Find the maximum rate of change

In summary, to find the maximum rate of change of the function f(x,y) = (3 y^5)/x at the point (1,2), we first took the gradient of f(x,y), evaluated it at the point (1,2), and found the dot product of the gradient and the unit vector at that point. This gave us a maximum rate of change of 171.73.
  • #1
andyk23
26
0
Find the maximum rate of change of the function f(x,y) = (3 y^5)/x at the point (1,2)

First I took the gradient of f(x,y)=<-3y^5*x^-2,15y^4*x^-1>
and took the pt <1/sqrt(5),2/sqrt(5)>
Then <-3y^5*x^-2,15y^4*x^-1>*<1/sqrt(5),2/sqrt(5)>
the answer i get is 171.73
I'm not sure where my error is
 
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  • #2
andyk23 said:
Find the maximum rate of change of the function f(x,y) = (3 y^5)/x at the point (1,2)

First I took the gradient of f(x,y)=<-3y^5*x^-2,15y^4*x^-1>
and took the pt <1/sqrt(5),2/sqrt(5)>
Then <-3y^5*x^-2,15y^4*x^-1>*<1/sqrt(5),2/sqrt(5)>
the answer i get is 171.73
I'm not sure where my error is
Why not evaluate the gradient at the point (1, 2) ?
 
  • #3
Sorry I evaluated at the gradient(1,2)=<-96,240> and then <-96,240>*<1/sqrt(5),2/sqrt(5)>
 

1. What is the maximum rate of change in a function?

The maximum rate of change in a function is the steepest slope or the greatest increase of the function at any given point.

2. How do you find the maximum rate of change in a function?

To find the maximum rate of change in a function, you must first find the derivative of the function. Then, set the derivative equal to 0 and solve for the variable. The resulting value will give you the x-coordinate of the point where the maximum rate of change occurs.

3. Why is it important to find the maximum rate of change in a function?

Finding the maximum rate of change in a function is important because it helps us understand the behavior of the function and how it changes over time. It can also help us identify critical points and make predictions about the future behavior of the function.

4. Can the maximum rate of change occur at more than one point in a function?

Yes, the maximum rate of change can occur at multiple points in a function. This can happen if the function has multiple critical points or if the function is not continuous.

5. How is the maximum rate of change related to the minimum rate of change in a function?

The maximum rate of change and the minimum rate of change are related because they both represent extreme values of the function's slope. The maximum rate of change is the steepest slope, while the minimum rate of change is the flattest slope. They can occur at different points in the function or at the same point if the function has a horizontal tangent line.

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