IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus
Limits and the Derivative
Approach limits numerically and graphically, then interpret the derivative as gradient and rate of change.
Learning goal
Estimate limits from tables and graphs and interpret a derivative at a point and as a gradient function.
Syllabus link
AA HL: SL 5.1 · Topic 5 Calculus.
Big idea
A limit describes nearby behaviour. A derivative describes instantaneous change; geometrically it is the gradient of the tangent.
Key relationship
\(\lim_{x\to a}f(x)=L\)
Core idea
A limit describes nearby behaviour. A derivative describes instantaneous change; geometrically it is the gradient of the tangent.
Estimate limits from tables and graphs and interpret a derivative at a point and as a gradient function.
Key results and method
- For a limit, approach the input value from both sides.
- For a derivative at a point, identify the tangent gradient.
- Interpret derivative units as output-units per input-unit.
Worked example
Estimate \(\lim_{x\to2}\frac{x^2-4}{x-2}\) from nearby values.
- Use values such as 1.9, 1.99, 2.01 and 2.1.
- The outputs approach 4 from both sides.
- The expression may be undefined at 2; the limit concerns nearby behaviour.
Practice
Try these questions before moving on.
- Explain the difference between \(f(a)\) and \(\lim_{x\to a}f(x)\).
- From a graph, estimate a two-sided limit at a hole.
- If \(f'(3)=-2\), interpret the sign and magnitude of the gradient.
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