Part III: Calculus I

Rates, limits, derivatives, and integrals, each one computed, checked, and connected to code.

18 chapters, 185 exercises.

Module 05From two values to an average rate8 exercisesBuild a difference quotient, explain each subtraction, simplify without setting the interval to zero, and attach appropriate rate units.
Module 06Limits: nearby behavior, not point substitution8 exercisesCompute limits by appropriate methods, distinguish one-sided behavior, interpret tables and graphs cautiously, and identify infinite behavior and asymptotes.
Module 07Continuity, existence, and precise limit reasoning8 exercisesTest continuity, apply the Intermediate Value Theorem with its assumptions, and construct simple epsilon–delta proofs.
Module 08The derivative and the tangent line8 exercisesCompute a derivative from its definition, separate function height from slope, build tangent lines, and distinguish differentiability from continuity.
W01From function inputs to executable contracts5 exercisesTranslate one object between algebra, geometry, and code; separate a domain restriction from a limiting extension; verify a tangent line and a state update.
Mixed review R1: notation to tangent lines8 exercisesUse this review after Module 08. The method is deliberately not supplied in the problem headings. Give exact answers when possible, preserve restrictions, and explain the main decision. The general rubric is ; a score alone is…
Module 09Derivative rules without losing the structure8 exercisesRecognize sums, products, quotients, and powers; differentiate them accurately; and explain why product derivatives contain two contributions.
Module 10The chain rule and elementary derivatives8 exercisesDifferentiate nested functions without missing inner factors, use elementary derivative formulas with their domains, and distinguish a product from a composition.
Module 11Implicit, inverse, and logarithmic differentiation8 exercisesDifferentiate equations that define a curve implicitly, find inverse derivatives without solving explicitly, and handle variable powers using logarithms.
Module 12Motion, higher derivatives, and related rates21 exercisesSeparate position, velocity, speed, and acceleration; construct related-rate equations; and interpret signs and units at a specified instant.
Module 13Linear approximation, extrema, and the Mean Value Theorem22 exercisesEstimate a nearby value with a tangent line, identify candidate extrema, and use the Mean Value Theorem with its hypotheses intact.
Module 14Curve analysis and optimization21 exercisesTranslate derivative signs into graph behavior and solve a constrained optimization problem from a mathematical model.
Module 15Newton’s method, indeterminate forms, and growth8 exercisesPerform and assess Newton iterations; recognize the hypotheses for L’Hôpital’s rule; and transform other indeterminate forms without illegal cancellation.
Module 16Antiderivatives, Riemann sums, and definite integrals8 exercisesDistinguish a family of antiderivatives from an integral value, construct a Riemann sum, and interpret signed accumulation with units.
Module 17The Fundamental Theorem, substitution, and area8 exercisesDifferentiate an accumulation function, evaluate a definite integral, change variables with correct bounds, and separate area from signed accumulation.
Module 18Volumes from slices, washers, and shells8 exercisesBuild a volume integral from a typical cross-section or shell, measure distances from the correct rotation axis, and compare alternative setups.
Module 19Length, surface area, work, fluid force, and mass8 exercisesTranslate a physical or geometric contribution into an integral and distinguish length, area, force, work, mass, and balance.
Cumulative assessment: Calculus I12 exercisesThis assessment covers the main skills in Modules 01–19. It is an original mixed assessment, not a reconstruction of any particular course’s exam. Work with solutions hidden. One or two sittings are reasonable; choose a time…

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checkpoint, review or assessment