Math Ladder · College

Calculus 1: Limits and Derivatives

Limits, the derivative as a rate of change, the rules of differentiation, and applications from optimization to the first integrals.

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54 topics in 5 units. Builds on Precalculus and Trigonometry. Leads to Calculus 2: Integrals and Series. It also opens the way to Mechanics on the Physics Ladder, and Int. Micro on the Economics Ladder. Not sure this is the right course? Find your rung.

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Unit 1 Limits

12 topics

Understand limits from graphs, compute them with algebra, and define them precisely.

  1. Limits from Both Sides
  2. One-Sided Limits
  3. Infinite Limits and Vertical Asymptotes
  4. Limit Laws
  5. Limits by Factoring and Canceling
  6. Limits by Rationalizing
  7. The Squeeze Theorem
  8. Special Trigonometric Limits
  9. Continuity
  10. The Intermediate Value Theorem
  11. Limits at Infinity and Horizontal Asymptotes
  12. The Precise Definition of a Limit

Unit 2 The Derivative

6 topics

Go from average to instantaneous rate of change, and define the derivative as a limit.

  1. From Average to Instantaneous Rate
  2. The Derivative at a Point
  3. Tangent Lines
  4. The Derivative as a Function
  5. Where Derivatives Fail to Exist
  6. Leibniz Notation and Units

Unit 3 Differentiation Rules

11 topics

Differentiate powers, products, quotients, compositions, and trig, exponential and log functions.

  1. Constant, Power and Sum Rules
  2. The Product Rule
  3. The Quotient Rule
  4. Derivatives of Trig Functions
  5. Higher Derivatives
  6. The Chain Rule
  7. Chain Rule with Several Layers
  8. Derivatives of Exponential and Log Functions
  9. Implicit Differentiation
  10. Derivatives of Inverse Functions and Inverse Trig
  11. Logarithmic Differentiation

Unit 4 Applications of Derivatives

11 topics

Use derivatives for motion, related rates, approximation, graph shape and optimization.

  1. Position, Velocity and Acceleration
  2. Related Rates
  3. Linear Approximation and Differentials
  4. Critical Points and Absolute Extrema
  5. The Mean Value Theorem
  6. Increasing, Decreasing and the First Derivative Test
  7. Concavity and the Second Derivative Test
  8. Curve Sketching
  9. Optimization
  10. L'Hôpital's Rule
  11. Newton's Method

Unit 5 Integrals

14 topics

Find antiderivatives, build definite integrals from sums, and use the Fundamental Theorem and substitution.

  1. Antiderivatives
  2. Antiderivatives with Initial Conditions
  3. Approximating Area with Rectangles
  4. Riemann Sums
  5. The Definite Integral
  6. Properties of Definite Integrals
  7. Accumulation Functions and FTC Part 1
  8. Evaluating Integrals with FTC Part 2
  9. Net Change and Total Distance
  10. Average Value of a Function
  11. Integration by Substitution
  12. Substitution in Definite Integrals
  13. Integrals Giving Exponentials and Logarithms
  14. Integrals Giving Inverse Trig Functions
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