sketch or check whether the curve crosses the axis, because a definite integral can cancel positive and negative regions. Differentiate trig functions in radians, not degrees.