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 $ \mu(\mu < v) $, then the time required to swim a distance $ L $ is $ L/(v - \mu) $ and the total energy $ E $ required to swim the distance is given by $$ E(v) = av^3 \cdot \dfrac{L}{v - \mu} $$ 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 of 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 $ \mu(\mu < v) $, then the time required to swim a distance $ L $ is $ L/(v - \mu) $ and the total energy $ E $ required to swim the distance is given by
$$ E(v) = av^3 \cdot \dfrac{L}{v - \mu} $$
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 of 50% greater than the current speed.

Key Concepts

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.

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.

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