Unit 3 · Core unit

Applications of Differentiation

Where the rules become arguments. Extremes, shape, optimization, and related rates all ask the student to justify a conclusion from derivative evidence.

What a strong answer looks like

A strong answer cites the evidence: which derivative changed sign, where, and what that establishes. A correct value with no justification is an incomplete answer here.

Practice this unit Course overview

Chapter introduction

Understand the idea before memorizing the rule

Where the rules become arguments. Extremes, shape, optimization, and related rates all ask the student to justify a conclusion from derivative evidence.

In Calculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

Lesson 1

Extremes and the Shape of a Graph

Concept explanation

The first derivative controls increase and decrease; the second controls concavity. A sign change in the first marks a local extreme, in the second an inflection point.

How to apply it

Find critical numbers, build a sign chart, and read the conclusions off it rather than guessing from a sketch.

Common mistake

Assuming a zero second derivative means an inflection point, or confusing concavity with direction.

Guided study move

Build a full sign chart for a cubic and state every conclusion it supports.

Lesson 2

Theorems That Guarantee Something

Concept explanation

The Extreme Value Theorem guarantees extremes exist on a closed interval; the Mean Value Theorem guarantees a tangent parallel to the secant.

How to apply it

Check the hypotheses first, then state precisely what the theorem gives, which is existence rather than location.

Common mistake

Applying a theorem without checking continuity or differentiability on the right interval.

Guided study move

State the hypotheses and conclusion of each theorem in your own words, then find a function where each fails.

Lesson 3

Optimization and Related Rates

Concept explanation

Both translate a described situation into an equation, then differentiate. Optimization differentiates with respect to the variable; related rates differentiate with respect to time.

How to apply it

Draw the picture, name the variables, write the relationship, and only then differentiate. Substitute given values last.

Common mistake

Substituting a changing quantity before differentiating, which wrongly freezes it as a constant.

Guided study move

Set up a related-rates problem completely without solving it, then check that every rate has a symbol.

Formula and visual reference

Connect the symbols to the picture

Chapter synthesis

Ideas to connect

Justifying with a sign change rather than a sketch
Checking endpoints alongside critical numbers
Substituting numbers only after differentiating

Modeling lab

Transfer the chapter to a new setting

A container of fixed volume is to use the least material. Set up the model, state the domain of the variable, and justify that your answer is a minimum.

Mastery checklist

• Find absolute extremes on a closed interval.
• Justify a local maximum from a first-derivative sign change.
• Locate inflection points and confirm concavity changes.
• Complete a related-rates problem from a described situation.

Check yourself

• Why must endpoints be checked separately for absolute extremes?
• Why is substituting a value before differentiating a mistake in related rates?

Modeling drill

A container of fixed volume is to use the least material. Set up the model, state the domain of the variable, and justify that your answer is a minimum.

Critical numberLocal extremeConcavityInflection pointMean Value TheoremOptimizationRelated ratesLinearization

Unit 2: The Derivative and Differentiation Rules

Unit 4: Integrals and the Fundamental Theorem