Unit 07 · Lesson 06 · Week 21 · Contextual Applications of Derivatives
Differentials and Propagated Error
Linearization uses the tangent line as a nearby approximation. Accuracy improves near the center point and depends on curvature and input distance. This lesson emphasizes differentials and propagated error through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in differentials and propagated error.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do derivatives describe motion, related quantities, sensitivity, and local approximation in authentic contexts?
Visual model
Linearization uses the tangent line as a nearby approximation. Accuracy improves near the center point and depends on curvature and input distance.
L(x)=f(a)+f'(a)(x-a)dy=f'(x)dx
Worked reasoning
Misconception check: A linearization is local; accuracy is not guaranteed far from the center.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Differentials and Propagated Error
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Find the linearization of f(x)=√x at a=4.
Answer: L(x)=2+(x−4)/4
Reasoning: f(4)=2 and f'(4)=1/4.
Reasoning: f(4)=2 and f'(4)=1/4.
Use that linearization to estimate √4.1.
Answer: 2.025
Reasoning: L(4.1)=2+0.1/4.
Reasoning: L(4.1)=2+0.1/4.
If f is concave down near a, does its tangent line generally over- or underestimate?
Answer: Overestimate
Reasoning: A concave-down graph lies below its tangent locally.
Reasoning: A concave-down graph lies below its tangent locally.
Write the differential relation.
Answer: dy=f'(x)dx
Reasoning: It approximates the output change.
Reasoning: It approximates the output change.
Why choose a convenient center?
Answer: Exact f(a) and f'(a) make the approximation efficient.
Reasoning: The tangent model is anchored at a.
Reasoning: The tangent model is anchored at a.
When does linearization lose accuracy?
Answer: Far from the center or where curvature is large.
Reasoning: The function departs from its tangent line.
Reasoning: The function departs from its tangent line.