Learn Interpret discontinuities, derivative information, signed accumulation and unequal-time measurements. Exact graph models and uncertain measured data are kept distinct. Public written practice requires review; this unit does not establish AP examination readiness or graphing-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.
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.
A supplied graph can include isolated values and excluded points.
Yes. The limit describes nearby values; a filled point gives the value at the input. A removable discontinuity has matching nearby limits but a different or missing value.
Table rows must follow the chain rule, not just share a row.
For f(g(x)), use f′ at g(x), then multiply by g′(x). Knowing f′(x) alone may not provide the needed value.
Integration and table estimates must preserve signs and units.
Multiply rate by interval width for each contribution, then combine. An integral of a signed rate is net change; integrating its absolute value instead measures total variation.
Loading saved mathematics work…
Graph and table synthesis: mixed practice
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Separate marked function values from nearby behavior and use a derivative graph without assuming it is the original function.
6 practice tasks.
Use input correspondence in chain and inverse rules and distinguish sampled secant slopes from exact derivatives.
6 practice tasks.
Use exact geometric areas, initial conditions and FTC to connect an integrand graph to its accumulation function.
6 practice tasks.
Build interval-weighted sums and identify the additional assumptions needed for exact totals or guaranteed bounds.
6 practice tasks.