Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.
Introduction to the Epsilon Delta Definition of a Limit.
Using the epsilon delta definition to prove a limit.
Calculus-Derivative: Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line).
Calculus-Derivative: Finding the slope (or derivative) of a curve at a particular point.
Finding the slope of a tangent line to a curve (the derivative). Introduction to Calculus.
More intuition of what a derivative is. Using the derivative to find the slope at any point along f(x)=x^2.
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.
Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".
A difficult but interesting derivative word problem.
Taking the derivative when y is defined implicitly.
More intuition behind the chain rule and how it applies to implicit differentiation.
Implicit differentiation example that involves the tangent function.
Intuition on what happens to the slope/derivative and second derivatives at local maxima and minima.
Understanding concave upwards and downwards portions of graphs and the relation to the derivative. Inflection point intuition.
Using the monotonicity theorem to determine when a function is increasing or decreasing.
2 examples of finding the maximum and minimum points on an interval.
Using the first and second derivatives to identify critical points and inflection points and to graph the function.
Find two numbers whose products is -16 and the sum of whose squares is a minimum.
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.
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.).
Minimizing the cost of material for an open rectangular box.
Finding the equation of the line tangent to f(x)=xe^x when x=1.
Another (simpler) example of using the chain rule to determine rates-of-change.
An introduction to indefinite integration of polynomials.
Examples of taking the indefinite integral (or anti-derivative) of polynomials.
Integration by substitution (or the reverse-chain-rule).
Introduction to Integration by Parts (kind of the reverse-product rule).
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.
More on why the antiderivative and the area under a curve are essentially the same thing.
More on why the antiderivative and the area under a curve are essentially the same thing.
Examples of using definite integrals to find the area under a curve.
More examples of using definite integrals to calculate the area between curves.
Solving a definite integral with substitution (or the reverse chain rule).
Example of using trig substitution to solve an indefinite integral.
Another example of finding an anti-derivative using trigonometric substitution.
Example using trig substitution (and trig identities) to solve an integral.
3 basic differential equations that can be solved by taking the antiderivatives of both sides.
Figuring out the volume of a function rotated about the x-axis.
The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis.
Taking the revolution around something other than one of the axes.
Using a polynomial to approximate a function at f(0).
Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series).
A glimpse of the mystery of the Universe as we approximate e^x with an infinite series.
Approximating cos x with a Maclaurin series.
The most amazing conclusion in mathematics!
Parts c&d of problem 1 in the 2008 AP Calculus BC free response.
Part 1d of the 2008 AP Calculus BC exam (free response).
Intuition of the gradient of a scalar field (temperature in a room) in 3 dimensions.
Let's evaluate the double integrals with y=x^2 as one of the boundaries.
Using a triple integral to find the mass of a volume of variable density.
Part 2 of an example of taking a line integral over a closed path.
Using a position vector valued function to describe a curve or path.
Visualizing the derivative of a position vector valued function.
Understanding the differential of a vector valued function.
Concrete example of the derivative of a vector valued function to better understand what it means.
Using line integrals to find the work done on a particle moving through a vector field.
Using a line integral to find the work done by a vector field example.
Understanding how to parametrize a reverse path for the same curve.
Showing that the line integral of a scalar field is independent of path direction.
Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.
Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.
Showing that the line integral along closed curves of conservative vector fields is zero.
Example of taking a closed line integral of a conservative field.
Using path independence of a conservative vector field to solve a line integral.
Using Green's Theorem to solve a line integral of a vector field.
Introduction to Parametrizing a Surface with Two Parameters.
Determining a Position Vector-Valued Function for a Parametrization of Two Parameters.
Partial Derivatives of Vector-Valued Functions.
Example of calculating a surface integral part 1.
Example of calculating a surface integral part 2.
Example of calculating a surface integral part 3.