Calculus II Tutorials
The Alternating Series Test
Direct and Limit Comparison Tests
The Alternating Series Test
An introduction to the Alternating Series test for series convergence, including a proof and some examples
The Integral Test
Direct and Limit Comparison Tests
The Alternating Series Test
A brief review of the integral test for series convergence, and its applicability to p-series
Direct and Limit Comparison Tests
Direct and Limit Comparison Tests
Absolute Convergence of Infinite Series
Statements and proofs of the direct and limit comparison tests for series convergence. Examples are also provided.
Absolute Convergence of Infinite Series
Ratio and Root Tests for Convergence of Infinite Series
Absolute Convergence of Infinite Series
A discussion of absolutely convergent series. Attention is given to why the terms of conditionally convergent series (those which are not absolutely convergent) can be rearranged to build new series that can converge to any desired number.
Ratio and Root Tests for Convergence of Infinite Series
Ratio and Root Tests for Convergence of Infinite Series
Ratio and Root Tests for Convergence of Infinite Series
The ratio and root tests for series convergence, and proofs of why they work. Examples are included, as well.
Introduction to Power Series
Ratio and Root Tests for Convergence of Infinite Series
Ratio and Root Tests for Convergence of Infinite Series
This video demonstrates how to build power series expansions of a number of functions by modifying the geometric power series in a number of ways. As a result, two fascinating infinite series are developed.
Intervals of Convergence for Power Series
Introduction to Parametric Equations: the Cycloid
Intervals of Convergence for Power Series
This video explains how to use the Ratio Test to determine the radius of convergence of a power series, and how to use this information to find the interval of convergence.
Taylor Series
Introduction to Parametric Equations: the Cycloid
Intervals of Convergence for Power Series
Power series expansions are developed for sin(x), cos(x), e(x), and ln(x), leading to the development of the definition of a Taylor Series for f(x), in general.
Introduction to Parametric Equations: the Cycloid
Introduction to Parametric Equations: the Cycloid
Introduction to Parametric Equations: the Cycloid
The parametric equations for the cycloid are developed, and the formula for the arc length of a circular sector is developed in the process.
Graphing Parametric Equations
Derivatives in the Context of Parametric Equations
Introduction to Parametric Equations: the Cycloid
Several examples of parametric equations are presented, and techniques are suggested that allow one to sketch the graphs of these equations.
Derivatives in the Context of Parametric Equations
Derivatives in the Context of Parametric Equations
Derivatives in the Context of Parametric Equations
This video shows how to find derivatives in the setting of parametric equations.
Calculus in the Context of Parametric Equations
Derivatives in the Context of Parametric Equations
Derivatives in the Context of Parametric Equations
Questions involving Calculus are answered in the setting of parametric equations. Examples involving the Cycloid are emphasized.
An Introduction to Polar Coordinates
Area and Arc Length in the Context of Polar Coordinates
Derivatives in the Context of Polar Coordinates
Rene Descartes invented the rectangular geometric coordinate system when he saw a rectangular grid pattern over his bed. This video begins with the geometric description of a parabola that would have been used by Euclid, and examines how we can use the genius of Descartes to provide the better known algebraic description of the parabola as the graph of a quadratic function in rectangular coordinates. We then introduce the polar coordinate system by imagining what Descartes might have invented if the his ceiling of his bedroom was decorated with a circular design. Hmmm...
Derivatives in the Context of Polar Coordinates
Area and Arc Length in the Context of Polar Coordinates
Derivatives in the Context of Polar Coordinates
How do you compute a derivative whenever your function is expressed in polar coordinates? Let @theMathSinger show you!
Area and Arc Length in the Context of Polar Coordinates
Area and Arc Length in the Context of Polar Coordinates
Finding the integral of sec(x) without using the traditional "trick"
This final video in the COVIDeo Calculus II series explores how to calculate area and arc length of functions that are expressed in polar coordinates.
Finding the integral of sec(x) without using the traditional "trick"
Finding the integral of sec(x) without using the traditional "trick"
Finding the integral of sec(x) without using the traditional "trick"
The integral of sec(x) is traditionally derived by using a "trick" that presumes a priori that the solution is already known. This video shows how find the integral of sec(x) by using a very unexpected u-substitution that leads to an integrand that can be evaluated using the method of partial fraction decomoposition.
1 Integral, 3 Techniques, and 4 Solutions
Finding the integral of sec(x) without using the traditional "trick"
1 Integral, 3 Techniques, and 4 Solutions
Students often believe that there is a particular technique that must be used to evaluate a given integral. Nothing could be further from the truth! In this video, a single integral is evaluated in 4 different ways by using 3 basic integration techniques. Watch them all, and be prepared to wow your teacher with your new integration skills!