AP Calculus BC Multiple Choice Practice Test (2016 AP CED Problems)

This paragraph introduces an AP Calculus BC video tutorial focusing on eight multiple-choice practice problems. It clarifies that the material is specific to BC calculus and not AB, drawing from the 2016-17 curriculum framework, also known as the Course and Exam Description (CED). The speaker advises viewers on how to approach the first problem, which involves finding the slope of a tangent line to a particle's path in the X-Y plane using parametric equations.

The second paragraph delves into the application of Euler's method to approximate the solution to a given differential equation with an initial condition. The method involves setting up a table to calculate successive approximations of the function's value at increments of the independent variable. The paragraph provides a step-by-step guide to finding the approximate value of the function at a specific point, emphasizing the importance of including the change in the independent variable (Δx) in the calculations.

The third paragraph addresses the calculation of an improper integral from zero to infinity. The integral involves an exponential function, and the solution process uses integration by parts. The speaker chooses appropriate functions for u and dv, carries out the integration, and then evaluates the limit as the upper bound of the integral approaches infinity. The calculation concludes with finding the value of K that makes the integral equal to one.

The fourth paragraph focuses on the Taylor series and the error bound for an alternating series. It discusses the nth degree Taylor polynomial and how to find the smallest number M that ensures the error between the function's value at a point and its polynomial approximation is less than or equal to M. The speaker calculates the error for the given Taylor polynomial and determines the correct answer choice based on the alternating series error bound.

The fifth paragraph is about constructing a third-degree Taylor polynomial for a function given specific values and derivatives in a table. The speaker points out that the polynomial must be centered around x equals one and uses the given derivatives to build the polynomial term by term, following the pattern of nth derivative over n factorial times (x - 1) to the power of n. The final answer is determined by simplifying the coefficients and comparing them to the provided answer choices.

The sixth paragraph examines the convergence of a given series using the alternating series test. The series is a sum from 1 to infinity of a term involving a negative power of n and a square root. The speaker determines that the series does not converge absolutely by comparing it to a known divergent series and then confirms that it does converge conditionally by applying the alternating series test criteria.

The seventh paragraph involves calculating the total distance traveled by a particle moving in the X-Y plane, given its velocity vector as a function of time. The speaker uses the concept of arc length in parametric equations to find the distance, noting that the problem requires calculator use and providing a brief overview of the process.

The final paragraph discusses the use of integral tests to determine the convergence of an infinite series. The series is defined by a term that involves a continuous, decreasing, and positive function g(x). The speaker uses the integral test to show that if the integral of g(x) from 1 to infinity converges, then the series also converges. The problem is solved by drawing rectangles under the graph of g(x) and comparing the sum of these rectangles to the integral's value, leading to the selection of the correct answer choice.