Unit 05 · Lesson 08 · Week 15 · Trigonometric Functions, Graphs & Periodic Models
Harmonic Comparisons and Model Limitations
Prepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. In this lesson, the central focus is harmonic comparisons and model limitations. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in harmonic comparisons and model limitations.
- Construct or analyze at least two representations related to harmonic comparisons and model limitations.
- Show symbolic work with correct domains, units, intervals, or angle measures.
- Explain why the selected method is valid and check the result against another representation.
Essential question
How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?
Multiple-representation model
Prepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior.
\lim_{x\to a}f(x)=L\frac{f(a+h)-f(a)}{h}m_{sec}=\frac{f(b)-f(a)}{b-a}
Worked reasoning
Misconception check: A limit describes nearby behavior and may exist even when the function value is missing or different.
Communication and reasoning
Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.
Khaemenes Academy · Open-Age Precalculus
Harmonic Comparisons and Model Limitations
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
Find the average rate of change of x² from 2 to 5.
Simplify [(x+h)²−x²]/h.
Evaluate lim_{x→3}(2x+1).
Evaluate lim_{x→2}(x²−4)/(x−2).
Can a two-sided limit exist if one-sided limits differ?
Name the discontinuity in (x²−1)/(x−1) at x=1.