Problem 33 - Calculus: Early Transcendental Functions

Question

The velocity of an object at various times is given. Use the data to estimate the distance traveled. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 40 & 42 & 40 & 44 & 48 & 50 & 46 \\ \hline \end{array}$$ $$\begin{array}{|l|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 46 & 42 & 44 & 40 & 42 & 42 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The total estimated distance traveled by the object is 1060 ft.

Step 1: Understanding the trapezoidal rule for numerical integration

The trapezoidal rule is a method in numerical integration which is used to estimate the definite integral of a function. It works by approximating the region under the curve as a trapezoid and then calculating the area. If the function values are \( f(x_1), f(x_2), ..., f(x_n) \) at points \( x_1, x_2, ..., x_n \), respectively, then the integral can be approximated as \( \frac{1}{2} \sum_{i=1}^{n-1} (x_{i+1}-x_i)(f(x_i)+f(x_{i+1})) \).

Step 2: Applying the trapezoidal rule

Apply the trapezoidal rule with \( f(x_i) = v(t_i) \) and the time steps being \( x_{i+1} - x_i = 1 \) s. Thus, the distance traveled can be approximated as: \( \frac{1}{2} \sum_{i=0}^{11} (v(t_i)+v(t_{i+1})) \).

Step 3: Calculation of distance traveled

Basing on the given velocities, the total distance traveled can be calculated as: \( \frac{1}{2} ((40+42)+(42+40)+(40+44)+(44+48)+(48+50)+(50+46)+(46+46)+(46+42)+(42+44)+(44+40)+(40+42)+(42+42)) \) ft. Calculate this sum to find the distance.

Key Concepts

Trapezoidal Rule

The trapezoidal rule is a basic yet effective method for numerical integration. It is particularly helpful when dealing with discrete data points. Imagine a graph of a function where data points are connected by straight lines, forming trapezoids between them. By calculating the area of these trapezoids, we approximate the area under the curve, or the integral, of the function.

This method is handy when the function is difficult to integrate analytically or when only discrete data is available, like in this case. If you have a set of points, the trapezoidal rule involves adding up the areas of each trapezoid. It is straightforward and involves multiplying the average of two consecutive function values by the time interval between them. This allows us to estimate the integral as follows:

• Divide the interval into smaller subintervals, using the given data points.
• Calculate each area of a trapezoid using the formula: \( \frac{1}{2} (b-a)(f(a)+f(b)) \), where \( b-a \) is the width of the subinterval, and \( f(a), f(b) \) are the function values at the endpoints of the subinterval.
• Sum these areas to get an approximation of the entire integral.

Velocity Data

Velocity data tells us how fast an object moves over time. In the context of estimating distance, velocity is crucial as it directly relates to how much ground an object covers. The given velocity data table indicates the object's speed at different times, provided in feet per second (ft/s).

Velocity data can look like this:

• There are specific times at which the velocity is recorded (e.g., 0 seconds, 1 second, 2 seconds, etc.).
• The velocity values change over time, reflecting how the object's speed fluctuates.

This data set serves as the basis for our distance estimation, guiding us to understand how the speed varies over the course of the time period. With each time interval, we can see how the function switches from one velocity to the next. To estimate the distance, these speeds will be critical as you apply methods like the trapezoidal rule.

Distance Estimation

Distance estimation based on velocity data is all about figuring out how far an object travels over a period of time, using its speed. This is where numerical integration, like the trapezoidal rule, comes into play. By knowing the object's velocity at certain moments, we can compute the total distance covered.

Here is how it works in our context:

• Start with the velocity readings at different time intervals.
• Using the trapezoidal rule, consider the velocity data points as vertices of trapezoids over each interval.
• Compute the area of each trapezoid with the formula: \( \frac{1}{2}(v(t_i) + v(t_{i+1})) \), where \( v(t_i), v(t_{i+1}) \) are the velocities at two successive points, and the interval is 1 second in this problem.
• Add these areas to estimate the total distance traveled over the entire time span.

By applying these steps, you transform discrete velocity data into an estimated total distance, making use of numerical integration techniques efficiently.