Problem 20 - Calculus: Early Transcendental Functions

Question

Use the given velocity function and initial position to estimate the final position \(s(b)\). $$v(t)=30 e^{-l / 4}, s(0)=-1, b=4$$

Step-by-Step Solution

Answer

The final position \(s(4)\) is: \(-120e^{-1} + 119\).

Step 1: Initial set up

We're given the velocity function \(v(t) = 30e^{-t / 4}\) and the initial condition \(s(0) = -1\). These details help define the position function.

Step 2: Calculate the definite integral

To find the position function from the velocity function, we calculate the definite integral of the velocity function from 0 to b: \(\int_{0}^{b} v(t) dt\). This yields \(\int_{0}^{b} 30e^{-t / 4} dt\). We'll apply the fundamental theorem of calculus and integrate.

Step 3: Integrate the function

Taking the integral, we get: \(-120e^{-t / 4}\). By evaluating this from 0 to b (b being 4), we get: \(-120e^{-4 / 4} - (-120e^{0})\), which simplifies to \(-120e^{-1} + 120\).

Step 4: Add the initial position

Now, we will add the initial position: \(-1 + (-120e^{-1} + 120)\).

Step 5: Simplify the result

By simplifying this, we finally get the position s(b) as \(s(4) = -120e^{-1} + 119\).

Key Concepts

Velocity Function

In calculus, the velocity function is essential for understanding motion. It describes how fast an object is moving and in which direction at any point in time. For this problem, our velocity function is given by:

• \( v(t) = 30e^{-t/4} \)

This function tells us:

• The velocity decreases exponentially over time due to the \(-t/4\) exponent.
• The coefficient '30' indicates the initial velocity at time \( t = 0 \).

Understanding this function allows us to predict how the position of an object should change over time if it starts at a known point.

Definite Integral

The definite integral is a fundamental calculus concept used to find the accumulated quantity over a specific interval. In other words, it helps determine the total change in position over time when dealing with velocity functions. To find the position change from time \( t = 0 \) to \( t = b \), we compute:

• \( \int_{0}^{b} v(t) \, dt = \int_{0}^{b} 30e^{-t/4} \, dt \)

The result of this integration provides the total distance the object has traveled from the start time \(0\) to the end time \(b\), minus any displacement due to changes in velocity over time.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, showing that they are inverse processes. It allows us to evaluate a definite integral by finding an antiderivative of the function. In this case, we calculated:

• \(-120e^{-t/4}\) as the antiderivative of \(30e^{-t/4}\).
• We then evaluated it at the bounds \(0\) and \(b\), to find the change in position.

Using this theorem, we turned a complex velocity function into a manageable calculation that reflects the position's total change over time.

Position Estimation

After calculating the definite integral, position estimation incorporates initial conditions to find the actual position at a certain time. Starting from the initial position \(s(0) = -1\), the total change from \( t = 0 \) to \( t = 4 \) needs to be added or subtracted depending on the direction of movement. Our final position is calculated as follows:

• \( s(b) = s(0) + (-120e^{-1} + 120) \)

By simplifying this expression, we determine the object's position at time \( b \). This method ensures accurate positioning by considering both initial location and cumulative changes over time.