# Average vs Instantaneous Rates of Change Problem

• lolzwhut?
In summary, the distance a person can walk in time t is given by d=√t, and the average velocity from t=1 to t=4 is 1/3 MPH while the estimated velocity at t=4 hours is 0.25 MPH.
lolzwhut?
Hello, all. I'm doing a review in the back of my Calculus book and need help on this particular problem. The book has the answer, but I don't understand how to get it. If someone can show me steps to solving this problem, it would be greatly appreciated.

Problem: The distance a person can walk in time t is given by d=√t where d is measured in miles and t is measured in hours.

A) Find the average velocity from t=1 to t=4.
B) Estimate the person's velocity at t=4 hours.

The answers to the problem are:
A) 1/3 MPH
B) 0.25 MPH

## Homework Equations

Velocity=distance/time

Ending words:
So if anyone can show me the steps on solving this problem for both parts A and B, it would be greatly appreciated to not just me, but my knowledge itself...

~Lolzwhut?

A) To find the average velocity from t=1 to t=4, we can use the formula v=d/t. We can calculate the total distance traveled by taking the difference in the distance at t=1 and t=4 and dividing it by the time interval of 3 hours. d(t=4)-d(t=1)=(√4)-(√1)=2-1=1Therefore, the average velocity from t=1 to t=4 is v=(1 miles)/(3 hours)=1/3 MPH.B) To estimate the person's velocity at t=4 hours, we can use the formula v=d/t. We can calculate the distance traveled at t=4 by taking the square root of 4.d(t=4)=√4=2 milesTherefore, the estimated velocity at t=4 hours is v=(2 miles)/(1 hour)=2/1=0.25 MPH.

## 1. What is the difference between average and instantaneous rates of change?

The average rate of change is the overall change in a quantity over a period of time, while the instantaneous rate of change is the change at a specific moment in time. Average rate of change is calculated by dividing the total change in the quantity by the total time elapsed, while the instantaneous rate of change is calculated using the derivative at a specific point.

## 2. How do you calculate the average rate of change?

The average rate of change is calculated by finding the difference between the final and initial values of a quantity, and then dividing that difference by the total time elapsed. This gives you the average rate of change over the entire period of time.

## 3. What is the importance of understanding average and instantaneous rates of change?

Understanding average and instantaneous rates of change is crucial in many scientific fields, particularly in physics and engineering. These concepts are used to analyze and predict the behavior of various phenomena, such as the speed of a moving object or the rate of change of a chemical reaction. They also form the basis for more advanced mathematical concepts, such as calculus.

## 4. Can you give an example of a real-world application of average and instantaneous rates of change?

One example is in the field of economics, where average and instantaneous rates of change are used to analyze changes in stock prices over time. By calculating the average rate of change, investors can determine the overall trend of a stock's performance. Instantaneous rates of change, on the other hand, can help predict short-term fluctuations in a stock's value.

## 5. How can I improve my understanding of average and instantaneous rates of change?

One way to improve your understanding is to practice solving problems and calculating rates of change using different methods. You can also seek help from a tutor or attend workshops or courses on the topic. Additionally, reading and studying examples and real-world applications can help deepen your understanding of these concepts.

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