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Lesson: Directional Derivatives and the Gradient Vector

Worksheet • 9 Questions

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’ 3 βˆ’ 2 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’ 1 βˆ’ 6 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 √ 3
  • B 𝑒 √ 3
  • C 3 𝑒 √ 3
  • D 2 𝑒 √ 3
  • E 4 𝑒

Q3:

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

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