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Translate finite rectangle sums into sigma notation and construct left, right and midpoint inputs on shifted intervals. Prerequisite: rate-times-width accumulation. Use supplied finite-sum identities and verify indexing by expanding a small case; no infinite-series theory is needed.
Oct 28, 2026
Estimate accumulated change by pairing sampled rates with actual interval widths. Prerequisites: signed quantities, units and basic algebra. Work with equal and unequal time gaps, distinguish estimates from exact constant-rate totals, and interpret the sign of a change.
Oct 28, 2026
Check your Chapter 11 skills with mixed primitives, domain checks, vertical translations and short reconstructions. Prerequisites: the five antiderivative and initial-value lessons. Verify proposed answers by differentiation and by their data; this checkpoint introduces no new integration method.
Oct 28, 2026
Recover signed velocity and position from simple acceleration laws. Use elementary antiderivatives and initial-value skills to assign separate constants, check units, and verify both measurements. One-dimensional motion only; position is distinct from total distance.
Oct 28, 2026
Choose and compute endpoint, midpoint and trapezoidal estimates from explicitly supplied data. Account for unequal widths, distinguish estimates from exact geometry, and justify error direction using whole-interval monotonicity or concavity. Prerequisites: partitions, integral properties and concavity. Eight independent questions.
Oct 28, 2026
Combine integrals with linearity, additivity and orientation. Check odd/even symmetry and convert pointwise rate bounds into bounds on accumulation. Prerequisite: signed geometric integrals. Ten independent questions; assume integrability as stated and justify each property.
Oct 28, 2026
Read signed integrals from exact rectangles, triangles and semicircles. Distinguish net accumulation from positive-region and total area, handle reversed bounds, and interpret velocity reversals. Prerequisite: the integral as a limit of sums. Ten independent questions using geometry.
Oct 28, 2026
Build the definite integral from increasingly fine signed rectangle sums. Use tagged partitions, interpret sampling independence, and evaluate short linear and quadratic sum limits. Prerequisites: equal partitions, finite-sum notation and continuity. Eight independent questions; no antiderivative shortcuts.
Oct 28, 2026
Checkpoint: formulate an objective, select feasible candidates, inspect boundaries and critique an incomplete optimization argument across fresh settings. Prerequisites: geometric, profit and nearest-point optimization and nonattainment. No new technique is introduced. Use results to identify the specific objective or prerequisite that needs review.
Oct 27, 2026
Decide whether an optimum is attained, approached at an excluded boundary, or absent on an unbounded domain. Prerequisites: feasible-domain modeling, extrema and end behavior. Check parameter values and boundary inclusion explicitly, and distinguish a limiting bound from a best feasible choice.
Oct 27, 2026
Translate nearest-point geometry into a distance objective, justify minimizing its square, and compare short supplied travel-time models with distinct speeds. Prerequisites: one-variable modeling, the chain rule and global-extremum reasoning. Distinguish boundary routes, stationary candidates and the units of distance versus time.
Oct 27, 2026
Build revenue and profit from supplied price and cost models, then justify continuous and whole-unit decisions using feasible candidates and boundaries. Prerequisites: marginal rates, one-variable modeling and derivative sign tests. Practice distinguishing sales volume, profit per unit and total profit; conclusions remain conditional on the stated model.
Oct 27, 2026
