Course Description:
Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists).
Lectures:
Lecture 1 - Limit Examples Part 4
More limit examples.
Lecture 2 - Epsilon Delta Limit Definition Part 1
Introduction to the Epsilon Delta Definition of a Limit.
Lecture 3 - Epsilon Delta Limit Definition Part 2
Using the epsilon delta definition to prove a limit.
Lecture 4 - Calculus: Derivatives 1
Calculus-Derivative: Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line).
Lecture 5 - Calculus: Derivatives 2
Calculus-Derivative: Finding the slope (or derivative) of a curve at a particular point.
Lecture 6 - Calculus: Derivatives 2.5
Calculus-Derivative: Finding the derivative of y=x^2
Lecture 7 - Derivatives Part 1
Finding the slope of a tangent line to a curve (the derivative). Introduction to Calculus.
Lecture 8 - Derivatives Part 2
More intuition of what a derivative is. Using the derivative to find the slope at any point along f(x)=x^2.
Lecture 9 - Derivatives Part 3
Determining the derivatives of simple polynomials.
Lecture 10 - Derivatives Part 4
Part 4 of derivatives. Introduction to the chain rule.
Lecture 11 - Derivatives Part 5
Examples using the Chain Rule.
Lecture 12 - Derivatives Part 6
Even more examples using the chain rule.
Lecture 13 - Derivatives Part 7
The product rule. Examples using the Product and Chain rules.
Lecture 14 - Derivatives Part 8
Why the quotient rule is the same thing as the product rule. Introduction to the derivative of e^x, ln x, sin x, cos x, and tan x.
Lecture 15 - Derivatives Part 9
More examples of taking derivatives.
Lecture 16 - Proof: d/dx(x^n)
Proof that d/dx(x^n) = n*x^(n-1).
Lecture 17 - Proof: d/dx(sqrt(x))
Proof that d/dx (x^.5) = .5x^(-.5).
Lecture 18 - Proof: d/dx(ln x) = 1/x
Taking the derivative of ln x.
Lecture 19 - Proof: d/dx(e^x) = e^x
Proof that the derivative of e^x is e^x.
Lecture 20 - Proofs of Derivatives of Ln(x) and e^x
Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".
Lecture 21 - Extreme Derivative Word Problem
A difficult but interesting derivative word problem.
Lecture 22 - Implicit Differentiation Part 1
Taking the derivative when y is defined implicitly.
Lecture 23 - Implicit Differentiation Part 2
A hairier implicit differentiation problem.
Lecture 24 - More Implicit Differentiation
2 more implicit differentiation examples.
Lecture 25 - More Chain Rule and Implicit Differentiation Intuition
More intuition behind the chain rule and how it applies to implicit differentiation.
Lecture 26 - Trig Implicit Differentiation Example
Implicit differentiation example that involves the tangent function.
Lecture 27 - Derivative of x^(x^x)
Calculus: Derivative of x^(x^x).
Lecture 28 - Maxima Minima Slope Intuition
Intuition on what happens to the slope/derivative and second derivatives at local maxima and minima.
Lecture 29 - Inflection Points and Concavity Intuition
Understanding concave upwards and downwards portions of graphs and the relation to the derivative. Inflection point intuition.
Lecture 30 - Monotonicity Theorem
Using the monotonicity theorem to determine when a function is increasing or decreasing.
Lecture 31 - Maximum and Minimum Values on an Interval
2 examples of finding the maximum and minimum points on an interval.
Lecture 32 - Graphing Using Derivatives
Graphing functions using derivatives.
Lecture 33 - Graphing with Derivatives Example
Using the first and second derivatives to identify critical points and inflection points and to graph the function.
Lecture 34 - Graphing with Calculus
More graphing with calculus.
Lecture 35 - Optimization with Calculus Part 1
Find two numbers whose products is -16 and the sum of whose squares is a minimum.
Lecture 36 - Optimization with Calculus Part 2
Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides.
Lecture 37 - Optimization with Calculus Part 3
A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.).
Lecture 38 - Optimization with Calculus Part 4
Minimizing the cost of material for an open rectangular box.
Lecture 39 - Introduction to Rate-of-change
Using derivatives to solve rate-of-change problems.
Lecture 40 - Equation of a Tangent Line
Finding the equation of the line tangent to f(x)=xe^x when x=1.
Lecture 41 - Rates-of-change Part 2
Another (simpler) example of using the chain rule to determine rates-of-change.
Lecture 42 - Ladder rate-of-change Problem
The classic falling ladder problem.
Lecture 43 - Mean Value Theorem
Intuition behind the Mean Value Theorem.
Lecture 44 - Indefinite Integrals Part 1
An introduction to indefinite integration of polynomials.
Lecture 45 - Indefinite Integrals Part 2
Examples of taking the indefinite integral (or anti-derivative) of polynomials.
Lecture 46 - Indefinite Integrals Part 3
Integration by doing the chain rule in reverse.
Lecture 47 - Indefinite Integrals Part 4
Integration by substitution (or the reverse-chain-rule).
Lecture 48 - Indefinite Integrals Part 5
Introduction to Integration by Parts (kind of the reverse-product rule).
Lecture 49 - Indefinite Integrals Part 6
Example using Integration by Parts.
Lecture 50 - Indefinite Integrals Part 7
Another example using integration by parts.
Lecture 51 - Another U-substitution Example
Finding the antiderivative using u-substitution.
Lecture 52 - Definite Integrals Part 1
Using the definite integral to solve for the area under a curve. Intuition on why the antiderivative is the same thing as the area under a curve.
Lecture 53 - Definite Integrals Part 2
More on why the antiderivative and the area under a curve are essentially the same thing.
Lecture 54 - Definite Integrals Part 3
More on why the antiderivative and the area under a curve are essentially the same thing.
Lecture 55 - Definite Integrals Part 4
Examples of using definite integrals to find the area under a curve.
Lecture 56 - Definite Integrals Part 5
More examples of using definite integrals to calculate the area between curves.
Lecture 57 - Definite Integral with Substitution
Solving a definite integral with substitution (or the reverse chain rule).
Lecture 58 - Integrals: Trig Substitution Part 1
Example of using trig substitution to solve an indefinite integral.
Lecture 59 - Integrals: Trig Substitution Part 2
Another example of finding an anti-derivative using trigonometric substitution.
Lecture 60 - Integrals: Trig Substitution Part 3
Example using trig substitution (and trig identities) to solve an integral.
Lecture 61 - Introduction to Differential Equations
3 basic differential equations that can be solved by taking the antiderivatives of both sides.
Lecture 62 - Solid of Revolution Part 1
Figuring out the volume of a function rotated about the x-axis.
Lecture 63 - Solid of Revolution Part 2
The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis.
Lecture 64 - Solid of Revolution Part 3
Figuring out the equation for the volume of a sphere.
Lecture 65 - Solid of Revolution Part 4
More volumes around the x-axis.
Lecture 66 - Solid of Revolution Part 5
Use the "shell method" to rotate about the y-axis.
Lecture 67 - Solid of Revolution Part 6
Using the disk method around the y-axis.
Lecture 68 - Solid of Revolution Part 7
Taking the revolution around something other than one of the axes.
Lecture 69 - Solid of Revolution Part 8
The last part of the problem in part 7.
Lecture 70 - Polynomial Approximation of Functions Part 1
Using a polynomial to approximate a function at f(0).
Lecture 71 - Polynomial Approximation of Functions Part 2
Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series).
Lecture 72 - Polynomial Approximation of Functions Part 3
A glimpse of the mystery of the Universe as we approximate e^x with an infinite series.
Lecture 73 - Polynomial Approximation of Functions Part 4
Approximating cos x with a Maclaurin series.
Lecture 74 - Polynomial Approximation of Functions Part 5
MacLaurin representation of sin x.
Lecture 75 - Polynomial Approximation of Functions Part 6
A pattern emerges!
Lecture 76 - Polynomial Approximation of Functions Part 7
The most amazing conclusion in mathematics!
Lecture 77 - Taylor Polynomials
Approximating a function with a Taylor Polynomial.
Lecture 78 - AP Calculus BC Exams: 2008 1 A
Part 1a of the 2008 BC free response.
Lecture 79 - AP Calculus BC Exams: 2008 1 B & C
Parts b and c of problem 1 (free response).
Lecture 80 - AP Calculus BC Exams: 2008 1 C & D
Parts c&d of problem 1 in the 2008 AP Calculus BC free response.
Lecture 81 - AP Calculus BC Exams: 2008 1 D
Part 1d of the 2008 AP Calculus BC exam (free response).
Lecture 82 - Calculus BC 2008 2 A
2a of 2008 Calculus BC exam (free response).
Lecture 83 - Calculus BC 2008 2 B & C
Parts 2b and 2c of the 2008 BC exam (free response).
Lecture 84 - Calculus BC 2008 2 D
Part 2d of the 2008 Calculus BC exam free-response section.
Lecture 85 - Partial Derivatives Part 1
Introduction to partial derivatives.
Lecture 86 - Partial Derivatives Part 2
More on partial derivatives.
Lecture 87 - Gradient
Introduction to the gradient.
Lecture 88 - Gradient of a Scalar Field
Intuition of the gradient of a scalar field (temperature in a room) in 3 dimensions.
Lecture 89 - Divergence Part 1
Introduction to the divergence of a vector field.
Lecture 90 - Divergence Part 2
The intuition of what the divergence of a vector field is.
Lecture 91 - Divergence Part 3
Analyzing a vector field using its divergence.
Lecture 92 - Curl Part 1
Introduction to the curl of a vector field.
Lecture 93 - Curl Part 2
The mechanics of calculating curl.
Lecture 94 - Curl Part 3
More on curl.
Lecture 95 - Double Integrals Part 1
Introduction to the double integral.
Lecture 96 - Double Integrals Part 2
Figuring out the volume under z=xy^2.
Lecture 97 - Double Integrals Part 3
Let's integrate dy first!
Lecture 98 - Double Integrals Part 4
Another way to conceptualize the double integral.
Lecture 99 - Double Integrals Part 5
Finding the volume when we have variable boundaries.
Lecture 100 - Double Integrals Part 6
Let's evaluate the double integrals with y=x^2 as one of the boundaries.
Lecture 101 - Triple Integrals Part 1
Introduction to the triple integral.
Lecture 102 - Triple Integrals Part 2
Using a triple integral to find the mass of a volume of variable density.
Lecture 103 - Triple Integrals Part 3
Figuring out the boundaries of integration.
Lecture 104 - (2^ln x)/x Antiderivative Example
Finding ?(2^ln x)/x dx.
Lecture 105 - Line Integrals
Introduction to the Line Integral.
Lecture 106 - Line Integral Example 1
Concrete example using a line integral.
Lecture 107 - Line Integral Example 2 Part 1
Line integral over a closed path (part 1).
Lecture 108 - Line Integral Example 2 Part 2
Part 2 of an example of taking a line integral over a closed path.
Lecture 109 - Position Vector Valued Functions
Using a position vector valued function to describe a curve or path.
Lecture 110 - Derivative of a Position Vector Valued Function
Visualizing the derivative of a position vector valued function.
Lecture 111 - Differential of a Vector Valued Function
Understanding the differential of a vector valued function.
Lecture 112 - Differential of a Vector Valued Function Example
Concrete example of the derivative of a vector valued function to better understand what it means.
Lecture 113 - Line Integrals and Vector Fields
Using line integrals to find the work done on a particle moving through a vector field.
Lecture 114 - Using a Line Integral to Find a Vector Field Example
Using a line integral to find the work done by a vector field example.
Lecture 115 - Parametrization of a Reverse Path
Understanding how to parametrize a reverse path for the same curve.
Lecture 116 - Scalar Field Line Integral Independent of Path Direction
Showing that the line integral of a scalar field is independent of path direction.
Lecture 117 - Vector Field Line Integral Dependent of Path Direction
Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.
Lecture 118 - Path Independence for Line Integrals
Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.
Lecture 119 - Closed Curve Line Integrals of Conservative Vector Fields
Showing that the line integral along closed curves of conservative vector fields is zero.
Lecture 120 - Example of Closed Line Integral of Conservative Field
Example of taking a closed line integral of a conservative field.
Lecture 121 - Second Example of Line Integral of Conservative Vector Field
Using path independence of a conservative vector field to solve a line integral.
Lecture 122 - Green's Theorem Proof Part 1
Part 1 of the proof of Green's Theorem.
Lecture 123 - Green's Theorem Proof Part 2
Part 2 of the proof of Green's Theorem.
Lecture 124 - Green's Theorem Example Part 1
Using Green's Theorem to solve a line integral of a vector field.
Lecture 125 - Green's Theorem Example Part 2
Another example applying Green's Theorem.
Lecture 126 - Introduction to Parametrizing a Surface with Two Parameters
Introduction to Parametrizing a Surface with Two Parameters.
Lecture 127 - Position Vector-Valued Function for a Parametrization of Two Parameters
Determining a Position Vector-Valued Function for a Parametrization of Two Parameters.
Lecture 128 - Partial Derivatives of Vector-Valued Functions
Partial Derivatives of Vector-Valued Functions.
Lecture 129 - Introduction to the Surface Integral
Introduction to the surface integral.
Lecture 130 - Calculating a Surface Integral Example Part 1
Example of calculating a surface integral part 1.
Lecture 131 - Calculating a Surface Integral Example Part 1
Example of calculating a surface integral part 2.
Lecture 132 - Calculating a Surface Integral Example Part 1
Example of calculating a surface integral part 3.
Lecture 133 - L'Hopital's Rule
Introduction to L'Hopital's Rule.
Lecture 134 - L'Hopital's Rule Example 1
L'Hopital's Rule Example 1.
Lecture 135 - L'Hopital's Rule Example 2
L'Hopital's Rule Example 2.
Lecture 136 - L'Hopital's Rule Example 3
L'Hopital's Rule Example 1.
Source: http://academicearth.org/courses/calculus
