AP Calculus BC Practice Exam 2012 - Multiple Choice questions 1-28

Van, the presenter, guides viewers through the AP Calc BC 2012 practice test. He begins with question one, which involves differentiating y = (sin x)^3 using the chain rule, and matches the result to choice E. For question two, he determines the particle's rest position by finding when the velocity in both the x and y components is zero, which occurs at t=2, leading to the position (-4, 12), corresponding to choice A. Question three deals with a piecewise function's definite integral from 0 to 4, which evaluates to zero, aligning with choice B. The arc length of y = ln(x) from x=1 to x=2 is found in question four, and Van uses the arc length formula to find the correct answer, choice A. In question five, Van evaluates the Maclaurin series for f(x) at x=3, which simplifies to 4/7, matching choice C.

The video continues with a discussion on u-substitution in integrals, where Van shows how to transform the integral of (x^2 - 3) with respect to x by setting u = x^2 - 3 and finding du. He also addresses the limits of integration in terms of u and solves the integral, which corresponds to choice C. For the next question, Van tackles an implicit differentiation problem involving arcsine x = ln y, solving for dy/dx and matching the result to choice A. He then discusses a tank problem where oil is being pumped in at a rate given by a table of values, using the data to find the amount of oil in the tank after 15 hours, which is choice C. Van also evaluates the convergence of various series by recognizing factorials, exponentials, and power functions, and determines that only one series converges, which is identified as choice T.

Van applies the ratio test to a series with the term (8^(n+1))/(n+1)!, showing that the series converges as the limit approaches zero. He then uses the limit comparison test to evaluate the convergence of another series, comparing it to 1/n^2, and concludes that the series converges. The video moves on to question 10, where Van uses the fundamental theorem of calculus to find the antiderivative of a function and evaluates it from 1 to 4, selecting choice E. For question 11, Van analyzes the function f(x) = sqrt(|x-2|) and determines that it is continuous but not differentiable at x=2, eliminating certain choices and concluding that the correct answer is choice A.

The video examines a scenario where a tank has an initial 50 liters of oil, and oil is being added at a variable rate. Van uses a Riemann sum with the given data points to find the total amount of oil in the tank after a specific time period, which is identified as choice C. He also discusses the convergence of series by comparing the terms and using the ratio test, concluding that one series converges, which is choice T. Van then explores a differential equation related to a tank's oil level and uses the second derivative test to find that a specific point on the graph represents a local maximum, leading to choice C.

Van rewrites a given power series to identify its convergence properties, determining that the series converges when |x - 4|^2 < 3, which corresponds to choice C. He then quickly identifies a logistic differential equation from the given options as choice E. For a function representing the area under a curve, Van discusses the function's properties and its derivatives, evaluating them at specific points and finding that the order from least to greatest matches choice A.

Using Euler's method, Van approximates the value of a function at x=2, starting with the function's value at x=1, and performs two steps of equal length to find the approximation, which is choice C. He then deals with a power series and recognizes it as a modified version of the sine function's power series, concluding that the function f(x) is sin(x)/x, which is choice C. Van also evaluates the antiderivative of a function from negative five to two and uses the geometric properties of the graph to find the area under the curve, leading to choice A.

Van sets up an equation to find the points on the graph of a function where the slope of the tangent line is one-half, using the quotient rule and algebraic manipulation to find the points x=0 and x=-4, which correspond to choice C. He then discusses how to find the horizontal asymptote of a function by examining the limit as x approaches infinity, identifying the correct function as choice E. For a power series that converges at x=5, Van uses the alternating series test to find the values of x that make the series converge, which is choice D.

Van identifies the differential equation that represents linear growth in a population as choice A, based on the constant rate of change. He then uses integration by parts to find the function f(x) that, when integrated with sine(x)dx, results in a specific expression, concluding that f(x) = x^4 is the correct answer, which is choice E. Van evaluates an improper integral by performing a u-substitution and taking the limit as b approaches infinity, finding the correct answer to be choice B.

Van calculates the slope of the line tangent to a polar curve at a specific theta value by using the relationships x = r*cos(θ) and y = r*sin(θ), and finds that the slope is 1/2, which corresponds to choice B. He then finds the values of p for which two series converge, solving inequalities to find the common interval for p, which is between 1/2 and 2, matching choice C. Lastly, Van evaluates a limit using l'Hôpital's rule after recognizing an indeterminate form, finding that the limit is 2, leading to the final answer of choice D.