Functions and Calculus
Functions and Calculus tests how well you read and sketch functions and apply differentiation and integration to rates, tangents, maxima, minima and areas. Ordinary Level questions are usually structured: find the derivative, set it to zero, then interpret.
Functions
- f(x) notation: f(3) means substitute x = 3.
- Know the shapes: line (ax + b), parabola (ax² + bx + c), cubic, exponential.
- Domain: allowed x-values; range: resulting y-values.
- A graph crosses the x-axis where f(x) = 0 and the y-axis at f(0).
Differentiation
For f(x) = ax^n, f′(x) = nax^(n−1). Differentiate term by term; constants differentiate to 0. The derivative gives the slope of the tangent. At a stationary point f′(x) = 0. Use the second derivative: f″(x) > 0 gives a minimum, f″(x) < 0 gives a maximum.
Integration
Integration reverses differentiation: ∫ ax^n dx = a x^(n+1)/(n+1) + C, for n ≠ −1. A definite integral gives a numerical value; the area between the curve and the x-axis from a to b is ∫_a^b f(x) dx, taking areas below the axis as positive in an area question.
Worked example
f(x) = x³ − 3x² + 2. Then f′(x) = 3x² − 6x = 3x(x − 2). Stationary points at x = 0 and x = 2. f″(x) = 6x − 6. At x = 0, f″(0) = −6 < 0, so a maximum; at x = 2, f″(2) = 6 > 0, so a minimum. The values are f(0) = 2 and f(2) = −2.
Traps
- Dropping the constant of integration in an indefinite integral.
- Forgetting the negative sign when differentiating a term such as 1/x = x^(−1).
- Treating a point where f′(x) = 0 as automatically a maximum.
- Mixing up f′(x) and f(x) when answering.