For a fish swimming at a speed v relative to the water, the...

For a fish swimming at a speed $v$ relative to the water, the energy expenditure per unit time is proportional to $v ^ { 3 } .$ It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current $u ( u < v ) ,$ then the time required to swim a distance $L$ is $L / ( v - u )$ and the total energy $E$ required to swim the distance is given by $$E ( v ) = a v ^ { 3 } \cdot \frac { L } { v - u }$$ where a is the proportionality constant. (a) Determine the value of $v$ that minimizes $E$ . (b) Sketch the graph of $E$ Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50$\%$ greater than the current speed.

For a fish swimming at a speed $v$ relative to the water, the energy expenditure per unit time is proportional to $v ^ { 3 } .$ It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current $u ( u < v ) ,$ then the time required to swim a
distance $L$ is $L / ( v - u )$ and the total energy $E$ required to
swim the distance is given by
$$E ( v ) = a v ^ { 3 } \cdot \frac { L } { v - u }$$
where a is the proportionality constant.
(a) Determine the value of $v$ that minimizes $E$ .
(b) Sketch the graph of $E$
Note: This result has been verified experimentally; migrating
fish swim against a current at a speed 50$\%$ greater than the
current speed.

Key Concepts

Energy Expenditure Modeling

This concept involves representing a physical process—in this case, the energy cost of swimming—as a mathematical function. By modeling the energy expenditure per unit time as being proportional to the cube of the swimming speed, and incorporating the impact of a counteracting current on the effective speed, the problem translates into a function that depicts how total energy consumption depends on the chosen swimming speed. Such models are crucial across physics and biology for quantifying and optimizing energy requirements in dynamic systems.

Calculus Optimization

Calculus optimization is the process of finding the maximum or minimum value of a function by analyzing its derivatives. In the context of minimizing total energy expenditure for a fish swimming against a current, optimization techniques are used to identify the swimming speed that results in the lowest possible energy cost. This involves setting the derivative of the energy function with respect to speed equal to zero and solving for the critical point, ensuring that the point indeed corresponds to a minimum.

Differentiation

Differentiation is a fundamental tool in calculus that allows us to determine the rate at which a function changes with respect to one of its variables. In optimization problems, computing the derivative of the energy function with respect to speed is essential. Here, the derivative provides information about how small changes in speed impact the total energy expended, ultimately guiding us to the optimal speed that minimizes energy consumption.

Graphical Analysis of Functions

Graphical analysis involves sketching or interpreting the behavior of functions to understand their properties, such as increasing or decreasing intervals, asymptotic behavior, and local extrema. For the energy expenditure function, plotting the graph helps visualize how energy changes with speed, highlight the minimum point, and illustrate the impact of parameters like the current speed. This visual representation aids in qualitatively understanding the solution to the optimization problem.

Transcript

00:01 In this question, we are told that as fish is swimming at some velocity, we're also up to the water.

00:07 And we know that it's the energy spent per time is proportional to the cube of the velocity.

00:15 And we want to verify that we want to find the value of the velocity that minimizes the amount of energy spent when we're swimming against the current.

00:24 And we're assuming that the current is less than the velocity.

00:28 And in this case, in order to do this question, all we really need to do, do is just take the derivative of this thing with respect to the velocity.

00:42 Because we know that u is a constant, a is a constant, and l is a constant.

00:47 So let's write down our formula again, like this instead, and we'll now take the derivative of this thing with respect to v.

01:09 In order to do this, we'll have to use the quotient rule here.

01:12 So that means we need to square the denominator, and what we're going to have is we're going to have the denominator times the derivative of the top, subtracted from the top, multiplied by the derivative of the bottom.

01:43 And since taking derivatives with respect to v, the u is just a constant.

01:52 So this means that we have e prime of v is equal to, okay, let's see what we have.

01:57 We have v minus u times 3al v squared, minus, just a lv cubed because the derivative of v is just one so this will be over v minus u the whole thing squared and what we can do is we can factor out this a times l times this a and l out from everything so this would be equal to a times l multiplied by 3 v squared times v minus u minus v cubed and this will be over v minus u squared...

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