Worksheet: Directional Derivatives

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In this worksheet, we will practice finding the directional derivative and the gradient vector of nice functions in 2 or 3 variables.

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 .

  • A 3 √ 2
  • B 4 √ 2
  • C √ 2
  • D 2 √ 2
  • E4

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 .

  • A 2 𝑒 √ 2
  • B 2 𝑒 √ 2
  • C 3 𝑒 √ 2
  • D 3 𝑒 √ 2
  • E 2 √ 2 𝑒

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 .

  • A 4 𝑒
  • B 3 𝑒 √ 3
  • C 2 𝑒 √ 3
  • D 4 𝑒 √ 3
  • E 𝑒 √ 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 .

  • A c o s 1 √ 3
  • B 3 √ 3 1 c o s
  • C √ 3
  • D √ 3 1 c o s
  • E 3 √ 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 .

  • A − √ 2
  • B √ 2 2
  • C √ 2
  • D − √ 2 2
  • E − 3 √ 2 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 .

  • A √ 3
  • B 2 √ 3 3
  • C √ 2 2
  • D √ 3 3
  • E 3 √ 2 2

Q7:

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

  • A 2 𝑥 √ 𝑥 + 𝑦 + 4 , 2 𝑦 √ 𝑥 + 𝑦 + 4
  • B 𝑦 √ 𝑥 + 𝑦 + 4 , 𝑥 √ 𝑥 + 𝑦 + 4
  • C 2 𝑦 √ 𝑥 + 𝑦 + 4 , 2 𝑥 √ 𝑥 + 𝑦 + 4
  • D 𝑥 √ 𝑥 + 𝑦 + 4 , 𝑦 √ 𝑥 + 𝑦 + 4
  • E 𝑥 √ 𝑥 + 𝑦 + 4 , 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?

  • A 𝑇 decreases the fastest in the direction of 𝑒 , 2 𝑒 , − 4 𝑒 − 1 − 6 8 .
  • B 𝑇 decreases the fastest in the direction of − 𝑒 , − 2 𝑒 , 4 𝑒 − 3 − 2 8 .
  • C 𝑇 decreases the fastest in the direction of − 𝑒 , − 2 𝑒 , 4 𝑒 − 1 − 6 8 .
  • D 𝑇 decreases the fastest in the direction of 𝑒 , 2 𝑒 , − 4 𝑒 − 3 − 2 8 .

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.

  • A 𝑓 increases the fastest in the direction of 2 9 √ 9 4 1 , 1 0 √ 9 4 1 and decreases the fastest in the direction of − 2 9 √ 9 4 1 , − 1 0 √ 9 4 1 .
  • B 𝑓 increases the fastest in the direction of − 9 √ 9 7 , − 4 √ 9 7 and decreases the fastest in the direction of 9 √ 9 7 , 4 √ 9 7 .
  • C 𝑓 increases the fastest in the direction of − 2 9 √ 9 4 1 , − 1 0 √ 9 4 1 and decreases the fastest in the direction of 2 9 √ 9 4 1 , 1 0 √ 9 4 1 .
  • D 𝑓 increases the fastest in the direction of 9 √ 9 7 , 4 √ 9 7 and decreases the fastest in the direction of − 9 √ 9 7 , − 4 √ 9 7 .

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