College Board-Aligned Original Notes
AP Calculus AB Unit 4 Topic 5: Local linearity and approximation
Use Local linearity and approximation across graphical, numerical, algebraic, and verbal representations.
Unit 4: Contextual Applications of Differentiation. College Board exam weighting listed for this unit: 10%-15% of exam score.
What to Know
- Check the conditions of a theorem or method before applying it.
- Show the setup before the calculation.
- Interpret the result in context, including units when the problem supplies them.
- Always connect this topic back to the larger unit: Contextual Applications of Differentiation.
Detailed Notes
Local linearity and approximation should be studied through multiple representations. A graph may show behavior quickly, an equation may make calculation possible, and a verbal interpretation explains what the result means.
In AP Calculus AB, AP questions often award credit for setup and reasoning, not just final answers. Write the expression, theorem, condition, or model before doing the computation.
When this topic appears in free response, check whether the question asks for a value, a rate, an interval, a comparison, or a justification. Use units and context to make the final answer precise.
Key Vocabulary
Derivative
Instantaneous rate of change of a function.
Differentiability
The property of having a derivative at a point.
Chain rule
A rule for differentiating composite functions.
Optimization
The process of finding maximum or minimum values under given conditions.
Quick Practice
How would you explain Local linearity and approximation in one or two AP-style sentences?
Name the concept, apply it to a specific example or source, and explain the reasoning that connects the evidence to your answer.
Related Topics in This Unit
- Identifying relevant mathematical information in verbal representations of real-world problems involving rates of change
- Applying understandings of differentiation to problems involving motion
- Generalizing understandings of motion problems to other situations involving rates of change
- Solving related rates problems