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), 144155. 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 2011 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2010 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2009 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2008 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2007 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2006 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2005 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2004 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2003 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2002 AP Calculus Free Response questions, in HTML format:
 AB
 AB (Form B)
 BC
 BC (Form B)
Also available in PDF:
 AB in PDF
 AB (Form B) in PDF
 BC in PDF
 BC (Form B) in PDF
 Solutions to the 2001 AP Calculus Free Response questions:
 AB
 BC
Also available in PDF:
 AB in PDF
 BC in PDF
 Solutions to the 2000 AP Calculus Free Response questions:
 AB
 BC
Also available in PDF:
 AB in PDF
 BC in PDF
 Solutions to the 1999 AP Calculus Free Response questions:
 AB
 BC
Also available in PDF:
 AB in PD
 BC in PDF
 Solutions to the 1998 AP Calculus Free Response questions:
 AB
 BC
Also available in PDF:
 AB in PDF
 BC in PDF
Back
to my home page.
This page was last modified on February 10, 2009.
Lou Talman; talmanl@mscd.edu