Learn

Learning preferences

Make learning more comfortable. Changes apply immediately.

Enlarges page content, including diagrams and their labels. You can also use your browser’s zoom.

Choose the colours that are easiest for you to distinguish. Applies to lesson diagrams and lab plots. Labels, shapes, hatching and line patterns also distinguish meanings. Source photographs and raw image brightness stay unchanged; mask-only views are available for image overlays.

ExampleComparisonSample link

Saved in this browser on this device. Preferences are separate from progress backups.

Mathematics · unit version 1

Calculus synthesis: regions, motion and approximation

Six focused investigations complete specific integration, spatial, planar-motion, polar and Taylor-data actions. Public worked practice and written rubrics support learning; they do not establish independent AP examination readiness or general calculator proficiency.

Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.

Readiness and preparation

Check your starting point

Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.

What must change when substituting in a definite integral?

A missing scale or unchanged bound can invalidate an otherwise familiar formula.

Compare reasoning and find a refresher

Transform the integrand, differential and endpoints consistently, or return to the original variable before applying its original bounds.

Review substitution and bounds

How does a polar point become a Cartesian point?

A radius function and its angle jointly describe a curve.

Compare reasoning and find a refresher

Use x=r cos θ and y=r sin θ. Negative r reverses the ray; the same Cartesian point can have different polar descriptions.

Review polar coordinates and area

Why does a Taylor coefficient contain a factorial?

A table of derivatives is not already a table of polynomial coefficients.

Compare reasoning and find a refresher

Differentiating (x−a)^k exactly k times produces k!, so matching the kth derivative requires coefficient f⁽ᵏ⁾(a)/k!. A separate remainder bound is needed to guarantee approximation error.

Review Taylor polynomial construction

Loading saved mathematics work…

Regions, trajectories and local models: mixed practice

The mixed assessment has separate response history and review criteria; it does not change lesson participation records.

Lesson sequence

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

  1. Complete the square before integrating

    Lesson details and objectives

    Turn a quadratic denominator into a scaled inverse-trigonometric integral and verify the result.

    • Complete a quadratic square and identify its positive scale.
    • Track substitution and definite bounds through an inverse-tangent primitive.
    • Distinguish positive denominators from singular cases before evaluating.

    6 practice tasks.

  2. Build section areas and split changing boundaries

    Lesson details and objectives

    Translate a region into section dimensions and choose boundary order separately on each interval.

    • Derive rectangular and semicircular cross-section areas from a supplied base width.
    • Find all boundary intersections and split geometric area where ordering changes.
    • Explain why a diameter, radius and section height produce different volume integrands.

    6 practice tasks.

  3. Recover planar position from velocity

    Lesson details and objectives

    Integrate components with an initial point and separate displacement, speed and traveled length.

    • Recover a planar position from component velocities and an initial point.
    • Compare displacement magnitude with the integral of speed, including retracing.
    • Use velocity and acceleration together to diagnose changing speed.

    6 practice tasks.

  4. Find polar tangents through Cartesian derivatives

    Lesson details and objectives

    Differentiate the Cartesian parameterization and test horizontal, vertical and singular cases.

    • Derive polar dy/dx from Cartesian component derivatives.
    • Classify horizontal and vertical tangents using numerator and denominator conditions.
    • Investigate simultaneous zero derivatives without dividing zero by zero.

    6 practice tasks.

  5. Split an overlap by its radial boundary

    Lesson details and objectives

    Locate intersections, identify which boundary is nearer, and integrate the actual shared region.

    • Find common Cartesian points without losing origin intersections at different angles.
    • Choose piecewise radial bounds for overlapping regions.
    • Distinguish overlap, union and subtraction of whole-region areas.

    6 practice tasks.

  6. Construct Taylor models from derivative evidence

    Lesson details and objectives

    Use a derivative table and a separately justified remainder bound, then transfer known series through substitution.

    • Construct a Taylor polynomial at a nonzero center from supplied derivative values.
    • Apply a separate interval-wide derivative bound to quantify approximation error.
    • Transform a familiar series while retaining its validity interval and endpoint qualifications.

    6 practice tasks.