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  • CALCULUS CONTENT
  • Calculus Home
      • A.1 First Calculations
      • A.2 Generalizing to Other Rates of Change
      • A.3 Introducing Linear Approximations
      • A.4 Practice Problems: Linear Approximations
      • A.5 Differentials; Steps Toward Determining 𝑑𝑓𝑑𝑥 at 𝑥 =𝑎
      • A.6 Estimating 𝑑𝑓𝑑𝑥 at 𝑥 =𝑎 Graphically
      • B.1 Average Velocity for a Trip
      • B.2 Average Velocity Over Any Interval
      • B.3 Generalizing Average Rate of Change
      • B.4 Lab 1 Introduction: Approximate & Bound Error for 𝑑𝑓𝑑𝑥
      • B.5 Lab 1 Activity: Approximate & Bound Error of 𝑑𝑓𝑑𝑥
      • A.1 Introduction to Limits
      • A.2 Tolerance & Pancake Story
      • A.3 Epsilon-Delta Formal Definition
      • A.4 Lab: Find Limit Using Approximations
      • A.5 Limits That Exist & Do Not
      • A.6 One-Sided Limits
      • A.7 Vertical Asymptotes
      • B.1 Limit Laws
      • B.2 Substitution to Find a Limit
      • B.3 Factor to Find a Limit
      • B.4 Use Conjugates to Find a Limit
      • B.5 Use Algebra to Find a Limit
      • B.6 The Squeeze Theorem
      • B.7 Special Trig Limits
      • C.1 Introduction to Limits at Infinity
      • C.2 The Epsilon Strip
      • C.3 Some Limits at Infinity That Exist; Some That Do Not
      • C.4 Some Limits at Negative Infinity That Exist; Some That Do Not
      • C.5 Limits of Polynomials
      • C.6 Limits of Rational Functions
      • C.7 Limits of Exponential and Logarithmic Functions
      • C.8 Limits at Infinity with Square Roots
      • D.1 Continuity at a Point and Over an Interval
      • D.2 Discontinuity Types: Removable Discontinuities
      • D.3 Continuous functions; Continuity theorems
      • D.4 The Intermediate Value Theorem
      • A.1 Derivatives of Constant, Linear, and Power Functions
      • A.2 Derivative of Exponential Functions
      • A.3 Derivatives of sin⁡𝑥 and cos⁡𝑥
      • B.1 The Product Rule
      • B.2 The Quotient Rule
      • C.1 Conceptual Understanding of the Chain Rule
      • C.2 The Chain Rule
      • C.3 Basic Practice with the Chain Rule
      • C.4 Deeper Work with the Chain Rule
      • C.5 Chain Rule - Used to Find Yet More Derivatives
      • D.1 Explicitly and Implicitly Defined Functions
      • D.2 Implicit Differentiation
      • E.2 Derivative of Inverse Functions
      • F.1 Derivative of ln⁡(𝑥)
      • Derivative Rules: Practice Problems & Solutions
      • Chain Rule: Practice Problems & Solutions
    • • Tangent & Normal Lines
    • • Maxima & Minima
    • • Curve Sketching
    • • Mean Value Theorem
    • • Approximations
    • • L'Hôpital's Rule
    • • Related Rates
    • • Optimization
    • • Definite Integration
    • • Indefinite Integration
    • • u-substitution
    • • List of all blog posts
    • • 4 Steps to Solve Related Rates Problems in Calculus - Part 1
    • • 4 Steps to Solve Related Rates Problems in Calculus - Part 2
    • • How to Solve Optimization Problems in Calculus
    • • 0 Divided by 0: Solve Limit Problems in Calculus, Part 1
    • • 0 Divided by 0: Solve Limit Problems in Calculus, Part 2
    • • Helping You with Calculus: Introducing Matheno
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I. Introductory Ideas

Let's build from everyday thinking to develop basic concepts and lay the groundwork we need to develop the fundamental ideas of Calculus.

Introductory Ideas chapter image

Topics in this Chapter

A. First Calculations; Linear Approximations, Differentials

  • A.1 First Calculations
  • A.2 Generalizing to Other Rates of Change
  • A.3 Introducing Linear Approximations
  • A.4 Practice Problems: Linear Approximations
  • A.5 Differentials; Steps Toward Determining 𝑑𝑓𝑑𝑥 at 𝑥 =𝑎
  • A.6 Estimating 𝑑𝑓𝑑𝑥 at 𝑥 =𝑎 Graphically

B. Average Rate of Change; Better Estimate for 𝑑𝑓𝑑𝑥 at 𝑥 =𝑎

  • B.1 Average Velocity for a Trip
  • B.2 Average Velocity Over Any Interval
  • B.3 Generalizing Average Rate of Change
  • B.4 Lab 1 Introduction: Approximate & Bound Error for 𝑑𝑓𝑑𝑥
  • B.5 Lab 1 Activity: Approximate & Bound Error of 𝑑𝑓𝑑𝑥
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