Oxford University Mathematician takes American AP Calculus BC Math Exam

Dr. Tom Crawford introduces himself and the context of the American AP Calculus BC exam, highlighting its popularity among high school students and its role in college applications. He explains the exam structure, which includes multiple-choice and free-response sections, and mentions his plan to tackle Part B of the free-response section without a graphical calculator.

Dr. Crawford begins solving the first free-response question, which involves analyzing a graph of a derivative function. He discusses the peculiarity of starting on question three and explains his approach to understanding the problem by copying the graph and annotating it. He then proceeds to calculate F of not and F of five, using the given derivative graph and integrating F Prime to find the function values at specific points.

In the next part, Dr. Crawford focuses on finding the x-coordinates of all points of inflection for the graph of the function f. He explains the concept of points of inflection and how they relate to changes in the gradient of the function. He uses the graph of f Prime to identify points where the gradient changes from positive to negative or vice versa, concluding with the identification of two points of inflection at x = 2 and x = 6.

Dr. Crawford moves on to Part C, where he defines a new function G as the difference between f and x. He calculates when G is decreasing by finding the intervals where G Prime is negative, using the graph of f Prime. He concludes that G is decreasing for x between 0 and 5, and again for x between 5 and 7, providing a clear analysis of the function's behavior.

In the final part of his exam walkthrough, Dr. Crawford addresses Part D, which involves finding the absolute minimum value of the function G on a given interval. He uses the information from Part C to determine that the minimum value occurs at x = 5. He then calculates the value of G at this point, considering the function's behavior and its derivative, and concludes with the absolute minimum value of G.

Dr. Crawford tackles a new question involving the average rate of change of a function's derivative over an interval and the application of the Intermediate Value Theorem. He approximates R Prime of 8.5 using the average rate of change between 7 and 10, showing the computations and units involved. He then confirms the existence of a time t for which R Prime of t equals -6, using the theorem and the continuity of the derivative function.

Dr. Crawford continues with a question about a conical ice sculpture, focusing on the radius and height's rate of change over time. He uses the given data to approximate the integral of R Prime of T using Riemann sums and calculates the height's rate of change at a specific time. He then finds the rate of change of the cone's volume with respect to time at a given moment, applying calculus concepts to the melting pattern of the ice sculpture.

In the final question, Dr. Crawford deals with power series, convergence, and volumes of solids. He uses the ratio test to find the interval of convergence for a given power series and calculates the volume of the solid formed by revolving an unbounded region around an axis. He also finds the volume of a solid by integrating a density function and term-by-term differentiation of a power series, applying calculus and series concepts to solve the problems.

Dr. Crawford concludes his video by reflecting on his experience taking the AP Calculus BC Exam. He marks his own answers, providing a self-assessment of his performance and discussing the difficulty level of the exam in comparison to other standardized tests. He shares his thoughts on the exam's content, the process of solving the problems, and his overall impression of the experience.