The temperature at any point in the plane is given by the function T(x,y)=100/(x^2+y^2+1). Find the direction of the greatest increase in temperature at the point (3,2). What is the magnitude of the greatest increase?
Could someone help me with this, please? Thanks.
To find the direction of the greatest increase in temperature at a given point, we need to calculate the gradient of the temperature function T(x, y) and evaluate it at the desired point (3, 2). The gradient will give us both the direction and magnitude of the steepest ascent.
The gradient vector is given by the partial derivatives of T with respect to x and y:
∇T = (∂T/∂x, ∂T/∂y)
To calculate the partial derivatives, we differentiate T(x, y) with respect to x and y separately.
∂T/∂x = -200x/(x^2+y^2+1)^2
∂T/∂y = -200y/(x^2+y^2+1)^2
Now, let's calculate the partial derivatives at the point (3, 2):
∂T/∂x = -200(3)/(3^2+2^2+1)^2 = -200(3)/14400 = -1/24
∂T/∂y = -200(2)/(3^2+2^2+1)^2 = -200(2)/14400 = -1/36
Therefore, the gradient vector at the point (3, 2) is:
∇T(3, 2) = (-1/24, -1/36)
Now, we know that the direction of the steepest ascent is in the direction of the gradient vector. So, the direction of the greatest increase in temperature at the point (3, 2) is (-1/24, -1/36).
To find the magnitude of the greatest increase, we can calculate the length of the gradient vector:
|∇T(3, 2)| = √((-1/24)^2 + (-1/36)^2) = √(1/576 + 1/1296) = √(4/2592) = 2/36 = 1/18
Therefore, the magnitude of the greatest increase in temperature at the point (3, 2) is 1/18.