AP Calculus Resources

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…

  1. 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)
  2. 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?
  3. 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?
  4. Defining the Natural Logarithm Function: Why do people use a certain integral as the definition of the natural logarithm function?
  5. 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?
  6. More on Improper Integrals:An example showing how the naive approach to improper integrals leads to unwanted trouble.
  7. 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.
  8. A Discontinuous Derivative: Why is the function x –> x^2 Sin[1/x] differentiable at the origin? Why is the derivative discontinuous there?
  9. Evaluating Limits: Why it is that we often really do set x = a when we evaluate Limit[f(x), x -> a]?
  10. On Implicit Differentiation: Implicit differentiation seems to cause a lot of confusion. Here is a discussion of some of the issues.
  11. Increasing Functions: How can a function be increasing on an interval even though its derivative vanishes somewhere in that interval?
  12. Functions “Increasing at a Point”: What does this mean? Revised on 08/07/07.
  13. 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?
  14. Error in Linearization: How much error is there in replacing a function with its linearization (or, if you prefer, its differential)?
  15. Taylor Polynomials: The Lagrange Error Bound
  16. Why Limit[(1 + h)^(1/h), h -> 0] Exists
  17. Separation of Variables: A discussion of what this formal technique really means and why it works.
  18. On An Antiderivative: Why is an absolute value needed in the antiderivative of 1/x?
  19. Substitution in Integrals: How it can get us in trouble.
  20. 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

  1. 2026 AB
  2. 2026 BC
  3. 2025 AB
  4. 2025 BC
  5. 2024 AB
  6. 2024 BC
  7. 2023 AB
  8. 2023 BC
  9. 2022 AB
  10. 2022 BC
  11. 2021 AB
  12. 2021 BC
  13. 2019 AB
  14. 2019 BC
  15. 2018 AB
  16. 2018 BC
  17. 2017 AB
  18. 2017 BC
  19. 2016 AB
  20. 2016 BC
  21. 2015 AB
  22. 2015 BC
  23. 2014 AB
  24. 2014 BC
  25. 2013 AB
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  27. 2012 AB
  28. 2012 BC
  29. 2011 AB
  30. 2011 AB Form B
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  33. 2010 AB
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  37. 2009 AB
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  41. 2008 AB
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  45. 2007 AB
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  49. 2006 AB
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  57. 2004 AB
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  61. 2003 AB
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  76. 1998 BC