Calculus II: series, curves and applications

Lesson sequence

Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.

1. Follow terms toward a limit

   Lesson details and objectives

   Separate a sequence, its continuous extension and its numerical evidence.

   • Evaluate sequence limits and diagnose oscillation.
   • Use monotone bounded convergence correctly.
   • State the distinction between boundedness and convergence.

   6 practice tasks.

2. An infinite sum is a limit of finite sums

   Lesson details and objectives

   Define convergence through partial sums and reject misleading termwise intuition.

   • Calculate partial sums and distinguish terms from totals.
   • Use necessary conditions without reversing them.
   • Compare numerical evidence with a proof of convergence.

   6 practice tasks.

3. Expose cancellation before taking a limit

   Lesson details and objectives

   Track finite endpoints in geometric and telescoping sums.

   • Derive geometric finite and infinite sums.
   • Use telescoping without dropping boundary terms.
   • State the ratio condition and distinguish an endpoint term from a limit.

   6 practice tasks.

4. Turn a positive tail into an integral comparison

   Lesson details and objectives

   Use the term test, p-series and remainder bounds with their hypotheses.

   • Apply the nth-term divergence test.
   • Check positivity, continuity and decrease for the integral test.
   • Bound a convergent tail by improper integrals.

   6 practice tasks.

5. Compare the tail to a known benchmark

   Lesson details and objectives

   Choose an inequality in the useful direction and apply limit comparison to positive terms.

   • Prove convergence or divergence using direct comparison.
   • Use finite positive limit comparison.
   • Distinguish sufficient comparisons from inconclusive ones.

   6 practice tasks.

6. Cancellation, absolute convergence and error

   Lesson details and objectives

   Check decreasing magnitudes and bound the remainder without ignoring absolute convergence.

   • Apply the alternating-series test with eventual monotonicity.
   • Distinguish absolute and conditional convergence.
   • Use a next-term bound with its assumptions.

   6 practice tasks.

7. Measure geometric decay in a complicated term

   Lesson details and objectives

   Use ratio and root limits and recognize their boundary failure.

   • Apply ratio and root tests to absolute values.
   • Treat limit one as inconclusive.
   • Select a useful test for factorials, powers and p-series.

   6 practice tasks.

8. Find the radius, then test each endpoint

   Lesson details and objectives

   Separate open-interval absolute convergence from boundary behavior.

   • Find centers and radii of power series.
   • Test endpoints using appropriate independent tests.
   • Handle zero and infinite radii.

   6 practice tasks.

9. Match derivatives with a polynomial

   Lesson details and objectives

   Construct Taylor coefficients and distinguish formal matching from convergence to a function.

   • Build Taylor and Maclaurin polynomials.
   • Track centers and factorial denominators.
   • Recognize that an infinite Taylor series need not represent its function.

   6 practice tasks.

10. Turn an approximation into a justified guarantee

    Lesson details and objectives

    Use derivative bounds and alternating bounds with correct degree and interval.

    • Apply the Lagrange remainder with regularity assumptions.
    • Choose a degree to meet a stated error tolerance.
    • Separate observed error from a proved upper bound.

    6 practice tasks.

11. Differentiate, integrate and approximate inside the radius

    Lesson details and objectives

    Use power-series operations with interval restrictions and separate endpoint justification.

    • Differentiate and integrate power series inside their radius.
    • Construct new representations from known ones.
    • Control truncation and justify endpoint claims separately.

    6 practice tasks.

12. Differentiate a curve through its parameter

    Lesson details and objectives

    Distinguish parameter velocity from geometric slope and handle zero horizontal speed.

    • Compute first and second derivatives for parametric curves.
    • Check regularity before dividing.
    • Interpret retracing and singular parameterizations.

    6 practice tasks.

13. Accumulate speed along a path

    Lesson details and objectives

    Use vector speed, displacement and parameter intervals without overlooking retracing.

    • Compute parametric arc length from velocity magnitude.
    • Distinguish distance from displacement magnitude.
    • State regularity and multiplicity assumptions.

    6 practice tasks.

14. Measure a region by small sectors

    Lesson details and objectives

    Interpret signed radii and choose a nonrepeating trace interval before integrating area.

    • Convert between polar and Cartesian coordinates.
    • Derive the half-r-squared sector integral.
    • Check intersections and tracing multiplicity.

    6 practice tasks.

15. Compute length when radius and direction both change

    Lesson details and objectives

    Derive polar speed from Cartesian derivatives and handle zero-radius points.

    • Use √(r²+(dr/dθ)²) for polar length.
    • Choose a trace interval and account for repetition.
    • Interpret cusp endpoints without dividing by zero.

    6 practice tasks.

16. Length uses slope, not vertical change alone

    Lesson details and objectives

    Derive graph length by refining chords and distinguish existence from elementary evaluation.

    • Use graph arc-length formulas under suitable smoothness.
    • Choose x or y parameterization appropriately.
    • Explain why a definite integral can be exact without an elementary primitive.

    6 practice tasks.

17. Rotate a curve and measure its skin

    Lesson details and objectives

    Use circumference times slant length rather than volume slices.

    • Derive a surface-of-revolution integrand.
    • Use distance to the axis as radius.
    • Check trace multiplicity and distinguish surface from volume.

    6 practice tasks.

18. Build a physical integral from local force

    Lesson details and objectives

    Specify units, displacement, density and depth before accumulating work or hydrostatic force.

    • Derive variable-force work and pumping models.
    • Integrate pressure over a submerged plate.
    • Separate teaching assumptions from calibrated predictions.

    6 practice tasks.