Lesson: Directional Derivatives

In this lesson, we will learn how to find the directional derivative and the gradient vector of nice functions in 2 or 3 variables.

Worksheet: Directional Derivatives • 9 Questions

Q1:

Find the directional derivative of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 1 2 2 at the point ( 1 , 1 ) in the direction of 𝑣 = 1 2 , 1 2 .

Q2:

Find the directional derivative of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑒 at the point ( 1 , 1 ) in the direction of 𝑣 = 1 2 , 1 2 .

Q3:

Find the directional derivative of 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑒 at the point ( 1 , 1 , 1 ) in the direction of 𝑣 = 1 3 , 1 3 , 1 3 .

Q4:

Find the directional derivative of 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑥 𝑦 𝑧 s i n at the point ( 1 , 1 , 1 ) in the direction of 𝑣 = 1 3 , 1 3 , 1 3 .

Q5:

Find the directional derivative of 𝑓 ( 𝑥 , 𝑦 ) = 1 𝑥 + 𝑦 2 2 at the point ( 1 , 1 ) in the direction of 𝑣 = 1 2 , 1 2 .

Q6:

Find the directional derivative of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 + 4 2 2 at the point ( 1 , 1 ) in the direction of 𝑣 = 1 2 , 1 2 .

Q7:

Compute the gradient of 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 + 𝑦 + 4 .

Q8:

The temperature 𝑇 of a solid is given by the function 𝑇 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑒 + 𝑒 + 𝑒 𝑥 2 𝑦 4 𝑧 , where 𝑥 , 𝑦 , 𝑧 are space coordinates relative to the centre of the solid. In which direction from the point ( 3 , 1 , 2 ) will the temperature decrease the fastest?

Q9:

In which direction does the function 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 + 𝑥 𝑦 increases the fastest from the point ( 2 , 3 ) ? In which direction does it decrease the fastest? Give your answer using unit vectors.

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