Single-variable calculus foundations

Lesson sequence

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1. Domains, composition and radian measure

   Lesson details and objectives

   Check whether an expression is defined before composing functions or measuring a changing angle.

   • Determine real domains from denominator and square-root restrictions.
   • Evaluate composition in the stated order and preserve its domain.
   • Connect radians to arc length and distinguish radians from degrees.

   6 practice tasks.

2. What a limit says—and what samples cannot prove

   Lesson details and objectives

   Read nearby behavior from tables and described graphs without confusing a limit with the value at a point.

   • Distinguish f(a) from a limit at a.
   • Compare left and right limits.
   • Explain why a finite table alone cannot prove a limit.

   5 practice tasks.

3. Limit laws and legitimate algebra

   Lesson details and objectives

   Use algebra to establish nearby behavior while preserving exclusions and checking quotient conditions.

   • Apply sum, product and quotient limit laws with their hypotheses.
   • Remove a common factor only away from its zero.
   • Rationalize a difference of square roots and distinguish 0/0 from an answer.

   5 practice tasks.

4. Bounding a function you cannot simplify

   Lesson details and objectives

   Establish a limit with upper and lower bounds, including oscillation that a sparse graph may conceal.

   • State the squeeze theorem with nearby bounds and equal bounding limits.
   • Construct bounds using |sin t|≤1.
   • Distinguish shrinking amplitude from persistent oscillation.

   5 practice tasks.

5. Continuity and guaranteed crossings

   Lesson details and objectives

   Check the three conditions for continuity and distinguish existence guarantees from locating or counting roots.

   • Check existence of value, existence of limit and their equality.
   • Choose a parameter to repair a removable discontinuity.
   • Apply IVT with continuity on a closed interval and a target between endpoint values.

   5 practice tasks.

6. Large inputs and unbounded outputs

   Lesson details and objectives

   Distinguish a finite limit at infinity from an infinite limit near a finite input, using signs and dominant terms.

   • Find rational-function end behavior by dividing by a dominant power.
   • Determine one-sided signs near a vertical asymptote.
   • Explain why infinity is not a real function value and why an asymptote can be crossed.

   5 practice tasks.

7. From secant rates to a derivative

   Lesson details and objectives

   Derive an instantaneous rate from a difference quotient, with a nonzero increment before taking its limit.

   • Form and simplify [f(a+h)−f(a)]/h.
   • Interpret the derivative as a local rate with units.
   • Distinguish a finite-step estimate from the exact limit.

   5 practice tasks.

8. When a tangent slope does not exist

   Lesson details and objectives

   Compare one-sided derivatives, jumps and vertical tangents; continuity alone does not promise a finite derivative.

   • Use left and right difference quotients to diagnose a corner.
   • Explain that differentiability implies continuity but not conversely.
   • Check both value matching and slope matching in piecewise functions.

   5 practice tasks.

9. Differentiate powers and combinations

   Lesson details and objectives

   Use established derivative rules without confusing differentiation with evaluation or multiplication of functions.

   • Apply the power, constant-multiple and sum rules.
   • Handle negative and fractional powers with domain restrictions.
   • Find all stationary inputs in a simple polynomial and distinguish stationarity from optimality.

   5 practice tasks.

10. A tangent line as a local prediction

    Lesson details and objectives

    Build a tangent model, quantify its error in a worked case and distinguish small input changes from globally linear behavior.

    • Construct L(x)=f(a)+f′(a)(x−a).
    • Use a linearization for a nearby estimate and retain units.
    • Analyze an exact remainder in a quadratic example and avoid unsupported error guarantees.

    5 practice tasks.