Grade 12Algebraic ThinkingOriginal practice

Derivatives Rates and Local Models: Explore and Explain Worksheet — Grade 12

Derivatives Rates and Local Models: Explore and Explain: Build a difference quotient from position values. Starts with 2 practice tasks. Answer guidance included. Print on Letter/A4 or save as PDF from the print window.

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Independent activity · About 12 minutes · Pencil · Calculus (optional advanced course)

Practice set 1 of 3. Each set keeps the same learning goal and task structures.

Sheet 3B72B929 · questions and answers share this ID

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Grade 12 · Mathematics

Derivatives Rates and Local Models: Explore and Explain

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Grade 12 · Mathematics · continued

Derivatives Rates and Local Models: Explore and Explain

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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. Explore the material. Talk through your reasoning with suitable support.

1.Find s(2), with units.

2.Simplify [s(2+h) − s(2)]/h for h ≠ 0.

Grade 12 · Mathematics · Answer key

Derivatives Rates and Local Models: Explore and Explain

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Open responses use criteria and an illustrative response. Other well-supported responses can be acceptable.

Grade 12 · Mathematics · Answer key · continued

Derivatives Rates and Local Models: Explore and Explain

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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.

Answer:8 m

2.Simplify [s(2+h) − s(2)]/h for h ≠ 0.

Answer:1h + 6

About this practice

Grade 12 · Mathematics · Matching answer key included · US Letter and A4

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Teaching guide, sample answers and curriculum references

Before you begin

Understand limits, continuity, polynomial expansion and slope units.

How to use this worksheet

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.

A common mistake to discuss

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

What to practice next

Continue to Derivatives Rates and Local Models: Practice Independently.

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

Source reference: Selected high-school course targets.

Published 2026-10-08; updated 2026-10-08. Read our editorial policy.

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Derivatives Rates and Local Models

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. 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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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.

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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.

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Calculus: Limits, Continuity & Derivative Rules

Part A: Calculus Computation. Evaluate each limit and derivative problem. Choose the correct mathematical result.

1.Evaluate the limit analytically: lim (x -> 3) [(x² - 9) / (x - 3)]

BUndefined (Division by zero)

2.Find the derivative f'(x) of the cubic polynomial function: f(x) = 4x³ - 5x² + 7x - 12

Af'(x) = 12x² - 10x + 7

Bf'(x) = 12x³ - 10x² + 7x

Cf'(x) = 4x² - 5x + 7

Df'(x) = x⁴ - x³ + 7

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Grade 12

HSF.IF.B.4 Key

Calculus: Limits, Continuity & Derivative Rules

Practice · Independent · 35 min

Selected optional calculus practice with evidence, conditions and a written explanation.

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