Curve Sketching
Use limits, derivatives and asymptotes to draw graphs.
Key idea
Use limits, derivatives and asymptotes to draw graphs.
In depth
Combine domain, intercepts, symmetry, asymptotes, increasing/decreasing from f′, concavity from f′′, and key points to draw a faithful graph. The goal is to show shape, not plot many points.
Worked example
For f = x³ − 3x: zeros 0, ±√3; max at (−1, 2); min at (1, −2); inflection at 0.
Real-world uses
- Drawing graphs of cost, profit and population models by hand before using software
Common mistake
Plotting points before analysing, missing features between them.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Curve Sketching?
Use limits, derivatives and asymptotes to draw graphs. It is a topic at the expert level in Calculus.
Where is Curve Sketching used in real life?
Real-world examples: Drawing graphs of cost, profit and population models by hand before using software.
What should I learn before Curve Sketching?
Make sure you are comfortable with L’Hôpital’s Rule, the previous topic in Calculus.
What is a common mistake with Curve Sketching?
Plotting points before analysing, missing features between them.
How can I practise Curve Sketching?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Curve Sketching?
Move on to Newton’s Method next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.