# Worksheet: Directional Derivatives and Gradient

In this worksheet, we will practice finding a derivative of multivariable functions in a given direction (directional derivative) and finding the gradient vector of the function.

**Q8: **

The temperature of a solid is given by the function , where , , are space coordinates relative to the center of the solid. In which direction from the point will the temperature decrease the fastest?

- A decreases the fastest in the direction of .
- B decreases the fastest in the direction of .
- C decreases the fastest in the direction of .
- D decreases the fastest in the direction of .

**Q9: **

In which direction does the function increases the fastest from the point ? In which direction does it decrease the fastest? Give your answer using unit vectors.

- A increases the fastest in the direction of and decreases the fastest in the direction of .
- B increases the fastest in the direction of and decreases the fastest in the direction of .
- C increases the fastest in the direction of and decreases the fastest in the direction of .
- D increases the fastest in the direction of and decreases the fastest in the direction of .

**Q20: **

Find a function so that the vector field is a gradient field.

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**Q21: **

Suppose with . Express the gradient (viewed as a matrix) in terms of the matrix , where , and a matrix of partial derivatives of .

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**Q22: **

Suppose with and that for a point . Express the gradient (viewed as a matrix) in terms of the matrix and a matrix of partial derivatives of .

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**Q23: **

Let where and . Given that there is a line in the - plane for which , find the equation of this line.

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