Lou Talman's AP Calculus Page

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Short Notes on Various Things

Some topics arise again and again in discussions of elementary calculus. Here are my thoughts about some of them, as well as some other things:

Continuity and Differentiability of Inverse Functions When is the inverse function of a continuous, differentiable function also continuous and differentiable? How does one prove what seems intuitively clear?
Defining the Natural Logarithm Function: Why do people use a certain integral as the definition of the natural logarithm function?
Improper Integrals: The definition of improper integrals sure causes my students a lot of trouble. Why isn't that definition what they think it ought to be?
More on Improper Integrals:An example showing how the naive approach to improper integrals leads to unwanted trouble.
Asymptotes: What is an asymptote? It may surprise you to learn that there is no definitive answer to this question. Here's my suggestion, and the reasons for it.
A Discontinuous Derivative: Why is the function x --> x^2 Sin[1/x] differentiable at the origin? Why is the derivative discontinuous there?
Evaluating Limits: Why it is that we often really do set x = a when we evaluate Limit[f(x), x -> a]?
On Implicit Differentiation: Implicit differentiation seems to cause a lot of confusion. Here is a discussion of some of the issues.
Increasing Functions: How can a function be increasing on an interval even though its derivative vanishes somewhere in that interval?
Functions "Increasing at a Point": What does this mean? Revised on 08/07/07.
The Definite Integral as an Accumulator: How can I use the idea of the definite integral as an accumulator to set up definite integrals in standard problems?
Error in Linearization: How much error is there in replacing a function with its linearization (or, if you prefer, its differential)?
Taylor Polynomials: The Lagrange Error Bound
Why Limit[(1 + h)^(1/h), h -> 0] Exists
Separation of Variables: A discussion of what this formal technique really means and why it works.
On An Antiderivative: Why is an absolute value needed in the antiderivative of 1/x?
Substitution in Integrals: How it can get us in trouble.
Differentiability for Multivariable Functions: What does the term "differentiable" mean for a function of two or more variables? Strictly speaking, this isn't an AP Calculus topic, but teachers of AP Calculus may nevertheless find it to be of interest.
"Simpson's Rule is Exact For Quintics," American Mathematical Monthly, 113(2006), 144-155. Abstract: In this article, we use tools accessible to freshman calculus students to develop exact—though usually uncomputable—expressions for the error that results in replacing a definite integral with its midpoint rule, trapezoidal rule, or Simpson's rule approximation. Among the tools we use is an extended version of the first mean value theorem for integrals. We obtain not only the classical estimates that appear in calculus books, but estimates for functions less smooth than the classical results require. We show, in particular, how to compute the exact error for a Simpson's rule approximation to an integral of a quintic polynomial.

Solutions to Free Response Questions

The College Board's copyright policies prohibit me from posting the questions themselves here. You can find them at AP Central.

Solutions to the 2010 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2009 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2008 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2007 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2006 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2005 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2004 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2003 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2002 AP Calculus Free Response questions, in HTML format:
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Solutions to the 2001 AP Calculus Free Response questions:
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Solutions to the 2000 AP Calculus Free Response questions:
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Solutions to the 1999 AP Calculus Free Response questions:
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Solutions to the 1998 AP Calculus Free Response questions:
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Back to my home page.

This page was last modified on February 10, 2009.

Lou Talman; talmanl@mscd.edu