Applications of Partial Derivatives and chain rule

In summary, at the given instant, the lengths a, b, and c of a rectangle are changing with time, with a rate of da/dt = db/dt = 1m/sec and dc/dt = -3m/sec. Using the chain rule for partial derivatives, the rate of change of the box's volume is calculated to be 3m^3/s. The correctness of this equation is confirmed, providing reassurance to the person asking the question.
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mit_hacker
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Homework Statement



The lengths a,b,c of a rectangle are changing with time. At the instant in question, a=1m, b=2m, c=3m and da/dt = db/dt = 1m/sec, and dc/dt = -3m/sec. At what rate is the box's volume changing at this instant?

Homework Equations



Chain rule for partial derivatives.

The Attempt at a Solution



∂V/∂t=(bc)(da/dt)+(ac)(db/dt)+(ab)(dc/dt)

Substituting the values gives 3m^3/s. Am I right?
Thanks for helping me out. The answer is not given in the book.
 
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  • #2
The equation is correct.
 
  • #3
Thanks!

Hey! Thanks a ton for the re-assurances! Makes me feel more confident about myself.
 

FAQ: Applications of Partial Derivatives and chain rule

How are partial derivatives used in real-world applications?

Partial derivatives are used in a wide range of fields such as economics, physics, engineering, and finance. They are used to find the rate of change of a function with respect to one of its variables while holding all other variables constant. This is useful in determining optimal solutions, predicting trends, and analyzing data in various industries.

Can you explain the chain rule and its role in partial derivatives?

The chain rule is a fundamental concept in calculus that allows us to calculate the derivative of a composite function. In the context of partial derivatives, the chain rule is used to find the rate of change of a multivariable function with respect to one of its variables by breaking it down into smaller functions and applying the chain rule to each one.

How is the concept of partial derivatives related to gradients?

The gradient is a vector that represents the direction and magnitude of the steepest increase of a function. In the context of partial derivatives, the gradient is closely related as it is composed of the partial derivatives of a multivariable function. This allows us to use the gradient to find the direction in which a function is changing the fastest.

Can you provide an example of a practical application of partial derivatives and the chain rule?

One example of a practical application of partial derivatives and the chain rule is in economics, specifically in production and cost analysis. The production function, which represents the relationship between inputs and outputs, can be optimized using partial derivatives and the chain rule to determine the most efficient levels of inputs to maximize output while minimizing costs.

How do partial derivatives and the chain rule relate to optimization problems?

Partial derivatives and the chain rule are essential tools in solving optimization problems. By finding the critical points of a multivariable function using partial derivatives, we can determine where the function is increasing or decreasing. This information, combined with the chain rule, allows us to find the maximum or minimum values of the function, which is crucial in optimization problems.

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