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 anti- derivative method and the Fundamental Theorem of Calculus (Theorem 1). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=3 \sin \pi t \text { on }[0,4] ; s(0)=1$$

Short answer

Answer: To determine when the object is moving in positive and negative directions, we can observe the graph of the velocity function on the interval [0,4]. The object is moving in a positive direction when the velocity function is above the horizontal axis, and moving in a negative direction when it is below the axis.

Step-by-step solution

Graph the velocity function on [0,4] and identify the direction

First, we are asked to graph the velocity function on the interval \([0,4]\). The given velocity function is \(v(t) = 3\sin(\pi t)\). Using a graphing tool, we can plot this function and observe where the velocity is positive and negative. The positive direction occurs when the velocity is above the horizontal axis, while the negative direction occurs when it's below the axis.

Determine the position function (anti-derivative method)

To determine the position function \(s(t)\) using the anti-derivative method, we find the indefinite integral of the velocity function \(v(t)\): $$s(t) = \int v(t) dt = \int 3\sin(\pi t) dt$$ Using substitution, let \(u = \pi t\), and \(du = \pi dt\). The integral becomes: $$s(t) = \frac{3}{\pi} \int \sin(u) du$$ The integral of \(\sin(u)\) is \(-\cos(u)\). Thus, $$s(t) = -\frac{3}{\pi} \cos(\pi t) + C$$ Now, use the initial condition \(s(0) = 1\) to find the constant C: $$1 = -\frac{3}{\pi} \cos(0) + C$$ $$C = 1 + \frac{3}{\pi}$$ So, the position function is: \(s(t) = -\frac{3}{\pi}\cos(\pi t) + 1 + \frac{3}{\pi}\)

Determine the position function (using the Fundamental Theorem of Calculus)

We can also find the position function using the Fundamental Theorem of Calculus: $$s(b) - s(a) = \int_a^b v(t) dt$$ Where \(s(b)\) is the position at time \(b\), \(s(a)\) is the position at time \(a\), and the integral represents the total change in position during the time interval \([a,b]\). To find the position function \(s(t)\), let \(a=0\) and \(b = t\). Then we have: $$s(t) - s(0) = \int_0^t v(t') dt'$$ $$s(t) = s(0) + \int_0^t v(t') dt' = 1 + \int_0^t 3\sin(\pi t') dt'$$ Using the same substitution method as before, let \(u = \pi t'\), and \(du = \pi dt'\). We have: $$s(t) = 1 + \frac{3}{\pi} \int_0^{\pi t} \sin(u) du$$ The integral of \(\sin(u)\) is \(-\cos(u)\). Thus, $$s(t) = 1 - \frac{3}{\pi}(\cos(\pi t) - 1) = -\frac{3}{\pi}\cos(\pi t) + 1 + \frac{3}{\pi}$$ This result agrees with the anti-derivative method in Step 2.

Graph the position function on [0,4]

We now have the position function: \(s(t) = -\frac{3}{\pi}\cos(\pi t) + 1 + \frac{3}{\pi}\). We can graph this function on the interval \([0,4]\) using a graphing tool, and this would give us a visual representation of the object's position over time.

Key concepts

Anti-derivative

In calculus, the concept of an anti-derivative is central to solving problems related to finding a function from its rate of change. An anti-derivative, also known as an indefinite integral, is essentially the reverse process of differentiation. It seeks a function whose derivative is the given function. For example, if we have a velocity function like \( v(t) = 3\sin(\pi t) \), we can find its anti-derivative to determine the position function of an object over time.

To find the anti-derivative, we perform integration. In the case of \( v(t) = 3\sin(\pi t) \), the indefinite integral is:
\[ s(t) = \int 3\sin(\pi t) \, dt = -\frac{3}{\pi}\cos(\pi t) + C \]
The constant \( C \) is determined using any initial condition provided such as the initial position \( s(0) = 1 \). Once we solve for \( C \), we acquire the full expression for the position function \( s(t) \).

This anti-derivative approach gives us a functional form, paving the path for analyzing the motion of an object given its velocity.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation (derivative) with integration (anti-derivative). It comprises two parts, but here we focus on the application for finding position from velocity.

When we have a function like the velocity of an object, the Fundamental Theorem helps us calculate the change in position over an interval \([a, b]\). Using the theorem, we can express it as the integral of the velocity function:
\[ s(b) - s(a) = \int_a^b v(t) \, dt \]
In our example, it confirms the calculation of the position function \( s(t) \) by integrating velocity over the time interval from 0 to \( t \):
\[ s(t) = s(0) + \int_0^t 3\sin(\pi t') \, dt' \]

By completing the integral and using the initial condition, we derive the position function which matches the earlier method using the anti-derivative. This theorem ensures that regardless of whether you start with a derivative (velocity) or integral, the connections remain intact, solidifying our understanding of calculus concepts.

Position function

The position function, denoted as \( s(t) \), beautifully captures the location of an object along a path or line over time. It is derived using the object's velocity, which represents the rate of change of position.

Given a velocity function such as \( v(t) = 3\sin(\pi t) \), the position function can be determined using either the anti-derivative approach or the Fundamental Theorem of Calculus. Starting from a known position \( s(0) = 1 \), we integrate the velocity function to find:
\[ s(t) = -\frac{3}{\pi}\cos(\pi t) + 1 + \frac{3}{\pi} \]
Each component of the position function lends insight:

• -\frac{3}{\pi}\cos(\pi t): Shows oscillation due to the sine function in velocity, reflecting changes in direction and speed.
• + 1 + \frac{3}{\pi}: This constant adjusts for initial conditions, setting a baseline for the graph.

The graph of \( s(t) \) over the interval \([0,4]\) visually depicts how the object's position evolves, oscillating between positive and negative velocities. Understandably, this function is critical in comprehending motion and dynamics within calculus applications.