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Calculus: Limits and First Derivatives: Tables, graphs, algebra and rates

Intro CollegeLevel 5
Cover of Calculus: Limits and First Derivatives: Tables, graphs, algebra and rates

About this book

See where a function is heading, then measure how fast it changes.

This workbook introduces the first two ideas of calculus: limits and derivatives. Learners read limits from tables and graphs, calculate them with factoring and conjugates, and test continuity. They then build the derivative from secant slopes, use the power, sum, product, quotient and chain rules, and write tangent lines. Every module follows the same sequence: explain, worked example, guided practice, independent practice, review and a checkpoint. Cumulative reviews, an exit check and a short retry follow. The answer key at the back explains each answer, names a common error and gives a reteaching cue.

  • One-sided and two-sided limits from tables and graphs
  • Limit laws, factoring, conjugates and continuity
  • The derivative as a limit of secant slopes
  • Power, sum, product, quotient and chain rules
  • Tangent lines, rates with units and local estimates

For college learners who can factor, simplify rational expressions and read function graphs.

Look inside

Page 5 of Calculus: Limits and First Derivatives. Step 2, Concept: Approach a value without assuming it
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Page 6 of Calculus: Limits and First Derivatives. Step 3, Worked Example: Approach a value without assuming it
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What this book covers

Covered

  • In scope: finite limits of polynomial, rational and square-root expressions; one-sided limits; continuity of simple piecewise functions; the limit definition of the derivative; power, sum, constant-multiple, simple product, quotient and chain rules; tangent lines and linear estimates.
  • Pages 53–54 then teach the quotient rule.

Not covered

Not in scope: ε–δ proofs, limits at infinity, derivatives of trigonometric, exponential and log functions, related rates, optimization and integrals.

Before you start

Before you start: you need the skills in College Algebra: Functions and Graphs and Quadratics: Patterns, Graphs, and Solutions.

Notice

Pictures in this book are simplified learning models. Check answers with the explanations at the back of the book.

The steps in this book

How the steps work
  1. Step 1: Readiness Check

    Check the skills you need before you begin. If something feels tricky, an adult can help you start in the right place.

  2. Step 2: Concept

    Read a short, clear explanation of the new idea and the words that go with it.

  3. Step 3: Worked Example

    Watch the idea being used, one step at a time, in a worked example.

  4. Step 4: Guided Practice

    Practice with prompts and help cues. The help gets smaller as you get stronger.

  5. Step 5: Independent Practice

    Practice on your own with different kinds of questions, and explain your thinking.

  6. Step 6: Review

    Look back at earlier learning so it stays strong, then check where to go next.

  7. Step 7: Answers and Explanations

    Check your answers. Each answer explains why, names a common mistake, and shows how to fix it.

What is in the book

The modules

  1. Module 1: Approach a value without assuming itPages 5–14
  2. Module 2: Calculate limits and assess continuityPages 15–24
  3. Module 3: Derivatives emerge from local changePages 25–34
  4. Module 4: Differentiate and interpret a local modelPages 35–44

Contents

  1. Readiness check, pp. 3–4. Before Module 1. Its table tells you where to start.
  2. Modules, pp. 5–44
  3. Checkpoints, pp. 14, 24, 34, 44. The last page of each module.
  4. Mixed review, pp. 45–48. Questions from across the book, mixed together.
  5. Exit check, pp. 49–50. A last check across the book.
  6. Reflect and repair, p. 51
  7. Try again, p. 52. Fresh questions like those in the exit check.
  8. Further practice, pp. 53–54. More pages of step 5, Independent Practice.
  9. Extra practice, pp. 55–56
  10. Answers and explanations, pp. 57–66. Notes in this part are headed “Why”, “Common mistake” and “Help”.

Before this book

Before you start: you need the skills in College Algebra: Functions and Graphs and Quadratics: Patterns, Graphs, and Solutions.

Missed p. 3 item 2 (slope between two points) → Linear Equations and Graphs

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