Preparatory Exercises

The Eight Leaf Rose Parametric Form

To see an animation of this parametric graph go to

animation

Interval of Convergence

Conditional Convergence

We say that the series is absolutely convergent if the series composed of the absolute values of the terms converges.

In number 9 you may need to use GST instead of p-series to complete the problem.

The p-series Test

Telescoping Series

The Harmonic Series

The Geometric Series

A Sequence of Partial Sums

Limits of sequences

Sequences

Integrals on unbounded intervals

Improper integrals for the unbounded function

Consider the integrals for the following functions on the interval [a,b] and using the above analysis determine whether the improper integral is bounded or unbounded.

Multiple root case

Purpose

The purpose of the cite is to encourage calculus students to go beyond the book and develop good study habits. This semester, the blog will deal primarily with the concepts of Calculus II.

Consider the problem of partial fractions. Assume that we have an integral involving a rational function which is quadratic in the denominator. We assume that the quadratic factors into distinct roots as shown.

It should be easy to verify that this formula works for all cases of distinct roots. Note that the divisor of the integral is just the distance between the two roots. The best technique is to write the roots in order with "alpha" as the larger root. Then we know that the divisor should be positive.

It should be clear that the above formula takes care of all such cases. You should verify this by computing the derivative. N.B. this technique is not intended to replace methods where the student is required to show all their work rather it should allow the student to understand a simple method and to see the form of the solution for such a problem. The student should show whatever work their instructor requires for a complete solution.

Solve the following problems by observation using the formula technique.