Curve Sketching

Level 1 - Calculus topic page in Applications Of Derivatives.

Principle

Curve sketching combines algebra, derivatives, and limits to describe the main features of a graph. The goal is not a perfect drawing but a reliable structural picture.

Notation

\(f'(x)\)

first derivative, used for increasing and decreasing behaviour

\(f''(x)\)

second derivative, used for concavity

\(x_c\)

critical point candidate

The Core Method

Find domain, intercepts, asymptotes if relevant, critical points from \(f'(x)=0\) or undefined derivative, and concavity from \(f''(x)\).

Increasing test

\[f'(x)\gt0\Longrightarrow f\text{ is increasing}\]

Concavity test

\[f''(x)\gt0\Longrightarrow f\text{ is concave up}\]

Use sign charts to organise intervals.

Worked Cases

Question

For \(f(x)=x^3-3x\), find the critical points and classify increasing intervals.

Answer

Differentiate: \(f'(x)=3x^2-3=3(x-1)(x+1)\). Critical points occur at \(x=-1\) and \(x=1\). The derivative is positive for \(x<-1\), negative for \(-1<x<1\), and positive for \(x>1\). Therefore the function increases on \((- \infty,-1)\) and \((1,\infty)\), and decreases on \((-1,1)\).

Examples

Question

For \(f(x)=x^4\), determine the concavity.

Answer

Compute \(f''(x)=12x^2\). Since \(12x^2\ge0\) for every \(x\), the graph is concave up everywhere, with zero second derivative only at \(x=0\).

Linear Approximation

Optimisation