math · Grade 12
Calculus: Limits, Continuity & Derivative Rules
Evaluate limits algebraically, compute instantaneous rates of change, and apply the Power Rule to determine derivative functions and tangent slopes.
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Learning goals
- Evaluate two-sided limits analytically using direct substitution, factoring, and algebraic rationalization
- Understand the derivative as the limit of the difference quotient representing the instantaneous rate of change
- Apply the Power Rule d/dx [x^n] = n·x^(n-1) and constant multiple rules to differentiate polynomial functions
Before you begin
Simplify algebraic expressions and understand a function's input and output.
Age and grade do not measure readiness. Adjust reading or adult support to preserve the intended learning goal.
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1. Practice
Practice previewCalculus: Limits, Continuity & Derivative Rules
Practice · Independent · 15 min
Evaluate limits algebraically, compute instantaneous rates of change, and apply the Power Rule to determine derivative functions and tangent slopes.
Teaching and response guide
- Model the target skill: evaluate two-sided limits analytically using direct substitution, factoring, and algebraic rationalization. Use the first task as a guided example before asking for an independent response.
- Check readiness first: Simplify algebraic expressions and understand a function's input and output.
- Discuss an incorrect response using this distinction: A function value and a limit are different; direct substitution may fail even when a limit exists.
Watch for: A function value and a limit are different; direct substitution may fail even when a limit exists.
Check the answer criteria for each task. A supported explanation may have more than one acceptable response. These activities offer evidence to discuss; completing a sheet alone does not establish mastery.
Possible next goals
Choose the next unit when its prerequisites fit your learner.