Calculus Lessons

Course Lesson List

Below are the lessons for Calculus. View All Lessons » to see all lessons from all courses.

Calculus

Functions and Limits - Function Review

Functions in Review

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Function Review

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Tags

  • Review
  • Functions
  • Function Analysis

Priority: Optional

Since Calculus is almost entirely about function analysis, this first lesson makes sure you're up to speed on all the major properties and mechanics you should already know about functions before starting Calculus.

  • Recap the most important properties of functions from prior courses
  • Review Domain, Range, and Function Inverses
  • Review basic how-to regarding graphing functions
Transformations of Functions Review

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Function Review

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Tags

  • Review
  • Functions
  • Function Analysis

Priority: Optional

Algebraically manipulating functions to translate or scale them is another skill we must brush up on. This lesson covers concepts from Algebra Two and Pre-Calculus, putting all function transformation knowledge in one place.

  • Recall the mechanics involved with function translations both horizontally and vertically, sometimes referred to as "slides" or "shifts"
  • Use multiplication and division to transform a function with stretching or compressing
Combining Functions Review

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Function Review

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Tags

  • Review
  • Functions
  • Function Analysis

Priority: Normal

This lesson reviews combining functions both by arithmetic (e.g. $h(x) = f(x) + g(x)$) and function composition (e.g. $h(x) = f(g(x))$).

  • Review how to apply arithmetic to functions as if they were variables
  • Thoroughly refresh on what function composition is all about, and being able to identify the "inner" and "outer" function
  • Be able to deconstruct a complex function into a series of compositions of simpler functions
Important Types of Functions

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Function Review

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Tags

  • Review
  • Functions
  • Function Analysis

Priority: Normal

Overall knowledge and familiarity of functions is crucial to success in Calculus. You'll be able to recognize families of functions and the characteristics of each, either by looking at the graph, or by seeing the function.

  • Know how to categorize functions, and each category's characteristics
  • Be able to identify and categorize very commonly used functions visually
  • Discern certain numerical characteristic about common function categories

Functions and Limits - Limits

Defining Limits and Finding Graph-Based Limits

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Limits

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Tags

  • Limits
  • Functions

Priority: Normal

In this inaugural discussion of limits, we seek to understand what limits are, and be able to find limits based on graphs.

  • Understand what limits are and know their notation
  • Evaluating finite limits based on graphs
  • Know the difference between one-sided and two-sided limits
  • Define two-sided limits based on both one-sided limits
  • Evaluate limits computationally using a calculator
Evaluating Limits with Limit Laws

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Limits

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Tags

  • Limits
  • Functions

Priority: High

As we come to be more familiar with limits, we want to treat them like algebra operations. This lesson starts with some detail about how limits are used purely in an algebra sense (without relying on a graph), as well as how to evaluate them

  • Recognize the technical definition of a one-sided limit
  • Know laws governing limits of sums and differences, products, quotients, and constants
  • Find limits using substitution and/or limit laws
  • Determine when a limit requires further analysis to evaluate, versus when it does not exist
Limits with Zero in the Denominator

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Limits

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Tags

  • Limits
  • Simplifying
  • Functions
  • Rational Expressions

Priority: VIP Knowledge

Using new knowledge about how limits work, we will practice evaluating limits of variable expressions. Most courses focus on rational functions for this purpose, so while we will look at several situations, the focus here will be on rational functions and expressions.

  • Recall how vertical asymptotes are formed
  • Distinguish between punctures and asymptotes
  • Use algebra and limit behavior to evaluate algebraic limit expressions that involve partial positives
  • Determine the limit of a rational expression at the point where its denominator is zero but its numerator is not zero
  • Determine the limit of a rational expression at the point where both its numerator and denominator are zero
Squeeze Theorem, End Behavior, and Limits at Infinity

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Limits

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Tags

  • Limits
  • Functions
  • Rational Expressions

Priority: Normal

In Algebra Two we studied the end behavior of polynomials. Here, using the concepts of infinity and limits, we can look at end behavior for any function. Specifically we will focus on rational functions. We will also examine what the "squeeze theorem" guarantees for well-behaved functions.

  • Use the squeeze theorem to evaluate otherwise indeterminate limits
  • Recall what we already know about polynomial end behavior
  • Understanding what limits mean as x goes to positive or negative infinity
  • Apply infinite limits to functions we are already familiar with such as exponential or logarithmic functions
  • Focus on determining end behavior of rational functions
Three Important Limits

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Limits

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Tags

  • Limits

Priority: Optional

This isolated topic is worth a quick look, since both AP exams and college professors often put limits of these forms on exams. The focus of this lesson is solely to expose you to what these limits are, and what they correctly evaluate to so that you can answer exam questions if they should appear.

  • Learn three specific limits that AP exams and college professors often use for exams
  • Focus on what the result of each limit is, not the derivation or justification, and be able to give the correct answer when asked about limits in these special forms

Functions and Limits - Continuity

Continuity

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Continuity

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Tags

  • Limits
  • Functions
  • Function Analysis

Priority: Normal

We've talked in the past in Algebra Two and Pre-Calculus about a loose definition of what it means for a function to be continuous. Now we'll talk about a rigorous definition, using limits.

  • Understand continuity at a point or over an interval conceptually and visually based on graphs
  • Define continuity of a function at a point using limits
  • Define continuity of a function over an interval using limits
  • Classify types of discontinuities, either based on graphs or based on function definitions
The Continuity Value Theorems

Calculus  $\rightarrow$  Functions and Limits  $\rightarrow$  Continuity

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Tags

  • Functions
  • Polynomials

Priority: Optional

When a function is known to be continuous. there are two value theorems that tell us about where the roots of the function are, or about the min or max point over a closed interval.

  • Learn and understand the intermediate value theorem
  • Learn and understand the extreme value theorem

Discovering Derivatives - The Concept of the Derivative

The Slope Problem

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  The Concept of the Derivative

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Tags

  • Functions
  • Derivatives

Priority: Normal

One of the major calculus tasks is to solve "the slope problem". This lesson seeks to define the issue we are addressing when we study differential calculus, and introduces us to some estimation techniques that will eventually evolve into calculus principles.

  • Define and understand the slope problem of Calculus
  • Examine ways we can estimate the instantaneous slope of a nonlinear function
  • Look at how and why we might want to interpret the slope of a function in a real life application
Average Rate of Change

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  The Concept of the Derivative

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Tags

  • Functions
  • Slope
  • Coordinate Plane

Priority: Normal

In order to further study the instantaneous slope of a function, it will be helpful to first study the average slope of a function over a specified interval. This lesson defines and works with this concept, which is commonly referred to as Average Rate of Change.

  • Define the Average Rate of Change over an interval
  • Understand why AROC only makes sense to examine over a fixed, finite interval
  • Learn how AROC can be used to approximate instantaneous slope
  • Briefly look at kinematics graphs and interpret AROC in context of a displacement or velocity graph
The Derivative as a Limit

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  The Concept of the Derivative

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Tags

  • Derivatives
  • Limits

Priority: High

Using what we know about the slope formula, AROC, and limits, we are ready to define a function's exact instantaneous slope at a point using a limit.

  • Know what the derivative of a function is and what it means
  • Define the derivative using the difference quotient
  • Use the difference quotient and limits to determine a function's instantaneous slope at a point
The Derivative as a Function

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  The Concept of the Derivative

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Tags

  • Derivatives
  • Functions

Priority: High

The limit approach to instantaneous slope from the last lesson got us exact slopes at a specific points. This lesson instead will show us how to obtain a function's "slope function" which will yield us the slope of the function at any point without having to repeatedly use the limit definition.

  • Obtain a function's "slope function" using derivatives
  • Use the difference quotient to get the "slope function" of a function for basic polynomial, radical, and rational functions
  • Understand why the generic slope function is better than finding specific slopes
Derivative Notation

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  The Concept of the Derivative

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Tags

  • Derivatives

Priority: Normal

There are two ways to write derivatives using math symbols. A derivative is a derivative, but while each way means the same thing, some derivative applications are easier to communicate with one versus the other. This lesson will explain what each way looks like and help you understand why we might favor one over the other depending on the situation.

  • See both ways we commonly notate derivatives, the "prime" and the "differential" notations
  • Understand why each notation has unique applications
  • Know the proper way to use each notation
Derivatives of Sums and Constants

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  The Concept of the Derivative

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Tags

  • Derivatives
  • Algebra Rules

Priority: High

Not coincidentally similar to limits, the derivative of a sum or difference of functions can be handled by taking things one piece at a time. This lesson outlines this for us with practice, and also looks at the derivative of a function that is multiplied by a constant.

  • Learn how the derivative of a function is affected if the function is multiplied by a constant
  • Learn why the derivative of a constant on its own is zero
  • Understand why and how we can take the derivative of sums and differences by handling one term at a time

Discovering Derivatives - Derivatives of Common Functions

Polynomial Derivatives: The Power Rule

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Derivatives of Common Functions

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Tags

  • Derivatives
  • Polynomials

Priority: VIP Knowledge

By far the most common type, polynomial derivatives have a simplistic, formulaic approach that we can use instead of performing the long hand limit definition method of derivatives. We will learn the massive time-saving formula approach in this lesson.

  • Learn the Power Rule for quickly taking the derivative of any single variable monomial
  • Take the derivative of entire polynomials by applying the power rule one term at a time
  • Apply the Power Rule for variables raised to fraction exponent powers
  • See and practice using the power rule correctly when the variable is in the denominator
Derivatives of Trig Functions

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Derivatives of Common Functions

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Tags

  • Derivatives
  • Trig Functions
  • Trigonometry

Priority: VIP Knowledge

Here we will learn the knowledge and tricks to take the derivative of sin, cos, and the other four trig functions. Note: this lesson covers the basics. If you are looking for more challenging practice problems involving trig functions, check out the upcoming lessons on Differentiation Techniques.

  • Learn how to take the derivative of any of the six trig functions
  • Use derivatives of trig functions alongside derivative rules we've already seen
Derivatives of Exponentials

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Derivatives of Common Functions

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Tags

  • Derivatives
  • Variable Exponents

Priority: VIP Knowledge

Here we will learn how to take the derivative of exponential terms, starting with the foundational natural exponentiation function $e^x$. After studying the derivative pattern of this vital function, we will see how to take the derivative of similar functions, including exponentials with bases other than $e$ as well as exponentials with constant multipliers on the variable, such as $e^{5x}$.

  • Learn how to take the derivative of $e^x$
  • Modify these new facts to take the derivative of things other than base $e$
  • Know the rule for the derivative of $e^{kx}$ where $k$ is a constant
Derivatives of Logarithms

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Derivatives of Common Functions

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Tags

  • Derivatives
  • Logarithms

Priority: High

The derivative of the natural logarithm function is incredibly useful for both semesters of Calculus. This lessons will start by showing us how to take the derivative of this fundamental function, before also showing us how to leverage old-school log rules for taking derivatives of complicated log expressions.

  • Learn how to take the derivative of $\ln (x)$
  • Modify the derivative of the natural log to be able to take the derivative of any base log
  • Extend our knowledge to know the derivative of an expression of the form $\log_b (ax + c)$ where $a$, $b$, and $c$ are constant coefficients
  • Leverage logarithm manipulation rules to find derivatives of otherwise complicated log expressions
Derivatives of Inverse Trig Functions

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Derivatives of Common Functions

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Tags

  • Derivatives
  • Trigonometry

Priority: Normal

We will learn how to take the derivative of the inverse trig functions we studied in trigonometry. While this lesson is very commonly tested, it is almost entirely a matter of memorization. You will only see this again infrequently in the future, but if you do, many teachers expect you to remember it.

  • Know the derivatives of arcsin(x), arccos(x), arctan(x)
  • See (but perhaps not memorize) the derivatives of arccot(x), arcsec(x), and arccsc(x)
  • Understand similarities and relationships among these derivatives to aide memorization
  • Learn a $u$ substitution formula to take the derivative of any inverse trig function when the input is not merely $x$ (previewing the Chain Rule)
Differentiating with Hyperbolic Trig Functions

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Derivatives of Common Functions

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Tags

  • Derivatives
  • Trigonometry

Priority: Optional

Hyperbolic trig functions are similar to the classic SOH-CAH-TOA trig functions, but they have different meanings and properties. Though they are not commonly used, some professors expect you to know what they are, and how to take the derivatives of both the functions themselves and their inverse functions. We'll start with what inverse trig functions are and what you're expected to know about them, and then lay down some formulas that you may or may not be asked to memorize depending on your professor's level of insanity.

  • Know the derivatives of sinh(x), cosh(x), and tanh(x)

Discovering Derivatives - Differentiation Techinques

Higher Derivatives

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Differentiation Techinques

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Tags

  • Derivatives
  • Functions
  • Derivative Techniques

Priority: High

Knowing derivatives is one thing, but what about the derivative of the derivative? What does that describe about a function? What about the derivative or the derivative of the derivative? This lesson formally discusses higher derivatives and in which situations we care about their use and interpretation.

  • Learn what compound derivatives mean and how we usually name and notate them
  • See (but not usually use) the limit definition of the second derivative
  • Understand important patterns and relationships between the function and its derivative, second derivative, third derivative, etc.
The Product and Quotient Rules

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Differentiation Techinques

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Tags

  • Derivatives
  • Functions
  • Derivative Techniques

Priority: High

Unlike limit laws, it is not true that the derivative of a product is the product of the derivatives. The derivative of a product has a pattern based approach, though, and we will learn and practice it here, before immediately moving on to how to take the derivative of a quotient.

  • Learn the product rule for differentiating a product of two pieces when you know the derivative of each piece
  • Learn the quotient rule for differentiating the quotient of two terms when you know the derivative of each term
  • Know expert tips and tricks to avoid very common pitfalls for using these techniques mistake-free
  • Because the quotient rule is more involved, learn how we can often turn a quotient problem into a multiplication problem, and then proceed with the product rule
The Chain Rule

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Differentiation Techinques

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Tags

  • Derivatives
  • Functions
  • Derivative Techniques

Priority: VIP Knowledge

The Chain Rule is incredibly important to derivatives because it gives us the ability to take the derivative of almost anything. This lesson will teach us how the Chain Rule works, and then get us exposed to the common ways in which we use it practically.

  • See and understand the concept of how the Chain Rule works
  • Apply the Chain Rule one step at a time by thinking about function composition
  • Practice seeing and using the chain rule with a level of automaticity
Implicit Differentiation

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Differentiation Techinques

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Tags

  • Derivatives
  • Derivative Applications
  • Derivative Techniques

Priority: High

In some ways, regular differentiation is just a subset of implicit differentiation. Implicit can also be viewed as an extension of the chain rule. This lesson shows you what it's all about, but most importantly, gives you the know-how to get the right answer for exams and future applications.

  • Understand implicit differentiation as an extension of the chain rule
  • Understand how implicit differentiation gives us generic results for unknown functions of $x$
  • Recognize when we do and do not necessarily need to use this approach
Logarithmic Differentiation

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Differentiation Techinques

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Tags

  • Derivatives
  • Derivative Applications
  • Logarithms
  • Derivative Techniques

Priority: Optional

When working with functions of $x$ that have $x$ as both a variable exponent and part of the base expression (e.g. $f(x) = x^x$, or $g(x) = [\cos(x)]^{x^2}$), the normal rules of differentiation do not work. In this case we can get the correct derivative expression by using a technique called Logarithmic Differentiation.

  • Learn the step by step method of Logarithmic Differentiation
  • Recognize when Logarithmic Differentiation is required to find a derivative
  • Understand what types of complicated functions are optionally (but much more easily) differentiated using Logarithmic Differentiation
Find The Derivative of Anything

Calculus  $\rightarrow$  Discovering Derivatives  $\rightarrow$  Differentiation Techinques

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Tags

  • Derivatives
  • Derivative Techniques

Priority: High

This lesson represents one of the major payouts of Calculus, both practically for your future and for exams. Here you will practice putting together all that you have learned, often applying concepts simultaneously, so that you can take the derivative of any one variable function that you could ever dream of.

  • Master knowing how to differentiate anything
  • Write the right amount of scratch work so that you never make a mistake
  • Understand when and why each required rule is needed, and in what order, for problems that require many steps.
  • Understand when the Chain Rule is needed, with and without needing another simultaneous rule such as the Product Rule or Quotient Rule
  • Learn to manipulate complicated log functions to simpler form before taking their derivative

Applications of Derivatives - Function Analysis with Derivatives

Differentiability

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Functions
  • Function Analysis

Priority: Normal

Not all functions have well-defined slope functions. This lesson explores the relationship between whether a function has a well-defined slope function and the characteristics of that function.

  • Understand what "differentiable" means and how it is similar and different from "continuous"
  • See visual cues from graphs that help us define and understand differentiability
  • Define differentiability using a limit definition
Relative Extrema and Intervals of Increase

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Functions
  • Function Analysis
  • Graphing

Priority: High

Critical points are places where a function can possibly have its slope change from positive to negative or vice versa. Using your new knowledge of derivatives, we will learn how to find the location of a critical point, as well as figuring out whether each critical point is a relative max, relative min, or neither, without having to rely on a graph.

  • Use derivatives to identify the critical points of a function.
  • Know the two things we look for when identifying critical points
  • Use derivatives to identify points that are potential relative maximum and minimum values of a function
  • Decide whether a critical point is a max, min, or neither by using the second derivative test
  • Use sign analysis on the number line to determine whether a critical point is a max or min
  • Find and concisely state a function's intervals of increase and decrease
Inflection Points and Concavity

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Functions
  • Function Analysis
  • Graphing

Priority: High

Similar to the last lesson, we can use derivatives to identify special points on a function. This time, instead of looking for max or min points, we will look for points where the function concavity changes. We will also seek to concisely express the intervals of concavity for a function, employing the aide of sign analysis, as we did in the last lesson.

  • Define the concavity of a function
  • Understand what inflection points are
  • Learn how to identify inflection points
  • Use sign analysis on the number line to determine whether a potential inflection point is truly an inflection point
  • Find and concisely state a function's intervals of concavity
Interpreting Graphs of Derivatives

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Functions
  • Function Analysis
  • Graphing

Priority: Normal

Without knowing exactly what a function is, we can identify key relationships between it and its derivatives, using only graphs. Here we will understand what those relationships are, and find an even more concise result when working with polynomial functions.

  • Identify visual relationships between the graph of a function and the graph of that function's derivative
  • For polynomial functions, find a specific relationship between the shape of a function and the shape of its first derivative as well as higher derivatives
Absolute Max and Min Values

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Functions
  • Function Analysis
  • Graphing

Priority: Normal

In this lesson, you will learn how to find the absolute largest or smallest function value on a closed interval, using only Calculus techniques.

  • Use derivatives and critical points to determine the maximum and minimum points on a closed interval
  • Know how and why to consider the endpoints of the interval
  • Tell whether or not a function has a global max or min using end behavior
Curve Sketching

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Functions
  • Function Analysis
  • Graphing

Priority: Normal

This lesson allows you to draw an unknown function using only clues about its behavior, with both specific points and derivative information.

  • Without explicitly defining a function, sketch a potential graph of it based on clues about it and its derivatives
  • Understand why there is no one right sketch for these questions, but also what can happen to make a sketch be incorrect
Mean Value Theorem and Rolle's Theorem

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Function Analysis with Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Functions
  • Function Analysis

Priority: Optional

The Mean Value Theorem (MVT) is the kind of thing that makes sense intuitively and is tested in fairly predictable ways, but is not very practical for real world applications. With that in mind, this lesson will make sure you understand the theorem and be able to answer typical exam questions based on it.

  • Learn what the Mean Value Theorem is and how you're expected to use it
  • Understand the geometric interpretation of the Mean Value Theorem

Applications of Derivatives - Other Applications of Derivatives

Tangent Line Equations

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivatives
  • Functions
  • Derivative Applications

Priority: High

On quizzes and exams, we are commonly asked to find not only the instantaneous slope at a point on a function, but also the linear equation of the straight line at that point. More advanced questions ask for equations of lines that are tangent to the function but pass through a point that is not on the function. Each question has its own short solution process, and this lesson will cover both cases.

  • Know how to find the tangent lines of a function at a point
  • Use the same technique to find the equation of a normal line at a point
  • Learn a guaranteed-to-work approach for finding equations of tangent lines that go through a specified point
Optimization

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Word Problems

Priority: High

One of the most common and useful applications of derivatives is finding the optimal solution to a problem - the one that maximizes or minimizes some quantity. This lesson will show you the common techniques used to do this, as well as the common ways this topic shows up on exams.

  • Understand why calculus is the most efficient way to find the max or min of something
  • Master the general technique for solving an optimization problem
  • Practice setting up optimization problems from scratch by relating several unknown quantities with a single variable
Equations of Motion (Derivatives Only)

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Function Analysis
  • Word Problems

Priority: Normal

Given the equation that describes an object's position at a given time, we can discern other measures of its motion using derivatives. This lesson will showcase the common ways in which you can expect professors and AP exams to test this topic.

  • Learn the Calculus-based relationships between acceleration, velocity, and location based on derivatives
  • Use graphs to understand relationships between acceleration, velocity, and position
  • Differentiate to find velocity from position, or acceleration from velocity
Marginality Applications

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Word Problems

Priority: Optional

This topic, which is fairly specific to business operations, will show you how to analyze and quantify profitability of producing $q$ number of goods based on how much it revenue and cost is involved, where revenue and cost each may vary based on $q$.

  • Examine the idea of producing goods to make a profit by defining profit, revenue, and cost as functions of $q$, the quantity of the good produced
  • Interpret the give revenue function $R(q)$ for $q$ number of goods produced, or create $R(q)$ from a price function if $R(q)$ is not given explicitly
  • Interpret the cost function $C(q)$, which is almost always given explicitly
  • Find and interpret the marginal cost and marginal revenue functions using differentiation, and know how these marginal functions may be used to make business decisions
  • Know how to calculate the ideal quantity of goods to produce to maximize the profit
L'Hopital's Rule

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivatives
  • Derivative Applications
  • Function Analysis

Priority: Normal

If you are evaluating a limit which yields an indeterminate result (such as 1/0, 0/0, $1^{\infty}$, $0^0$, etc.), there is a good chance L'Hopital's Rule will give you the answer quickly. This lesson will teach you more specifically when to employ the rule, and, of course, how to use it properly.

  • Review and master being able to recognize all indeterminate forms
  • Understand when and why L'Hopital's rule is required
  • Know how to use L'Hopital's Rule
The Derivative of a Function's Inverse

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivatives
  • Derivative Applications

Priority: Optional

Though a fairly isolated topic, the derivative of a function's inverse has a formulaic approach that allows you to find the answer without finding the function inverse function. This lesson shows how this works.

  • Understand the motivation for wanting to use a formula approach for this situation rather than explicitly finding the inverse
  • Learn the formulaic approach to talking the derivative of a function's inverse
Linearization and Newton's Method

Calculus  $\rightarrow$  Applications of Derivatives  $\rightarrow$  Other Applications of Derivatives

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Tags

  • Derivative Applications
  • Solving Equations

Priority: Optional

Linearization and Newton's Method are two quick commonly studied derivative applications. Both of these methods are what we call "numerical methods" because rather than finding an exact solution, these methods can be used to find very precise (but not exact) decimal solutions.

  • Learn the method of linearization, which approximates function values using the derivative
  • Use Newton's Method to iteratively find the roots of equations that have no algebraic exact solution

Understanding Integration - The Calculus Area Problem

Intro to Integration

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  The Calculus Area Problem

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Tags

  • Integrals

Priority: High

To get us started with integration calculus, we will first understand the problem we are trying to solve, and some basic properties about the area enclosed between a function and the $x$ axis.

  • Understand the "area problem" that integration calculus seeks to solve, similar to how differential calculus solves the slope problem
  • Know the difference between positive and negative net area
Rectangular Approximation

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  The Calculus Area Problem

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Tags

  • Integrals
  • Numerical Approximation

Priority: Normal

As a stepping stone for the processes that answer "The Area Problem" exactly and definitively, we will examine a common technique used to approximate the area under a curve

  • Learn the method of approximating area under a curve using a finite number of rectangles
  • Understand the three ways the rectangles could be oriented in an approximation: left, right, and midpoint
  • Use function concavity and the number of rectangles used to draw conclusions about the accuracy of the approximation
Other Area Approximations

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  The Calculus Area Problem

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Tags

  • Integrals
  • Numerical Approximation

Priority: Normal

While rectangle approximations yield reasonable estimates, there are two similar methods which are slightly more accurate at the expense of being slightly more complicated. Here we will look at everything we need to know about the Trapezoidal Rule and Simpson's Rule.

  • Get a first look at the trapezoidal approximation approach by seeing how it works visually
  • Learn the trapezoidal approximation approach by learning the formula
  • Learn and use the formula for Simpson's approach
  • For both approximations, learn how to quantify the error bounds in estimates as compared to the true value
Riemann Sums

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  The Calculus Area Problem

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Tags

  • Integrals
  • Limits

Priority: Normal

The concept of Riemann Sums is to integrals what the difference quotient was for derivatives - it's the "long way" of getting the exact answer we seek. Using limits and finite sum formulas, we'll be able to get an exact answer for the area under a curve.

  • Understand the concept of Riemann Sums as an extension of the rectangle approximation method
  • Practice setting up infinite sums with limits that represent the exact area under a function from starting point $a$ to ending point $b$
  • Use limits, summation laws, and finite sum formulas to compute Riemann Sums

Understanding Integration - Antiderivatives

Integration and Antiderivatives

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Integrals

Priority: VIP Knowledge

One of the tools we will need to find specific areas is the process of finding a function's antiderivative. This first lesson will conceptually familiarize us with what they are and what we need to know about them.

  • Learn what antiderivatives are and their relation ship to functions
  • Understand the relationship between antiderivatives and derivatives and why they are very very nearly (but not quite) inverse operations
  • Similar to derivatives, know the linear operator rules for working with sums, differences, and constant coefficients
  • Learn what the process of indefinite integration means
The Definite Integral

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Integrals
  • Functions
  • Graphing

Priority: VIP Knowledge

This lesson more precisely defines the operation of finding the area under a curve between two $x$ values. We'll look at properties of definite integrals, and relate definite integrals to antiderivatives using a very important theorem.

  • Learn what a definite integral represents conceptually and visually
  • Split or combine integrals of a function using given information and integration limits
  • Know what happens when you switch the limits of integration and why
  • Relate functions and their Antiderivatives using the a Fundamental Theorem of Calculus
  • Use the FTC to evaluate definite integrals
The Fundamental Theorem of Calculus

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Integrals
  • Functions

Priority: VIP Knowledge

The definite integral of a known function does not require a graph or computer, but rather can be expressed algebraically using the antiderivatives and the Fundamental Theorem of Calculus. This lesson shows us how to execute on definite integrals of known functions, which we will continue practicing as we learn more about specific function families' antiderivative processes.

  • Relate functions and the area under their curve using Antiderivatives and the Fundamental Theorem of Calculus
  • Use the FTC to evaluate definite integrals
  • Understand Part 2 of the FTC, involving simultaneous derivatives and integrals
The Power Rule for Antiderivatives

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Integrals
  • Polynomials

Priority: VIP Knowledge

We recently learned that integration involves finding antiderivatives, so we need to know how to find them for any given function type. Here we will start with polynomials, using a Power Rule for integration similar to the one that exists for derivatives.

  • Learn the Power Rule for polynomial antiderivatives
  • Be able to specifically state why it behaves like an inverse to the Power Rule for derivatives
  • Practice taking definite and indefinite integrals of polynomials
Antiderivatives of Exponentials

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Integrals
  • Variable Exponents

Priority: High

Leveraging the derivative knowledge we have about exponentials, we will understand how to find antiderivatives of them as well. We'll start with the base case antiderivative of $e^x$, before working on antiderivatives of both $e^{kx}$ and $a^{kx}$, where $a$ and $k$ are real number constants.

  • Using what we know about derivatives, understand how to find the antiderivative of terms of the form $e^{kx}$
  • Learn how to integrate exponential terms when the base is not $e$
  • Practice taking definite and indefinite integrals of exponential functions
Antiderivatives Requiring Logarithms

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Integrals
  • Logarithms

Priority: Normal

There are a few cases where logarithms get thrown in the mix during our study of integration. First we'll see how to use logarithms to integrate $x^{-1}$, since the Power Rule cannot give a definitive answer for this term. Then, we'll look at the antiderivatives of logarithm expressions on their own.

  • See why the Power Rule fails for integrating $x^{-1}$ and find its antiderivative by recalling the function that has a derivative of $x^{-1}$
  • Practice taking definite and indefinite integrals of logarithmic terms and $x^{-1}$
  • Learn the best way to deal with integrating logarithm terms, first with natural logarithms and then with any-base logarithms
Trigonometric Antiderivatives

Calculus  $\rightarrow$  Understanding Integration  $\rightarrow$  Antiderivatives

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Tags

  • Trig Functions
  • Integrals

Priority: Normal

Here we will see some trig function antiderivatives that will align with what we already know about trig function derivatives. We'll also look at the antiderivative of each of the six trig functions, though some of them are not common or useful due to their complexity.

  • Leverage our knowledge of derivatives to instantly know six specific antiderivatives
  • See (but hopefully not memorize or even use) four other trig function antiderivatives

Integration Techniques - Common Methods of Integration

Integrals with Absolute Value

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Common Methods of Integration

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Tags

  • Integrals
  • Algebra Manipulation
  • Integration Techniques

Priority: Normal

Whenever expressions containing absolute value appear in an integral, a minor but important step is needed to make sure that the result of integration is correct. This lesson will show you what's needed for both indefinite and definite integration.

  • Learn the principle behind splitting integrals of absolute value based on the definition of absolute value
  • For indefinite integrals, understand the need and process for representing results as piecewise functions
  • For definite integrals, understand the net-area reasons and implications of splitting the integral into two integrals
Basic U-Substitutions

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Common Methods of Integration

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Tags

  • Integrals
  • Algebra Manipulation
  • Integration Techniques

Priority: VIP Knowledge

The method of U Substitutions is a very common technique in integration. This lesson shows you how to do it, and when the method is and is not applicable. We will learn the mechanics to apply this method for both indefinite and definite integrals.

  • Learn the integration method of U Substitutions, and when it does and does not work
  • Specific to U Substitutions, learn the Mr. Math method to write scratch work to avoid calculation errors
  • Practice integrating indefinitely using the U Substitution method
  • Learn the minute yet important differences between indefinite and definite integrals when it comes to the U Substitution approach
Advanced U-Substitutions

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Common Methods of Integration

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Tags

  • Integrals
  • Algebra Manipulation
  • Integration Techniques

Priority: High

While a vast majority of the cases you'll need U substitution for will closely resemble the problems we saw in the last lesson, there are specific cases where U substitution is used in unintuitive ways. This lesson will demonstrate these less common uses for U substitutions.

  • Reinforce what we learned about the mechanics of U substitution in the previous lesson
  • Utilize the substitution method for untraditional situations such as manipulated linear expressions
  • Learn how to apply U substitution for integrals involving rational functions with the variable in two places
Integration By Parts

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Common Methods of Integration

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Tags

  • Integrals
  • Integration Techniques

Priority: High

Another technique we commonly use to solve more complex integration problems is the method of Integration By Parts (IBP). First we'll focus on indefinite integration to learn how to use the formula both on its own and recursively, and then we'll see what changes for definite integrals.

  • Memorize and use integration by parts formula
  • Recognize when this is the correct approach
  • Use Integration By Parts formula recursively
  • Apply the tabular approach to integrating certain common recursive forms
  • After gaining mastery of the process, learn how to use IBP correctly with definite integration

Integration Techniques - Advanced Integration Techniques

Partial Fraction Decomposition

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Advanced Integration Techniques

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Tags

  • Integrals
  • Integration Techniques
  • Algebra Manipulation

Priority: Normal

Though this concept is based entirely in algebra, splitting complex fractions into pieces doesn't really have a convenient use beside integrating. We'll learn the fraction decomposition technique, and then apply it while integrating.

  • Algebraically reduce fractions using partial fraction decomposition
  • Understand when this algebra technique is useful for integration, and also when it cannot be applied
  • Practice applying this technique for both indefinite and definite integrals
  • Apply the technique for linear factors of multiplicity 1 (all levels)
  • Apply the technique for linear factors with multiplicity, and for irreducible quadratic factors (BC and College level only)
Trigonometric Substitution

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Advanced Integration Techniques

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Tags

  • Integrals
  • Integration Techniques
  • Trig Functions

Priority: Normal

When integrating a function that contains one of three special root expressions, there is a trig function substitution that works when a U Sub fails. This lesson will teach you which expressions this method applies to, and then how to execute the method.

  • Learn the $x$-substitution method with trig functions for integrating special root expressions
  • Learn the three common root expressions this technique is good for, and which trig functions are the right fit in each case
  • When these expressions are present, know to first check to see if a U Sub will suffice (it's less work)
  • Practice this method using both indefinite and definite integration
Integration by Division

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Advanced Integration Techniques

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Tags

  • Integrals
  • Integration Techniques
  • Algebra Manipulation
  • Rational Expressions

Priority: Normal

Integrating rational expressions is among one of the greater puzzle challenges of integral calculus, because similar looking situations may require vastly different approaches. This lesson combines long division with knowledge from recent lessons and gives us a better overview of how to handle any feasible rational expression integral.

  • See the common ways in which monomial division masks the fact that integration could be done with the Power Rule
  • Classify rational expressions using properties of each the numerator and denominator to help choose an integration method
  • See and get familiar with common patterns so that you will be able to classify problems quickly for test day
Advanced Trig Integrals

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Advanced Integration Techniques

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Tags

  • Integrals
  • Integration Techniques
  • Trig Functions

Priority: Optional

Through the use of iterative Integration By Parts, it is possible to develop formulaic approaches to integrals that involve trig functions raised to large powers. We'll look at these types of integrals and how they might appear on exams, though they are uncommon.

  • Become familiar with the forms of trig function expressions that we will solve using a formula based approach
  • Learn the methods and formulas for integrating these types of problems
Improper Integrals

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Advanced Integration Techniques

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Tags

  • Integrals
  • Integration Techniques
  • Limits

Priority: Normal

Normal integrals are performed on smooth, continuous intervals. If an integral is performed where either one of the limits is a point where the function is undefined, or the interval between the limits contains a singularity, it is improper. This lesson shows you how to evaluate improper integrals, if possible.

  • Know the definition of an improper integral, and how to recognize one
  • For each of the two types of improper integrals, know how to proceed with trying to evaluate them
Choosing an Integration Approach

Calculus  $\rightarrow$  Integration Techniques  $\rightarrow$  Advanced Integration Techniques

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Tags

  • Integrals
  • Integration Techniques

Priority: High

This wrap-up integration lesson takes an important step back to think about how we should approach general problems. Often on tests, we know what method to use, because we are being tested on one or two methods at a time. If a totally random integral is throw at you, it's helpful to know what to try first and what to look for. This lesson develops your expertise in the "art" of knowing how to efficiently evaluate any integral they throw at you.

  • Have a general approach and "pecking order" of methods ready if you're going to integrate a random problem
  • See common integrals that students find confusing because the required method is not immediately clear

Applications of Integration - Graphical Applications of Integrals

Area Between Two Curves

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Graphical Applications of Integrals

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Tags

  • Integrals
  • Integration Applications

Priority: VIP Knowledge

By now we know that definite integration yields the area under a function's graph. This lesson outlines a simple way to use definite integration to find the area enclosed between two functions.

  • Learn how to find the area enclosed between two functions that do not intersect
  • Learn how to find the area enclosed between two functions that intersect
  • Find the area between curves for functions of $y$ (College Calc and BC only)
Volume of Revolution by Disks or Washers

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Graphical Applications of Integrals

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Tags

  • Integrals
  • Integration Applications
  • Volume

Priority: Normal

Rotating a finite area in the coordinate plane around the $x$ or $y$ axis in three dimensions creates a volume - and using the technique explained in this lesson, we can measure that volume. First we'll better visualize and understand the 3-d volume that we get from revolution of area, and then we'll learn how to calculate it.

  • Understand conceptually what is meant by creating a volume of revolution
  • Learn the disk method of calculating the volume of revolution created by a single function.
  • Learn the similar "washer" method for calculating the volume of revolution created by the area bound between two functions
  • Be able to apply this technique for functions of $x$ and for functions of $y$ depending on the axis of rotation
  • Know which situations this method works best for
Volume of Revolution by Cylindrical Shells

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Graphical Applications of Integrals

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Tags

  • Integrals
  • Integration Applications
  • Volume

Priority: Normal

The disk / washer method we just learned works well for most functions, but a second, entirely different method is cleaner and more efficient for certain situations. This lesson first goes over this second method (cylindrical shells), as well as how and when this method is the preferred approach.

  • Understand the "Russian Doll" concept of integrating to obtain volume
  • Learn and practice applying the method of cylindrical shells for functions of $x$
  • Learn and practice applying the method of cylindrical shells for functions of $y$
  • Understand which method is preferable (washers vs shells) based on the problem
Arc Length of a Function

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Graphical Applications of Integrals

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Tags

  • Integrals
  • Integration Applications
  • Function Analysis

Priority: Normal

Although it is a less common need, Calculus also allows us to find the arc length of a function between two points. This lesson will show you where the integral formula for function arc length comes from, and how to use it.

  • Derive and understand the integration formula to find the arc length of a function
  • Memorize and use the formula
  • Understand which types of functions do and do not lend themselves well to this operation
Surface of Revolution

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Graphical Applications of Integrals

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Tags

  • Integrals
  • Integration Applications
  • Function Analysis
  • Area

Priority: Normal

A direct application of the function arc length formula that we learned in the prior lesson is the ability to calculate the surface area of a solid of revolution. This is conceptually similar to volume of revolution concepts we recently learned, but instead yields the surface area.

  • Use the arc length formula to derive the formula for the surface area of a volume of revolution
  • Learn tricks for memorizing the formula, if you are required to memorize it
3D Volumes via Cross Sections

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Graphical Applications of Integrals

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Tags

  • Integrals
  • Integration Applications
  • Volume

Priority: Optional

Volumes of revolution are not the only way to obtain volume using single variable integration. Volumes that are defined by perpendicular cross sections over a region can be calculated when those cross sectional shapes are related to the shape of the region. While not all professors cover this potentially confusing and hard-to-picture topic, many do!

  • Understand what kinds of volumes are described this way
  • Draw the appropriate picture in the coordinate plane to aide your work, even thought the actual solid is nearly impossible to draw
  • Be able to set up and evaluate the integral that gives the required volume

Applications of Integration - Integration Applications to Functions

Unknown Function Initial Conditions

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Integration Applications to Functions

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Tags

  • Integrals
  • Derivatives
  • Function Analysis

Priority: Normal

When we integrate indefinitely, our answer contains an unknown constant (the infamous "plus c"). However, if we are also given more information about the answer, we can solve for the value of that coefficient in that case. This is called initial conditions, and this lesson shows how to work with this type of information.

  • Learn how to replace the random "+C" integration constant with a specified constant in cases where enough information is provided to do so
  • See the similarities and differences between applying initial conditions to $f(x)$ and $F(x)$ versus to $f'(x)$ and $f(x)$
Average Value of a Function

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Integration Applications to Functions

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Tags

  • Integrals
  • Integration Applications
  • Function Analysis

Priority: Normal

This lesson describes the methodology for finding the average value of a function over an interval, and helps us interpret what the average value represents.

  • Understand what the "average function value" means conceptually
  • See the formula for average function value and understand where it comes from
  • Practice applying the average function value formula
Integrals as Accumulation Over Time

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Integration Applications to Functions

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Tags

  • Derivative Applications
  • Integration Applications
  • Function Analysis
  • Word Problems

Priority: Normal

As we saw with the last lesson on equations of motion, differentiation and integration can be used to relate quantities of net change over time. This lesson will use the same ideas as the last lesson but in the context of unit analysis and non-motion applications.

  • Use derivatives and Integrals depending on the situation to translate between rate functions and net amounts
  • Apply basic curve fitting techniques to data and integrate the result to find accumulated amounts
  • Practice using this concept for problems in which you must create your own rate function from a description
Equations of Motion (Derivatives and Integrals)

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Integration Applications to Functions

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Tags

  • Derivative Applications
  • Integration Applications
  • Function Analysis
  • Word Problems

Priority: High

Using both differentiation and integration, we can examine the physics relationships between location, velocity, and acceleration. This lesson will explore the relationships among all three, both graphically and algebraically.

  • Learn the Calculus-based relationships between acceleration, velocity, and location
  • Use graphs to understand relationships between acceleration, velocity, and position
  • Differentiate to find velocity from position, or acceleration from velocity
  • Integrate to find velocity from acceleration, or position from velocity
  • Solve initial value or initial condition problems

Applications of Integration - Intro to Differential Equations

First Look at Differential Equations

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Intro to Differential Equations

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Tags

  • Derivatives
  • Derivative Applications
  • Differential Equations

Priority: Normal

Entire college courses are dedicated to studying differential equations, but we're often asked to understand them at a basic level in a Calculus course (including AP courses). This lesson will introduce us to the idea of what a differential equation is, and how we can prove or disprove proposed solutions to such equations.

  • Define and understand what a differential equation is
  • Know how to verify whether or not a proposed solution of a differential equation is indeed a solution
Separable Differential Equations

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Intro to Differential Equations

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Tags

  • Derivatives
  • Derivative Applications
  • Differential Equations

Priority: Normal

Solving differential equations is generally left to its own course, but there are two types of differential equations that we are often expected to be able to solve in a Calculus course. The first of these is called Separable Differential Equations, because we will be able to separate variables on each side of the equation and integrate. This lesson will outline the specifics of how this process works.

  • Understand what makes a differential equation "separable"
  • Be able to identify differential equations as separable or not separable
  • Be able to setup and solve separable differential equations
The Logistic Differential Equation

Calculus  $\rightarrow$  Applications of Integration  $\rightarrow$  Intro to Differential Equations

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Tags

  • Derivatives
  • Derivative Applications
  • Differential Equations

Priority: Normal

The second of the two types of differential equations that we are often expected to know how to solve in a Calculus course as opposed to a proper differential equations course is the Logistic Differential Equation, which can model realistic population growth with a given carrying capacity.

  • Recognize when you are presented with a differential equation that is in the Logistic form
  • Be able to write your own Logistic differential equation from a word problem or description
  • Know how to organically solve a Logistic differential equation using algebra and calculus manipulation techniques
  • Know how to solve a Logistic differential equation based on the prescribed solution formula

Sequences, Series, and their Applications - Infinite Sequences and Series

Infinite Sequences and Series Overview

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Infinite Sequences and Series

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Tags

  • Review
  • Sequences
  • Series
  • Limits

Priority: High

Commonly in a Pre-Calculus course, students will have studied sequences and series to an extent that leaves them familiar with arithmetic and geometric sequences. This lesson recaps the material that you would be expected to know going into a Calculus course that covers series analysis - whether or not you learned it in a prior course.

  • Understand that sequences are infinitely long lists, and understand how they are defined
  • Recall the common arithmetic and geometric sequence types
  • Learn how to work with generic sequences that have terms which are defined relative to their term number
  • Define series in terms of sequence language
  • Recall how to calculate partial sums of arithmetic and geometric series
  • Understand what infinite series are, even if it is difficult to understand their result
  • Derive and understand how and when infinite geometric series converge to a finite result
Common Infinite Series

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Infinite Sequences and Series

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Tags

  • Series
  • Limits

Priority: High

From Pre-Calculus knowledge, which we recalled in the last lesson, we have some light familiarity with infinite geometric sequences. This lesson introduces some other common types of infinite series, and the properties of each type, so that by the end of this lesson, we'll have a strong understanding of what infinite series tend to look like.

  • Quickly revisit the infinite geometric series, which, until now, is the only one we have been familiar with
  • Further understand what convergence and divergence means for an infinite series
  • Define the alternating and p-series common series types
  • Examine series that we call "telescopic" and learn techniques for computing the finite sum of these types of series
Basic Convergence Tests

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Infinite Sequences and Series

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Tags

  • Series
  • Limits

Priority: Normal

One of the major tasks we are charged with when we study series is determining whether a given infinite series converges or diverges (i.e. has a finite sum or not). This lesson will introduce a few common ways to determine whether or not a series converges.

  • Define and understand how The Divergence Test works, and what the results of it specifically mean
  • Learn the Comparison Test and Limit Comparison Test and when it is useful
  • Practice using these tests in situations that would otherwise be difficult to analyze with other (forthcoming) convergence tests
The Integral Test

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Infinite Sequences and Series

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Tags

  • Series
  • Limits

Priority: Normal

When the infinite series we are analyzing can be defined using continuous functions such as polynomial, exponential, or logarithm functions, we can apply the Integral Test. This is a quick and decisive way to identify convergence or divergence of such series, using an improper integral.

  • Learn how the Integral Test works and know how to interpret its results
  • Gain familiarity with which types of series this test can apply to
  • Know when this test is a better choice than some of the tests we have learned up to this point, and vice versa

Sequences, Series, and their Applications - Advanced Series and Convergence

The Ratio Test

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Advanced Series and Convergence

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Tags

  • Series
  • Limits

Priority: High

The Ratio Test is yet another means of analyzing whether an infinite series converges to a finite sum. This test is also useful for looking at Taylor Series, which is a near-future topic.

  • Understand what the Ratio Test says and how to use it
  • Know the types of series that this test excels at examining
  • Understand the limitations of this test and what to do if the test is inconclusive
The Root Test

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Advanced Series and Convergence

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Tags

  • Series
  • Limits

Priority: Normal

The Root Test is a convergence test that is not widely applicable, but very useful in certain situations. In some ways, this makes it easier to know when this is the best test to use, once we practice working some problems. This lesson discusses what it is and when it is particularly useful.

  • Understand what the Root Test says and how to use it
  • Know the types of series that this test excels at examining
  • Understand the limitations of this test and what to do if the test is inconclusive
Alternating Series Convergence

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Advanced Series and Convergence

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Tags

  • Series
  • Limits

Priority: Normal

Series with sequential terms that alternate between being positive and negative are referred to simply as alternating series. Because of this behavior, the criteria for determining whether or not a series converges is slightly different for these types of series, as it is "easier" for them to be convergent in a sense (in fact some converging alternating series would diverge if all the terms were instead positive). This lesson will help us understand the approach we should always take when working with alternating series.

  • Recall from an earlier lesson exactly what we mean when we refer to a series as "alternating"
  • Learn the two-part criteria for determining the convergence of an alternating series
  • Define convergence of an alternating series as either conditional or absolute, depending on how its behavior would change if all the terms were instead positive

Sequences, Series, and their Applications - Polynomial Approximations with Series

Power Series

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Polynomial Approximations with Series

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Tags

  • Polynomials
  • Series
  • Limits

Priority: Normal

Up until now, the series we have examined have been defined without variables. In other words, while we could describe the pattern of the terms by using $n$, the term number, there was no free variable that could take on any value. Power series are infinite series of the form $\Sigma a_n (x-a)^n$, and whether or not they converge may depend on our choice of $x$. This lesson introduces Power Series, and ways to examine their convergence.

  • Understand what Power Series are and the difference that it makes to have a variable in the series
  • Learn how to determine which values of $x$ cause a Power Series to converge or diverge
  • Learn the two ways that we are asked to describe the range of $x$ values that allow convergence, as a radius of convergence and as an interval of convergence
  • Know how to validate the behavior at the interval end points, and determine whether or not those endpoints allow absolute or conditional convergence
Taylor Polynomials

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Polynomial Approximations with Series

Web Lesson Coming Soon! »

Tags

  • Polynomials
  • Numerical Approximation
  • Limits
  • Series
  • Derivative Applications

Priority: High

Using derivatives and Power Series, we can approximate any continuous function using a polynomial. The more terms we include, the more like the original function the polynomial will look. This phenomenon, known as Taylor Polynomials, is the subject of this lesson.

  • See visually how any function can be approximated by a polynomial, centered at any $x$ value you choose
  • Understand how our polynomial approximation becomes more and more like the function it's trying to replicate as we include more and more terms
  • Define the "order" of a Taylor polynomial based on the degree of polynomial we want as a result
Maclaurin Series

Calculus  $\rightarrow$  Sequences, Series, and their Applications  $\rightarrow$  Polynomial Approximations with Series

Web Lesson Coming Soon! »

Tags

  • Polynomials
  • Numerical Approximation
  • Limits
  • Series
  • Derivative Applications

Priority: Normal

Taylor polynomial approximations can be centered around any $x$ value, but when they are centered at $x=0$, we call the resulting infinite Taylor series a Maclaurin Series. These series have special properties, which we will study in this lesson.

  • Define a Maclaurin Series as a special instance of a Taylor Series
  • Know (and possible memorize) the Maclaurin series for common functions
  • Derive Maclaurin series for derivatives, integrals, and transformations of common functions by applying those operations term-wise to the corresponding series