Problem 93 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

SPEED AND DISTANCE A car is driven so that after \(t\) hours its speed is \(S(t)\) miles per hour. a. Write down a definite integral that gives the average speed of the car during the first \(N\) hours. b. Write down a definite integral that gives the total distance the car travels during the first \(N\) hours. c. Discuss the relationship between the integrals in parts (a) and (b).

Step-by-Step Solution

Answer

a. \( \frac{1}{N} \int_{0}^{N} S(t) \, dt\) b. \( \int_{0}^{N} S(t) \, dt \) c. Average speed is total distance divided by time interval.

Step 1: Define the Average Speed Integral

The average speed of the car during the first N hours is given by the formula for the average value of a continuous function. The integral for the average speed is: \ \[ \frac{1}{N} \int_{0}^{N} S(t) \, dt \]

Step 2: Define the Total Distance Integral

The total distance traveled by the car during the first N hours is given by the integral of the speed function over that time interval. The integral for the total distance is: \ \[ \int_{0}^{N} S(t) \, dt \]

Step 3: Discuss the Relationship Between the Integrals

The relationship between the integrals in parts (a) and (b) is that the average speed is simply the total distance divided by the time interval N. Mathematically, \ \[ \text{Average Speed} = \frac{1}{N} \times \text{Total Distance} = \frac{1}{N} \int_{0}^{N} S(t) \, dt \]

Key Concepts

average speed

Understanding how to compute the average speed of a car over a certain period is essential in calculus. When you drive a car, your speed might vary over time. To find the average speed over the first N hours, we use the formula for the average value of a continuous function. This is done using a definite integral. The average speed integral is calculated as follows:

\(\frac{1}{N} \int_{0}^{N} S(t) \, dt \)

Here, \( S(t) \) represents the speed of the car at time \( t \), and \( N \) is the total number of hours.

If you integrate the speed function over the interval from 0 to N and then divide by N, you get the average speed. This concept helps understand how the speed changes over time but expressed as a single value that communicates overall performance.

total distance

Total distance measures how far the car travels over a specific period, accounting for the changes in speed. To compute the total distance over the first N hours, we use another definite integral, which encompasses the continuous change in speed. The integral for total distance is:

\(\int_{0}^{N} S(t) \, dt \)

Essentially, this integral sums up all the small distances covered at each moment over the given time period.

Remember, speed is a rate of change. By integrating the speed function \( S(t) \), you find the total area under the curve, representing the entire distance traveled over N hours.

definite integrals

Definite integrals are a fundamental tool in calculus used to find the accumulated quantity, such as total distance or area under a curve. They are represented as:

\(\int_{a}^{b} f(t) \, dt \)

This notation means you're summing up values of the function \( f(t) \) from \( t = a \) to \( t = b \).

In the context of the car problem, the definite integrals for average speed and total distance both involve integrating the speed function \( S(t) \) over the interval from 0 to N.

• For average speed: The definite integral is scaled by \( \frac{1}{N} \), adjusting for the total time.
• For total distance: The same integral sums up the speed over time directly.

Understanding definite integrals helps in grasping how these different calculations relate, showing the connection between average measures and total accumulations.