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Calculus I · Unit 2B · hub

Unit 2B: Applications of Derivatives

Study Unit 2B: Applications of Derivatives through a complete sequential course with explanations, visuals, checks, reviews, practice exams, and answer keys.

Core textbook pathChoose the next lesson from the ordered unit map.

Core textbook

The complete Unit 2B path

Begin with interpretation, move through approximation and modeling, and finish by defending conclusions with units, domains, assumptions, and reasonableness checks.

What this unit teaches

Turn derivatives into explanations, estimates, and decisions.

Interpret motion and sensitivity; build linear and Newton approximations; connect related quantities; analyze complete graphs; optimize feasible designs; evaluate indeterminate limits; and test models against their assumptions.

Prerequisites

Unit 2A derivative foundations and fluent algebra.

You should be able to compute common derivatives, solve equations, read graphs, and track units. Use the bridge diagnostic and linked Unit 2A refreshers when a calculation skill needs repair.

Section

Orientation and the Unit 2A bridge

Begin with the modeled quantity and the question it must answer; differentiation belongs in the middle of the translation, not at the beginning.

  1. 01Unit 2B: Applications of Derivativeshub
  2. 02How to Translate Derivative Word Problemslesson
Section

Interpretation, motion, and rates

Use the sign, size, units, and zeros of derivatives to tell a time-aligned story about motion or another changing quantity.

  1. 03Interpreting Higher Derivativeslesson
  2. 04How to Interpret a Derivative in Contextlesson
  3. 05Position, Velocity, and Accelerationlesson
  4. 06Speeding Up and Slowing Downlesson
  5. 07Reading Function and Derivative Graphslesson
  6. 08Rates from Data and Unitslesson
  7. 09Higher Derivatives and Motion Reviewreview
Section

Local linearity, differentials, and Newton's method

A differentiable curve behaves like its tangent line over a small enough neighborhood, with concavity explaining the direction of error.

  1. 10Local Linearitylesson
  2. 11Why a Tangent Line Can Approximate a Curvelesson
  3. 12Using Linearizationlesson
  4. 13Differentialslesson
  5. 14Error Estimation with Differentialslesson
  6. 15Newton's Methodlesson
  7. 16Approximation and Newton's Method Reviewreview
Section

Theorems, extrema, and curve shape

Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.

  1. 25Extrema and Critical Numberslesson
  2. 26Graph Analysis as a Coherent Storylesson
  3. 27The Closed Interval Methodlesson
  4. 28Rolle's Theoremlesson
  5. 29The Mean Value Theoremlesson
  6. 30Increasing and Decreasing Intervalslesson
  7. 31The First Derivative Testlesson
  8. 32Concavity and Inflection Pointslesson
  9. 33The Second Derivative Testlesson
  10. 34Complete Curve Sketchinglesson
  11. 35Derivative Theorems and Shape Reviewreview
Section

Optimization

Separate the objective from the constraint, reduce to one feasible variable, and interpret the winning candidate in the original design.

  1. 36Optimization Modelinglesson
  2. 37The Anatomy of an Optimization Problemlesson
  3. 38Closed-Interval Optimizationlesson
  4. 39Geometric Optimizationlesson
  5. 40Business and Scientific Optimizationlesson
  6. 41Optimization Reviewreview
Section

L'Hopital's Rule and indeterminate forms

Identify the limiting form before differentiating; transformation and simpler limit laws come before L'Hopital's Rule.

  1. 42L'Hopital's Rulelesson
  2. 43L'Hopital's Rule Is Not Quotient Cancellationlesson
  3. 44L'Hopital's Rule for 0/0lesson
  4. 45L'Hopital's Rule for infinity/infinitylesson
  5. 46Repeated L'Hopital Applicationslesson
  6. 47Products, Differences, and Powerslesson
  7. 48When Not to Use L'Hopital's Rulelesson
  8. 49L'Hopital's Rule Reviewreview
Section

Modeling studio

Treat each derivative model as a conditional claim whose variables, units, assumptions, calibration range, and limitations remain visible.

  1. 50Medication Concentration, Peak Timing, and Model Limitslesson
  2. 51Reaction Time, Braking, and Vehicle Stopping Distancelesson
  3. 52Radar and Camera Tracking with Changing Angleslesson
  4. 53Manufacturing Tolerances and Error Propagationlesson
  5. 54Marginal Cost, Revenue, Profit, and Elasticitylesson
  6. 55Derivative Applications Modeling Studio Reviewreview
Section

Review, practice, exams, and reference

Mixed applications remove the method label; classify the model, commit to a complete attempt, and diagnose the first wrong decision.

  1. 56Unit 2B Review: Applications of Derivativesreview
  2. 57Common Derivative Application and Optimization Errorsreview

Practice around the path

Reviews, quizzes, diagnostics, and exams

Use these after a section or whenever a worked example reveals a specific gap. The answer keys are separated so an honest first attempt stays easy.

diagnosticDerivative Applications Bridge DiagnosticCheck the prerequisite skills needed for Applications of Derivatives and follow targeted review links before beginning the unit.reviewHigher Derivatives and Motion ReviewReview higher derivatives and motion review with mixed practice and links back to the exact lessons behind each skill.quizInterpretation and Motion Concept QuizTest conceptual understanding of interpretation and motion concept quiz with a short self-check and model responses.reviewApproximation and Newton's Method ReviewReview approximation and newton's method review with mixed practice and links back to the exact lessons behind each skill.quizApproximation Concept QuizTest conceptual understanding of approximation concept quiz with a short self-check and model responses.reviewRelated Rates ReviewReview related rates review with mixed practice and links back to the exact lessons behind each skill.quizRelated Rates Concept QuizTest conceptual understanding of related rates concept quiz with a short self-check and model responses.reviewDerivative Theorems and Shape ReviewReview derivative theorems and shape review with mixed practice and links back to the exact lessons behind each skill.quizDerivative Theorems and Shape Concept QuizTest conceptual understanding of derivative theorems and shape concept quiz with a short self-check and model responses.reviewOptimization ReviewReview optimization review with mixed practice and links back to the exact lessons behind each skill.quizOptimization Concept QuizTest conceptual understanding of optimization concept quiz with a short self-check and model responses.reviewL'Hopital's Rule ReviewReview l'hopital's rule review with mixed practice and links back to the exact lessons behind each skill.quizL'Hopital Concept QuizTest conceptual understanding of l'hopital concept quiz with a short self-check and model responses.practiceDerivative Applications Modeling StudioPractice derivative applications modeling studio with mixed problems, staged guidance, and source-linked solutions.reviewDerivative Applications Modeling Studio ReviewReview derivative applications modeling studio review with mixed practice and links back to the exact lessons behind each skill.reviewUnit 2B Review: Applications of DerivativesReview unit 2b review: applications of derivatives and optimization with mixed practice and links back to the exact lessons behind each skill.reviewCommon Derivative Application and Optimization ErrorsReview common derivative application and optimization errors with mixed practice and links back to the exact lessons behind each skill.practiceUnit 2B Cumulative Applications PracticePractice unit 2b cumulative applications practice with mixed problems, staged guidance, and source-linked solutions.examUnit 2B Practice Exam AComplete Unit 2B Practice Exam A as a cumulative Calculus I assessment, then use the separately published answer key for review.examUnit 2B Practice Exam BComplete Unit 2B Practice Exam B as a cumulative Calculus I assessment, then use the separately published answer key for review.referenceUnit 2B Formula and Strategy ReferenceUse this applications of derivatives reference for formulas, notation, domains, units, and problem-solving strategy.

Check your work

Published exam answer keys

Every exam has a separately routed, numbered key. Finish the exam first, then compare one item at a time.

Go deeper

Focused integral explorations

These articles zoom in on one modeling choice, theorem, or interpretation. They are enrichment around the textbook path, not a replacement for it.

Advanced explorationAdvanced Notes: Analysis Behind the ApplicationsExplore advanced notes: analysis behind the applications in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationSensitivity, Elasticity, and Condition NumbersExplore sensitivity, elasticity, and condition numbers in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationNewton's Method: Convergence and Failure ModesExplore newton's method: convergence and failure modes in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationConvexity, Tangent Inequalities, and Global MinimaExplore convexity, tangent inequalities, and global minima in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Advanced explorationCauchy's Mean Value Theorem and L'Hopital's RuleExplore cauchy's mean value theorem and l'hopital's rule in an optional advanced note connecting Calculus I to later analysis and mathematical theory.Focused articleCurve sketching from derivative evidenceA concise in-depth exploration connected to the main unit.

Finish the derivative sequence

Connect applications back to derivative foundations

Use Unit 2A whenever an application reveals a differentiation gap, then return here and complete the model with units, assumptions, interpretation, and a reasonableness check.

Review Unit 2A foundations →

Unit 2B Hub: Applications of Derivatives

Unit 2A taught how derivatives are defined and computed. Unit 2B turns those calculations into information and decisions. The central question is no longer merely "what is the derivative?" but "what does this derivative let us conclude?"

In ordinary language

The practical meaning

A derivative is useful when its sign, size, units, or zeros answer a real question. Positive or negative may describe direction. A zero may identify a candidate peak. A second derivative may show whether growth is speeding up or slowing down. A tangent line may replace a difficult function near one convenient input.

Main sequence

• Interpretation, higher derivatives, motion, and rates from data. • Local linearity, differentials, uncertainty, and Newton's method. • Related rates and geometric modeling. • Extrema, the Mean Value Theorem, increasing/decreasing behavior, concavity, and curve analysis. • Optimization in geometry, science, engineering, and business. • L'Hopital's Rule and indeterminate forms. • Modeling studios, advanced notes, cumulative review, and exams.

Source note

The application-first organization was informed by the modern activities in Active Calculus; the physical intuition in Calculus Made Easy; and the public-domain applied traditions in Greenhill and Granville--Smith. BetterGrades prose, problems, models, checks, and digital specifications are independently written unless explicitly attributed.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.