Problem 16 - Calculus Early Transcendentals

Question

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=-t^{3}+3 t^{2}-2 t \text { on }[0,3] ; s(0)=4$$

Step-by-Step Solution

Answer

Question: Graph the velocity function, \(v(t)\), on the interval \([0,3]\) and determine when the object is moving in the positive and negative direction. Then, find the position function, \(s(t)\), using both the antiderivative method and the Fundamental Theorem of Calculus. Verify that the two results agree, and graph the position function on the same interval.

Step 1: a. Graphing the velocity function and finding the direction

To graph the velocity function on the interval \([0,3]\), we can use a graphing calculator or software. For determining when the object is moving in the positive and negative direction, we must look for the points when \(v(t) > 0\) (positive direction) and \(v(t) < 0\) (negative direction).

Step 2: b. Find the position function using the antiderivative method and the Fundamental Theorem of Calculus

For the antiderivative method, we integrate the velocity function \(v(t)\): $$s(t) = \int v(t) dt = \int (-t^3 + 3t^2 - 2t) dt$$ For the Fundamental Theorem of Calculus, we use the given initial position, \(s(0) = 4\): $$s(t) = 4 + \int_0^t (-u^3 + 3u^2 - 2u) du$$ Now, we compute the integrals for both methods and check if they agree with each other.

Step 3: c. Graph the position function

To graph the position function on the interval \([0,3]\), we can use a graphing calculator or software.

Key Concepts

Graphing Velocity Functions

Understanding the motion of an object requires analyzing its velocity function, which represents the rate of change of an object's position over time. In calculus, graphing the velocity function provides crucial insights into the dynamics of the system under consideration.

Velocity functions can often be graphed using technology like graphing calculators or computer software, making it easier to visualize the changes in velocity over a specified interval. For instance, in the given problem, the function to be graphed is a cubic polynomial, which can show as a curve moving upward or downward as time progresses on the interval [0,3]. When graphing, critical points, where the velocity changes its sign from positive to negative (or vice versa) should be clearly marked. These points indicate a change in the direction of motion and are pivotal in understanding the object's movement.

Determining Motion Direction

The direction of motion for an object can be determined from the velocity function by identifying when the velocity is positive or negative. In our scenario, we interpret a positive velocity as the object moving in the forward or positive direction, and a negative velocity as moving in the reverse or negative direction.

By setting up inequalities, such as \( v(t) > 0 \), and solving for \( t \), we can find the intervals where the motion is in the positive direction. Conversely, intervals where the velocity is less than zero, expressed as \( v(t) < 0 \), will tell us when the object is moving in the negative direction. In calculus, finding these intervals often involves solving polynomial inequalities, which in turn, involves identifying roots and test intervals.

Antiderivative Method

The antiderivative method plays a significant role in calculus, especially when working with velocity functions to find position functions. Since the antiderivative, also known as the indefinite integral, of a velocity function gives us the position function (except for a constant of integration), this method is crucial for motion analysis.

To apply the antiderivative method to our problem, we integrate the velocity function without limits. We would carry out the integration term by term, adding a constant of integration that we later determine using initial conditions. In essence, the antiderivative method serves to 'undo' the process of differentiation, thus providing the formula for the position function from the velocity function.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of anti-differentiation with definite integrals and is a key player in translating velocity functions into position functions. According to the theorem, if a function is continuous over an interval and has an antiderivative within that interval, the definite integral of the function over that interval is equal to the difference in the values of the antiderivative at the endpoints of the interval.

Applying this to motion, if we are given the velocity function and initial position of an object, we can use a definite integral to find its position at any later time. As seen in the exercise, the position function at time \( t \) is the initial position plus the definite integral of the velocity function from 0 to \( t \). The result will conform to the antiderivative found by the alternative method. This is important as it reinforces the validity of the calculation and underlying physical interpretation.

Graphing Position Functions

Once we have expressed the object's position as a function of time using the antiderivative method or the Fundamental Theorem of Calculus, we can graph the position function. The graph demonstrates the object's position relative to time on a chosen interval.

Just like with graphing velocity functions, the position function's graph is crucial for visualizing the object's movement over time. The slopes at various points indicate the object's velocity, while points where the slope is zero hint at a momentary stop or change in direction. For instance, in our exercise, graphing the position function \( s(t) \) on the interval [0,3] with the initial position \( s(0)=4 \) would reveal how the object moves along a line over time and can help us notice patterns of motion such as acceleration and deceleration.