# When are we ever …

The following answer to a question we’ve all been asked was posted yesterday on a private Facebook page for AP Calculus readers. The author, Allen Wolmer is a teacher and AP Calculus reader. He teaches at Yeshiva Atlanta High School, in Atlanta, Georgia. I reprint it here with his kind permission. Thank you Allen.     A…

# Determining the Indeterminate 2

The other day someone asked me a question about the implicit relation . They had been asked to find where the tangent line to this relation is vertical. They began by finding the derivative using implicit differentiation: The derivative will be undefined when its denominator is zero. Substituting y = 0 this into the original…

# The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus or FTC, as its name suggests, is a very important idea. It is not sufficient to present the formula and show students how to use it. Show them where it comes from. Here is an approach to demonstrate the FTC. I try to sneak up on the result by proposing…

# The Definition of the Definite Integral

From the last post, it seems pretty obvious that as the number of rectangles in a Riemann sum increases or, what amounts to the same thing, the width of the sub-intervals decreases, the Riemann sum approaches the area of the region between a graph and the x-axis. The figures below show left-Riemann sums for the…

# Riemann Sums

In our last post we discussed what are called Riemann sums. A sum of the form  or the form (with the meanings from the previous post) is called a Riemann sum. The three most common are these and depend on where the is chosen. Left-Riemann sum, L, uses the left side of each sub-interval, so…

# Flying to Integrationland

Here is a problem similar to the one in the last post, but with foibles of its own. The speed of an airplane in miles per hour is given at half-hour intervals in the table below. Approximately how far does the airplane travel in the three hours given in the table? How far is it…