Unit 08 · Lesson 06 · Week 24 · Analytical Applications & Optimization
Second Derivative Test and Curve Analysis
Derivative sign charts organize global behavior. Critical numbers, endpoints, and nondifferentiable points divide the domain into intervals where monotonicity and concavity can be tested. This lesson emphasizes second derivative test and curve analysis through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in second derivative test and curve analysis.
- 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 reveal increasing behavior, extrema, shape, and optimal decisions?
Visual model
Derivative sign charts organize global behavior. Critical numbers, endpoints, and nondifferentiable points divide the domain into intervals where monotonicity and concavity can be tested.
f'(c)=0f''(x)>0f''(x)<0
Worked reasoning
Misconception check: A critical number is a candidate for an extremum, not a guarantee.
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
Second Derivative Test and Curve Analysis
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Find critical numbers of f(x)=x³−3x.
Answer: x=−1 and x=1
Reasoning: f'=3x²−3=0.
Reasoning: f'=3x²−3=0.
Where is the same function increasing?
Answer: (−∞,−1)∪(1,∞)
Reasoning: f'>0 outside the critical numbers.
Reasoning: f'>0 outside the critical numbers.
Find f''(x) for f(x)=x³−3x.
Answer: 6x
Reasoning: Differentiate f'=3x²−3.
Reasoning: Differentiate f'=3x²−3.
Where is the graph concave up?
Answer: (0,∞)
Reasoning: f''=6x>0 for x>0.
Reasoning: f''=6x>0 for x>0.
State the Mean Value Theorem conclusion.
Answer: Some c has f'(c)=[f(b)−f(a)]/(b−a).
Reasoning: An interior tangent parallels the endpoint secant.
Reasoning: An interior tangent parallels the endpoint secant.
Why are endpoints checked for absolute extrema?
Answer: An absolute extremum on a closed interval can occur at an endpoint.
Reasoning: Critical points alone are insufficient.
Reasoning: Critical points alone are insufficient.