Mathematics · Grade 12 · Calculus (optional advanced course)

Derivatives Rates and Local Models

Build a difference quotient from position values. Calculate a derivative and compare average rates. Interpret a tangent and apply the model domain.

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Derivatives as rates & slopes →Differentiate & connect representations →

Learning goals

Build a difference quotient from position values.
Calculate a derivative and compare average rates.
Interpret a tangent and apply the model domain.

Before you begin

Understand limits, continuity, polynomial expansion and slope units.

Age and grade do not measure readiness. Adjust reading or adult support to preserve the intended learning goal.

Follow the sequence

Complete practice

Derivatives Rates and Local Models

Position and instantaneous rate

Invented practice data or scenario; not a report of a real study.

An idealized position is s(t) = 1t² + 2t + 0 metres, for 0 ≤ t ≤ 4 seconds. Treat position on a number line, not distance travelled. Study the difference quotient at t = 2 for nonzero increments h with 2 + h in the domain.

Worked example: For p(t)=t² at t=1, [p(1+h)−p(1)]/h = 2+h for h≠0, so p′(1)=2. The value p(1)=1 is different. Use the material to work through each task. Show the requested response.

1.Find s(2), with units.

Practice preview

Grade 12

Key

Derivatives Rates and Local Models

Practice · Independent · 45 min

Build a difference quotient from position values. Calculate a derivative and compare average rates. Interpret a tangent and apply the model domain. A complete set with exploration, practice and application. Use the linked unit for the learning sequence and teaching guide.

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1. Explore

Derivatives Rates and Local Models: Explore and Explain

Position and instantaneous rate

Invented practice data or scenario; not a report of a real study.

Worked example: For p(t)=t² at t=1, [p(1+h)−p(1)]/h = 2+h for h≠0, so p′(1)=2. The value p(1)=1 is different. Explore the material. Talk through your reasoning with suitable support.

1.Find s(2), with units.

Practice preview

Grade 12

Key

Derivatives Rates and Local Models: Explore and Explain

Explore · Independent · 12 min

Build a difference quotient from position values. Calculate a derivative and compare average rates. Interpret a tangent and apply the model domain. A focused learn activity. Use the linked unit for the learning sequence and teaching guide.

Choose another practice set

Preview / Print →

2. Practice

Derivatives Rates and Local Models: Practice Independently

Position and instantaneous rate

Invented practice data or scenario; not a report of a real study.

Use the material to work through each task. Show the requested response.

1.Take the limit as h → 0 to find s′(2). State its unit.

Practice preview

Grade 12

Key

Derivatives Rates and Local Models: Practice Independently

Practice · Independent · 20 min

Build a difference quotient from position values. Calculate a derivative and compare average rates. Interpret a tangent and apply the model domain. A focused practice activity. Use the linked unit for the learning sequence and teaching guide.

Choose another practice set

Preview / Print →

3. Apply

Derivatives Rates and Local Models: Apply and Create

Position and instantaneous rate

Invented practice data or scenario; not a report of a real study.

Use what you learned to explain, construct or apply an idea.

1.Write a tangent-line equation at t = 2 and interpret the slope.

Practice preview

Grade 12

Key

Derivatives Rates and Local Models: Apply and Create

Apply · Independent · 25 min

Build a difference quotient from position values. Calculate a derivative and compare average rates. Interpret a tangent and apply the model domain. A focused apply activity. Use the linked unit for the learning sequence and teaching guide.

Choose another practice set

Preview / Print →

Teaching and response guide

Begin with: Find s(2), with units.
Connect the response to: Take the limit as h → 0 to find s′(2). State its unit.
Ask students to apply the idea: Write a tangent-line equation at t = 2 and interpret the slope.

Watch for: A derivative is a rate, not a position; an algebraic zero must lie within the model domain.

Check the answer criteria for each task. A supported explanation may have more than one acceptable response. These activities offer evidence to discuss; completing a sheet alone does not establish mastery.