The temperature at a point 
 is given by
where 
 is measured in 
 and 
, 
, and 
 in meters.
1. Find the rate of change of the temperature at the point 
 in the direction toward the point 

Answer: 
 
2. In what direction does the temperature increase fastest at 
?
Answer: 

3. Find the maximum rate of increase at 
.
Answer: 




The problem you're tackling involves the gradient of the temperature function \( T(x, y, z) \). The gradient will help us find the rate of change in various directions and the direction of the maximum temperature increase.

Let's go through each part step by step:

### 1. **Find the rate of change of the temperature at the point \((2, 1, 2)\) in the direction toward the point \((3, 3, 3)\)**

The **rate of change of temperature in a given direction** is given by the directional derivative, which is calculated using the gradient of the temperature function and the unit vector in the direction of the point.

#### Steps:
- Find the gradient of \( T(x, y, z) \): \( \nabla T(x, y, z) \).
- Find the unit vector in the direction from \((2, 1, 2)\) to \((3, 3, 3)\).
- Compute the directional derivative using the formula:
  \[
  D_{\mathbf{u}} T = \nabla T(x, y, z) \cdot \mathbf{u}
  \]
  where \( \mathbf{u} \) is the unit vector in the given direction.

### 2. **In what direction does the temperature increase fastest at \((2, 1, 2)\)?**

The direction of **fastest temperature increase** is always in the direction of the gradient vector, \( \nabla T(x, y, z) \), evaluated at the point \((2, 1, 2)\).

### 3. **Find the maximum rate of increase at \((2, 1, 2)\)**

The **maximum rate of increase** is the magnitude of the gradient at the point \((2, 1, 2)\), which is given by:
\[
\text{Maximum rate of increase} = |\nabla T(2, 1, 2)|
\]

Could you please provide the temperature function \( T(x, y, z) \) so we can compute these values accurately?

1. Find the rate of change of the temperature at the point (2, 1, 2) in the direction toward the point (3, 3, 3). 2. In what direction does the temperature increase fastest at (2, 1, 2)? 3. Find the maximum rate of increase at (2, 1, 2).

Explore how to find the rate of change of temperature at the point (2, 1, 2) toward (3, 3, 3), determine the direction of fastest temperature increase, and compute the maximum rate of increase. This problem covers the gradient and directional derivative in multivariable calculus.

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