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What are Related Rates? Exploring the Concept on Khan Academy

Grace HelbigVerified
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Summary：Learn about related rates in calculus and their real-world applications in finance and investment. Explore examples and problem-solving techniques on Khan Academy.

What are Related Rates? Exploring the Concept on Khan Academy

Related rates refer to mathematical problems that involve calculating the rate at which one quantity changes in relation to the rate at which another quantity changes. These problems are a common topic incalculusand are important in understanding how different variables affect one another. In this article, we will delve into the concept of related rates and explore some examples to illustrate how they are used.

Understanding Related Rates

Related rates problems can be quite challenging, involving complex mathematical formulas and equations. However, at their core, they are simply problems that require us to find the rate at which one variable changes in relation to another variable. For example, if we have a balloon that is inflating at a rate of 2 cubic feet per minute, we might be asked to find the rate at which the radius of the balloon is increasing. This is a related rates problem because the rate at which the radius changes is related to the rate at which the balloon is inflating.

Solving Related Rates Problems

To solve related rates problems, we typically use the chain rule of differentiation to find the derivative of one variable with respect to another variable. This allows us to express the rate of change of one variable in terms of the rate of change of another variable. In the example of the inflating balloon, we can use the formula for the volume of a sphere (V=4/3πr^3) to find the derivative of the volume with respect to time. We can then use this derivative to find the rate at which the radius is increasing.

Real-World Applications of Related Rates

Related rates problems are not just theoretical exercises. They have many real-world applications, particularly in the field offinanceandinvestment. For example, investors use related rates to calculate the rate of return on different investments. By comparing the rate of return on one investment to the rate of return on another investment, investors can make informed decisions about where to put their money.

Another example of related rates in finance is the calculation of interest rates. When banks lend money, they charge interest based on the rate at which the borrower will pay back the loan. This rate is related to the risk of the loan, as well as the current economic conditions. By understanding the relationship between these variables, banks can set interest rates that are appropriate for different types of loans.

Conclusion

Related rates are an important concept in calculus and have many real-world applications in finance and investment. By understanding how different variables affect one another, we can make informed decisions about where to put our money and how to manage our finances. While related rates problems can be challenging, with practice and perseverance, anyone can learn to master them.

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