# Application of derivatives problem?

• mirs
In summary, the conversation discusses a biologist's experiment to determine the number of calories burned by a salmon swimming upstream against a current. The biologist's energy equation is given as (kdv^5)/(v-v0), where v is the salmon's swimming speed. The conversation then talks about the salmon's progress and the time taken to cover a distance d. The conversation also mentions finding the critical value of v using the energy equation and using it to calculate the time of travel.
mirs

## Homework Statement

A biologist determines experimentally that the number of calories burned by a salmon swimming a distance d in miles upstream against a current v0 in miles per hour is given by

Energy = kdv^5/v − v0

where v is the salmon’s swimming speed relative to the water it is in. This means that the salmon’s progress upstream is at the rate of v − v0 miles per hour, so that the distance d is covered in a time of

t=d/v − v0

If v0 = 2 mph and d = 20 miles, and the salmon, being smart, swims so as to minimize the calories burned, how many hours will it take to complete the journey?

## Homework Equations

Requires derivatives

## The Attempt at a Solution

I really just don't understand where to start. I subbed t into the equation and ended up with ktv^5/v-v0, took the derivative of that with respect to t and got 5kv^4 but I have no idea what I really don't know where to go from there.

mirs said:

## Homework Statement

A biologist determines experimentally that the number of calories burned by a salmon swimming a distance d in miles upstream against a current v0 in miles per hour is given by

Energy = kdv^5/v − v0

Do you mean ##\frac{kdv^5} v - v_0##, which is what you wrote, or ##\frac{kdv^5}{v − v0}##?

where v is the salmon’s swimming speed relative to the water it is in. This means that the salmon’s progress upstream is at the rate of v − v0 miles per hour, so that the distance d is covered in a time of

t=d/v − v0

Same problem here. Obviously you mean ##\frac d {v-v_0}##. Use parentheses!
If v0 = 2 mph and d = 20 miles, and the salmon, being smart, swims so as to minimize the calories burned, how many hours will it take to complete the journey?

## Homework Equations

Requires derivatives

## The Attempt at a Solution

I really just don't understand where to start. I subbed t into the equation and ended up with ktv^5/v-v0, took the derivative of that with respect to t and got 5kv^4 but I have no idea what I really don't know where to go from there.

Try finding the critial value of ##v## using the energy equation. Then use that to get the time of travel.

Yes! I meant (kdv^5)/(v−v0).

LCKurtz said:
Try finding the critial value of ##v## using the energy equation. Then use that to get the time of travel.

OK, so I'm guessing I don't plug t into the energy equation in the first place? because I did that and found the derivative to be 5ktv^4

LCKurtz said:
Try finding the critial value of ##v## using the energy equation. Then use that to get the time of travel.

mirs said:
OK, so I'm guessing I don't plug t into the energy equation in the first place? because I did that and found the derivative to be 5ktv^4

Why are you guessing? Why not just try what I suggested?

## What is the application of derivatives problem?

The application of derivatives problem is a type of mathematical problem that involves finding the rate of change or slope of a function at a specific point. It is commonly used in various fields such as physics, economics, and engineering to analyze real-world situations.

## How do you solve an application of derivatives problem?

To solve an application of derivatives problem, you need to follow certain steps. First, identify the function and the variable that is changing. Then, find the derivative of the function using the rules of differentiation. Finally, plug in the given values to find the slope or rate of change at the specific point.

## What are some common real-world applications of derivatives?

Some common real-world applications of derivatives include finding the maximum or minimum values of a function, determining the velocity and acceleration of an object, and optimizing production processes in economics.

## What is the difference between an optimization problem and an application of derivatives problem?

An optimization problem is a type of application of derivatives problem where the goal is to find the maximum or minimum value of a function. However, not all application of derivatives problems are optimization problems. Some may involve finding the slope or rate of change of a function at a specific point.

## How can I improve my problem-solving skills for application of derivatives problems?

To improve your problem-solving skills for application of derivatives problems, it is important to practice regularly and familiarize yourself with the different rules of differentiation. You can also seek help from textbooks, online resources, and tutors to gain a better understanding of the concepts and techniques used in solving these types of problems.

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