Gradient Problem: Find Rate of Change, Direction, & Maximum Increase

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In summary, the conversation discusses finding the rate of change of T at point P(-1,1,-1) in the direction of u=<8,10,-8>, determining the direction in which the temperature increases fastest, and calculating the maximum rate of increase at P. The homework equations used include the gradient of T=(16x-7y+7yz, -7x+7xz, 7xy). The solution involves multiplying the gradient by the unit vector and finding its magnitude.
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munkhuu1
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Homework Statement


T(x,y,z)=8x^2-7xy+7xyz
a. find the rate of change of t at point p(-1,1,-1) in the direction u=<8,10,-8>
b. which direction does the temperature increase fastest
c. find the maximum rate of increase at P.

Homework Equations


gradient of T=(16x-7y+7yz, -7x+7xz, 7xy)


The Attempt at a Solution


a. i multiplied gradient by unit vector <8,10,-8>/square root of 228 and i got -2.914
please tell if this is right.
b. i just found gradient T(-1,1,-1)=<-30,14,-7>
c. i found the magnitude of b which was 33.84
 
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  • #2
yep, all good. Nice work.
 

FAQ: Gradient Problem: Find Rate of Change, Direction, & Maximum Increase

1. What is a gradient?

A gradient is a mathematical concept that represents the rate of change of a function at a particular point. It is a vector that contains both the direction and magnitude of the steepest slope of a function at that point.

2. How do you find the gradient of a function?

The gradient of a function can be found by taking the partial derivatives of the function with respect to each of its variables. These partial derivatives are then combined into a vector to give the gradient.

3. What is the significance of the gradient in a function?

The gradient represents the direction of greatest increase of the function at a specific point. This information is important in optimization problems, as it helps to determine the direction in which the function is changing the most rapidly.

4. How do you find the maximum increase of a function using the gradient?

The maximum increase of a function can be found by setting the gradient equal to zero and solving for the variables. This will give the critical points of the function, where the maximum increase occurs. The maximum increase can then be found by evaluating the function at these critical points.

5. Can the gradient be used to find the minimum decrease of a function?

Yes, the gradient can be used to find both the maximum increase and the minimum decrease of a function. The minimum decrease can be found by setting the gradient equal to zero and solving for the variables, and then evaluating the function at these critical points.

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