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Worksheet: Directional Derivatives

Q1:

The temperature 𝑇 of a solid is given by the function 𝑇 ( π‘₯ , 𝑦 , 𝑧 ) = 𝑒 + 𝑒 + 𝑒 βˆ’ π‘₯ βˆ’ 2 𝑦 4 𝑧 , where π‘₯ , 𝑦 , 𝑧 are space coordinates relative to the center 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 ο€Ή 𝑒 , 2 𝑒 , βˆ’ 4 𝑒  βˆ’ 1 βˆ’ 6 8 .
  • B 𝑇 decreases the fastest in the direction of ο€Ή βˆ’ 𝑒 , βˆ’ 2 𝑒 , 4 𝑒  βˆ’ 3 βˆ’ 2 8 .
  • C 𝑇 decreases the fastest in the direction of ο€Ή βˆ’ 𝑒 , βˆ’ 2 𝑒 , 4 𝑒  βˆ’ 1 βˆ’ 6 8 .
  • D 𝑇 decreases the fastest in the direction of ο€Ή 𝑒 , 2 𝑒 , βˆ’ 4 𝑒  βˆ’ 3 βˆ’ 2 8 .

Q2:

Find the directional derivative of 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑒 2 𝑦 𝑧 at the point ( 1 , 1 , 1 ) in the direction of 𝑣 = ο€Ώ 1 √ 3 , 1 √ 3 , 1 √ 3  .

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

Q3:

Compute the gradient of 𝑓 ( π‘₯ , 𝑦 ) = √ π‘₯ + 𝑦 + 4 2 2 .

  • A ο€Ώ 2 π‘₯ √ π‘₯ + 𝑦 + 4 , 2 𝑦 √ π‘₯ + 𝑦 + 4  2 2 2 2
  • B ο€Ώ 𝑦 √ π‘₯ + 𝑦 + 4 , π‘₯ √ π‘₯ + 𝑦 + 4  2 2 2 2
  • C ο€Ώ 2 𝑦 √ π‘₯ + 𝑦 + 4 , 2 π‘₯ √ π‘₯ + 𝑦 + 4  2 2 2 2
  • D ο€Ώ π‘₯ √ π‘₯ + 𝑦 + 4 , 𝑦 √ π‘₯ + 𝑦 + 4  2 2 2 2
  • E ο€» π‘₯ √ π‘₯ + 𝑦 + 4 , 𝑦 √ π‘₯ + 𝑦 + 4  2 2 2 2

Q4:

Find the directional derivative of 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ 𝑒 2 𝑦 at the point ( 1 , 1 ) in the direction of 𝑣 = ο€Ώ 1 √ 2 , 1 √ 2  .

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

Q5:

Find the directional derivative of 𝑓 ( π‘₯ , 𝑦 ) = 1 π‘₯ + 𝑦 2 2 at the point ( 1 , 1 ) in the direction of 𝑣 = ο€Ώ 1 √ 2 , 1 √ 2  .

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

Q6:

Find the directional derivative of 𝑓 ( π‘₯ , 𝑦 ) = √ π‘₯ + 𝑦 + 4 2 2 at the point ( 1 , 1 ) in the direction of 𝑣 = ο€Ώ 1 √ 2 , 1 √ 2  .

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

Q7:

Find the directional derivative of 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ + 𝑦 βˆ’ 1 2 2 at the point ( 1 , 1 ) in the direction of 𝑣 = ο€Ώ 1 √ 2 , 1 √ 2  .

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

Q8:

In which direction does the function 𝑓 ( π‘₯ , 𝑦 ) = π‘₯ 𝑦 + π‘₯ 𝑦 2 3 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 ο€Ώ 2 9 √ 9 4 1 , 1 0 √ 9 4 1  and decreases the fastest in the direction of ο€Ώ βˆ’ 2 9 √ 9 4 1 , βˆ’ 1 0 √ 9 4 1  .
  • B 𝑓 increases the fastest in the direction of ο€Ώ βˆ’ 9 √ 9 7 , βˆ’ 4 √ 9 7  and decreases the fastest in the direction of ο€Ώ 9 √ 9 7 , 4 √ 9 7  .
  • C 𝑓 increases the fastest in the direction of ο€Ώ βˆ’ 2 9 √ 9 4 1 , βˆ’ 1 0 √ 9 4 1  and decreases the fastest in the direction of ο€Ώ 2 9 √ 9 4 1 , 1 0 √ 9 4 1  .
  • D 𝑓 increases the fastest in the direction of ο€Ώ 9 √ 9 7 , 4 √ 9 7  and decreases the fastest in the direction of ο€Ώ βˆ’ 9 √ 9 7 , βˆ’ 4 √ 9 7  .

Q9:

Find the directional derivative of 𝑓 ( π‘₯ , 𝑦 , 𝑧 ) = π‘₯ 𝑦 𝑧 s i n at the point ( 1 , 1 , 1 ) in the direction of v = ο“’ 1 √ 3 , 1 √ 3 , 1 √ 3 ο““ .

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