Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Limits, Continuity and First Principles

Connect limits, continuity, differentiability and the derivative from first principles.

AA HL · AHL 5.12

Learning goal

Use the first-principles derivative, reason about continuity/differentiability and work with higher derivatives.

Syllabus link

AA HL: AHL 5.12 · Topic 5 Calculus.

Big idea

The derivative is a limit of secant gradients. Differentiability is a stronger local condition than continuity.

Key relationship

\(f\'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)

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Core idea

The derivative is a limit of secant gradients. Differentiability is a stronger local condition than continuity.

Use the first-principles derivative, reason about continuity/differentiability and work with higher derivatives.

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Key results and method

\[f\'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\]
\[\text{differentiable at }a\Rightarrow\text{continuous at }a\]
  1. Form the difference quotient.
  2. Simplify before taking the limit.
  3. For continuity/differentiability, compare left/right behaviour and slope where needed.
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Worked example

Question

Use first principles to differentiate \(f(x)=x^2\).

Solution
  1. \(\frac{(x+h)^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h\).
  2. Let \(h\to0\).
Answer: \(f\'(x)=2x\).
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Practice

Try these questions before moving on.

  1. Differentiate a linear or quadratic function from first principles.
  2. Give an example of a continuous function that is not differentiable at a point.
  3. Use higher-derivative notation correctly.

Search the worksheet library for more practice →

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