# 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.

Q1:

Find the directional derivative of at the point in the direction of .

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Q2:

Find the directional derivative of at the point in the direction of .

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Q3:

Find the directional derivative of at the point in the direction of .

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Q4:

Find the directional derivative of at the point in the direction of .

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Q5:

Find the directional derivative of at the point in the direction of .

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Q6:

Find the directional derivative of at the point in the direction of .

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Q7:

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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 .

Q10:

Consider the function , given by . Which of the statements below is incorrect?

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Q11:

Compute the gradient of the function .

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Q14:

Compute the gradient of the function .

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Q16:

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Q19:

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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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