AP Calculus AB 2008 Multiple Choice (Calculator) - Questions 76-92

Vincent introduces the AP Calculus AB 2008 multiple-choice section, focusing on question 76. He explains how to determine where a function F is increasing by looking at the derivative f'(x) and identifying where it's greater than zero. The discussion covers interpreting the graph of f'(x) to find intervals of increase, and the importance of not using a calculator for every question to avoid slowing down. A detailed analysis of a sloppy question and its answer choice B is provided, along with a general approach to tackling such problems efficiently.

The summary delves into question 77, which involves a piecewise function and the evaluation of true statements about its behavior. Vincent explains the concept of limits from the left and right, and how they relate to the existence of a limit at a certain point. The importance of the limit being equal on both sides for its existence is highlighted. The summary also covers questions 78 and 80, where the focus is on finding intervals where a function is increasing using derivatives and a graphing calculator, and determining points of inflection by analyzing the second derivative.

Vincent tackles questions 81 and 82, which involve the Fundamental Theorem of Calculus (FTC) and a motion problem, respectively. For question 81, he demonstrates how to use FTC to find the value of an antiderivative at a specific point. In question 82, the focus is on finding the acceleration of a particle at a given time by taking the derivative of the velocity function. The use of a calculator to efficiently solve these problems is emphasized.

The summary explains how to find the area between two curves for question 83 by identifying points of intersection and setting up integrals. For question 84, Vincent discusses the use of the first derivative test to find relative maxima and minima, highlighting how to interpret the graph of the derivative. The concept of horizontal tangent lines is clarified as a distraction in this context.

Vincent explores question 85, which involves using Rolle's theorem to deduce that a function must not be differentiable at some point within a given interval if certain conditions are met. The explanation covers the three conditions of Rolle's theorem and how the non-existence of a value C, where the derivative equals zero, leads to the conclusion that the function is not differentiable at some point K within the interval.

The summary for question 86 discusses how to determine which graph could represent a position function based on given velocity values. Vincent explains the process of elimination using the slope and function values at specific points. For question 87, he demonstrates how to find the position of an object at a given time using the integral of the velocity function and the initial position.

Vincent addresses a related rates problem in question 88, where the rate of change of the surface area of a sphere is calculated given the rate of change of the radius. The use of the chain rule and the importance of including the rate of change of the radius (dr/dt) are highlighted. For question 90, he explains how to analyze a table of values for a function that is twice differentiable, with a focus on the slope and concavity of the function.

The final summary covers question 92, which involves calculating the total population within a rectangular boundary of a city with a river on one side. Vincent explains how to use the concept of population density and definite integrals to find the total population by summing up the population slices along the river's edge.