Optimum values and w/ chain rule

In summary, two boats, one heading west at 15 km/h and the other heading south at 12 km/h, reach the same dock at different times. By calculating the derivative of the distance between the boats with respect to time, it can be determined that the boats were closest to each other at 21 minutes and 36 seconds.
  • #1
Alain12345
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A boat leaves dock at 2:00 PM, heading west at 15 km/h. Another boat heads south at 12 km/h and reaches the same dock at 3:00 PM. When were the boats closest to each other?

I've solved other problems similar to this one, but I can't seem to figure this one out. I'm not sure what I have to do with the times that are given... I know you guys like to see work done to show that I've tried it, but all I have is a crappy diagram...

thanks.
 
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  • #2
On the assumption that this is parametric, that would make boat A:
x(t) = -15t
y(t) = 0

And Boat B:
x(t) = 0
y(t) = -12t+12

therefore,
d = ((x_a(t)-x_b(t))^2+(x_a(t)-x_b(t))^2)^0.5

Plug in the equations and simplify this(I have pity on you if you don't have a graphing calculator for this) and find the derivative of the distance with respect to time, which you can graph as you would any other derivative function. At that point you should know how to use the first derivative test for local extrema.

I got 21 minutes, 36 seconds. I'd definitely doublecheck, though, since I'm still in AP Cal AB and could very well be pulling a great deal of this out of my ass.
 
Last edited:
  • #3
Since the derivative of x2 with respect to t is 2x(dx/dt), minimizing the distance is the same as minimizing the square of the distance. You don't need the 1/2 power.
 

1. What is the chain rule and why is it important in finding optimum values?

The chain rule is a formula used in calculus to find the derivative of a composite function. It is important in finding optimum values because it allows us to analyze how small changes in one variable affect the overall output of the function.

2. How do you apply the chain rule to find the optimum value of a function?

To apply the chain rule, you must first identify the composite function and then differentiate each individual function within the composite. Then, multiply the derivatives together and simplify to find the final derivative. Set the derivative equal to zero and solve for the variable to find the optimum value.

3. Can the chain rule be used for any type of composite function?

Yes, the chain rule can be applied to any type of composite function, including exponential, logarithmic, and trigonometric functions. However, it may require some manipulation and knowledge of specific derivative rules to find the final derivative.

4. How does the chain rule help us understand the behavior of a function near its optimum value?

The chain rule allows us to analyze the rate of change of the function at different points near its optimum value. This can help us determine if the optimum value is a maximum or minimum and how sensitive the function is to changes in its variables near the optimum.

5. Are there any alternative methods to finding optimum values without using the chain rule?

Yes, there are alternative methods such as using first and second derivative tests, graphing the function, or using optimization algorithms. However, the chain rule is often the most efficient and straightforward method for finding optimum values.

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