Calculus AB Unit 5 Review: Particle Motion

This paragraph discusses the initial position and motion of a particle moving along the x-axis. The position function X(T) is given by T^3 - 12T + 5, and the initial position is determined to be 5 units at T=0. The particle's position at the origin is found by solving X(T)=0, yielding two times T=0.42 and T=3.23. The average velocity over the interval from T=1 to T=4 is calculated to be 9 units. At T=4, the particle's speed is analyzed by comparing velocity and acceleration, concluding it is increasing due to both being positive at this time.

The paragraph explores when the particle changes direction by examining the velocity function, which is derived from the position function. It is determined that the particle changes direction at T=2, as the velocity changes sign from negative to positive. The interval where the particle moves to the left is identified as T being greater than or equal to 0 and less than 2. The total distance traveled over the interval from T=1 to T=4 is calculated by splitting the interval at the point of direction change and summing the absolute changes in position, resulting in a total distance of 37 units.

This section analyzes the velocity of a rocket at specific times. It is deduced that there must be an interval where the velocity is 0 based on the intermediate value theorem, given the continuous nature of the velocity function and the recorded velocities. The rocket's position is noted to decrease when the velocity is negative, specifically between T=14 and T=25. An approximation for the rocket's acceleration at T=5 is calculated using the average rate of change, resulting in an acceleration of -1 feet per second squared.

The paragraph calculates the rocket's average acceleration over the interval from T=0 to T=12 using the change in velocity over time, resulting in an average acceleration of -13 feet per second squared. It also discusses the speed and acceleration of a particle at T=5, concluding that the speed is decreasing due to the velocity and acceleration having opposite signs. Fundamental concepts of motion, such as rest, direction change, constant velocity, and increasing or decreasing velocity, are explained in relation to their effects on position.

The paragraph interprets a velocity graph to determine the particle's motion. It identifies when the particle is moving left or right based on the sign of the velocity and when the particle is at rest when the velocity is zero. The time when the particle changes direction is found to be at T=9, where the velocity changes sign. The greatest speed of the particle is determined to be at T=15 with a speed of 6 units, and the acceleration at T=1 is calculated using the slope of the velocity graph, resulting in an acceleration of 1.5 units.

This section verifies the conditions of the mean value theorem for a given function and finds a value C that satisfies the theorem's conclusion. The function is continuous on the closed interval from 0 to 3 and differentiable on the open interval, satisfying the theorem's conditions. The derivative of the function is set equal to the average rate of change, and the value of C is found to be 0.897, confirming the theorem's application.

The paragraph examines the continuity and differentiability of a function on the interval from 1 to 4. The function is continuous and differentiable everywhere on this interval, except at x=-2, which is outside the interval of interest. The mean value theorem is confirmed to apply, guaranteeing a value C within the interval where the derivative equals the average rate of change, and this value is found to be 2.43.

The paragraph checks the hypotheses of the mean value theorem for a piecewise function. The function is continuous on the closed interval from 0 to 2 and differentiable on the open interval, with the exception of x=1, where the function has a discontinuity. The conditions for the mean value theorem to apply are discussed, including the need for the function to be continuous at endpoints and differentiable on the interval, leading to the determination of specific values for a, M, and B that satisfy these conditions.

This section applies the mean value theorem to a real-world scenario involving a trucker who was cited for speeding. The trucker's average speed over a two-hour period is calculated, and the mean value theorem is used to justify that the trucker must have been driving at the average speed of 79.5 miles per hour at some point during the trip, which exceeds the speed limit.

The paragraph uses the first derivative test to identify relative extreme points of a function that is continuous and twice differentiable except at x=1. A relative minimum is found at x=1 due to a sign change in the derivative. The maximum value of the function on the interval from -1 to 4 is determined by evaluating the function at its relative extreme point and at the endpoints, concluding that the maximum value is 3 at x=4.

The final paragraph describes the process of sketching a function with specific characteristics based on the given information about its derivatives. Key points are plotted, and the function's behavior between these points is described, including whether it is increasing or decreasing and concave up or down. The sketch includes a cusp at x=1 and attempts to reflect the slopes indicated by the derivative at various points.