Calculus 1 | Index

Calculus 1 is the study of change. It asks how fast a quantity is changing at one instant and how small changes add up to a total.

The course covers limits and continuity, derivatives and their rules, related rates and optimization. It then turns to antiderivatives, the Fundamental Theorem of Calculus, u-substitution, and areas and volumes.

It’s for AP Calculus AB students, the first half of AP Calculus BC, and college Calculus I students.

What to know first: Precalculus, especially functions, the difference quotient and limits, and Trigonometry, especially the unit circle and trig identities.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Limits and Continuity

  1. Parent Functions — review the graphs, domains and ranges of the basic function families
  2. Limit Laws (coming soon) — rules for finding limits of sums, products, quotients and powers
  3. Evaluating Limits Algebraically (coming soon) — find limits by direct substitution, factoring and rationalizing
  4. Infinite Limits (coming soon) — what it means when a limit grows without bound, and how it shows a vertical asymptote
  5. Limits at Infinity (coming soon) — find end behavior and horizontal asymptotes with limits
  6. Squeeze Theorem (coming soon) — find a limit by trapping a function between two others
  7. Continuity (coming soon) — the three conditions for continuity and the types of discontinuity
  8. Intermediate Value Theorem (coming soon) — show that a continuous function reaches every value between two outputs
  9. Epsilon-Delta Definition of a Limit (coming soon) — the formal definition of a limit and short epsilon-delta proofs

Also see: introduction to limits and one-sided limits in Precalculus, Unit 8.

Unit 2: Definition of the Derivative and Basic Rules

  1. Instantaneous Rate of Change (coming soon) — how the slopes of secant lines lead to the slope of a tangent line
  2. Definition of the Derivative (coming soon) — find a derivative from the limit of the difference quotient
  3. Sketching the Graph of a Derivative (coming soon) — draw the graph of f′ from the graph of f
  4. Differentiability (coming soon) — where a function has no derivative: corners, cusps, vertical tangents and breaks
  5. Power Rule (coming soon) — differentiate powers and polynomials with the power, constant multiple and sum rules
  6. Equation of a Tangent Line (coming soon) — write the equation of the line tangent to a curve at a point
  7. Higher-Order Derivatives (coming soon) — find second and third derivatives and what they tell you
  8. Position, Velocity and Acceleration (coming soon) — describe motion along a line with derivatives

Also see: the difference quotient in Precalculus, Unit 1, and Average Rate of Change (Algebra 1).

Unit 3: Product, Quotient and Chain Rules; Implicit Differentiation

  1. Product Rule (coming soon) — differentiate a product of two functions
  2. Quotient Rule (coming soon) — differentiate a quotient of two functions
  3. Chain Rule (coming soon) — differentiate a function inside another function
  4. Implicit Differentiation (coming soon) — find dy/dx when y isn’t written by itself
  5. Derivatives of Inverse Functions (coming soon) — find the derivative of an inverse function from the original function

Unit 4: Derivatives of Exponential, Logarithmic, Trig and Inverse Trig Functions

  1. Derivatives of Trig Functions (coming soon) — the derivatives of sine, cosine, tangent and the other three trig functions
  2. Derivatives of Exponential Functions (coming soon) — differentiate e to the x and other exponential functions
  3. Derivatives of Logarithmic Functions (coming soon) — differentiate ln x and logarithms with other bases
  4. Logarithmic Differentiation (coming soon) — use logarithms to differentiate products, quotients and powers like x to the x
  5. Derivatives of Inverse Trig Functions (coming soon) — the derivatives of arcsine, arccosine, arctangent and the others

Also see: logarithms and the number e in Algebra 2, Unit 7, and inverse trig functions in Trigonometry, Unit 5.

Unit 5: Related Rates, Linearization and L’Hôpital’s Rule

  1. Related Rates (coming soon) — find how fast one quantity changes from how fast a related quantity changes
  2. Linear Approximation (coming soon) — estimate values with the tangent line, also called linearization
  3. Differentials (coming soon) — use dy and dx to estimate changes and measurement errors
  4. L’Hôpital’s Rule (coming soon) — find limits of the forms 0/0 and ∞/∞ with derivatives
  5. Indeterminate Forms (coming soon) — rewrite products, differences and powers like 0 · ∞ and ∞ − ∞ so L’Hôpital’s rule applies

Unit 6: Analyzing Functions and Optimization

  1. Mean Value Theorem (coming soon) — what the mean value theorem and Rolle’s theorem guarantee about a function
  2. Critical Points (coming soon) — find critical numbers, where f′ is zero or undefined
  3. First Derivative Test (coming soon) — find where a function increases or decreases and locate its local extrema
  4. Absolute Extrema (coming soon) — use the extreme value theorem to find the absolute maximum and minimum on a closed interval
  5. Concavity and Inflection Points (coming soon) — use the second derivative to find concavity and inflection points
  6. Second Derivative Test (coming soon) — classify critical points as local maximums or minimums with f″
  7. Curve Sketching (coming soon) — sketch a graph using f, f′ and f″ together
  8. Optimization (coming soon) — solve maximum and minimum word problems with derivatives
  9. Newton’s Method (coming soon) — approximate a zero of a function with tangent lines

Unit 7: Antiderivatives, Riemann Sums and the Fundamental Theorem of Calculus

  1. Antiderivatives and Indefinite Integrals (coming soon) — reverse the derivative rules and add the constant of integration
  2. Riemann Sums (coming soon) — approximate the area under a curve with left, right and midpoint rectangles
  3. Trapezoidal Rule (coming soon) — approximate the area under a curve with trapezoids
  4. Definite Integrals (coming soon) — define the definite integral as a limit of Riemann sums and find signed area
  5. Properties of Definite Integrals (coming soon) — use the rules for definite integrals and find areas with geometry
  6. Fundamental Theorem of Calculus, Part 1 (coming soon) — differentiate an accumulation function defined by an integral
  7. Fundamental Theorem of Calculus, Part 2 (coming soon) — evaluate a definite integral with an antiderivative
  8. Average Value of a Function (coming soon) — find the average height of a function over an interval
  9. Displacement and Total Distance (coming soon) — use integrals of velocity to find net change and total distance traveled

Also see: sigma notation in Algebra 2, Unit 8.

Unit 8: U-Substitution

  1. U-Substitution (coming soon) — find indefinite integrals by changing the variable
  2. U-Substitution with Definite Integrals (coming soon) — change the limits of integration when you substitute
  3. Integrals That Give Natural Logarithms (coming soon) — integrate 1/x, tan x and other functions whose antiderivatives involve ln
  4. Integrals That Give Inverse Trig Functions (coming soon) — integrals whose antiderivatives are arcsine, arctangent or arcsecant

Unit 9: Area Between Curves and Volumes

  1. Area Between Curves (coming soon) — find the area between two curves with top minus bottom or right minus left
  2. Volumes by Cross Sections (coming soon) — find the volume of a solid with square, triangle or semicircle cross sections
  3. Disk Method (coming soon) — find the volume of a solid of revolution with disks
  4. Washer Method (coming soon) — find the volume of a solid of revolution that has a hole

Also see: the shell method in Calculus 2, Unit 3.

For Teachers: Standards Alignment

Lessons follow the AP Calculus AB course framework and a typical first-semester college Calculus I course. The slope fields and separable equations from AP Calculus AB Unit 7 are in Calculus 2, Unit 4. A few topics, such as the epsilon-delta definition of a limit, differentials and Newton’s method, are taught in college but not tested on the AP exam.

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