Unit 12 · Lesson 02 · Week 33 · Limits & Calculus Readiness
Difference Quotients and Secant Lines
Prepare for calculus by reasoning about average and instantaneous change, limits, continuity, secant lines, and local behavior. In this lesson, the central focus is difference quotients and secant lines. 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 difference quotients and secant lines.
- Construct or analyze at least two representations related to difference quotients and secant lines.
- 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 local behavior, average and instantaneous change, limits, continuity, and accumulation prepare learners for calculus?
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
Difference Quotients and Secant Lines
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.