Collections of Old Exams
Some Short Notes
I am currently revising some of these, and will post the revised versions at some future time. I’ve no promises about when, though…
- Cosines of Square Roots: An examination of some surprising behavior exhibited by the function that we get from taking cosines of (positive) square roots. (04/23/2018)
- Using Definite Integrals to Solve a Separable Initial Value Problem:. How can I use definite integrals to solve an initial value problem than involves a separable differential equation?
- 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.
Solutions to FRQs
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