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 𝑓(𝑥,𝑦)=𝑥+𝑦1 at the point (1,1) in the direction of 𝑣=12,12.

  • A2
  • B32
  • C42
  • D4
  • E22

Q2:

Find the directional derivative of 𝑓(𝑥,𝑦)=𝑥𝑒 at the point (1,1) in the direction of 𝑣=12,12.

  • A2𝑒2
  • B22𝑒
  • C2𝑒2
  • D3𝑒2
  • E3𝑒2

Q3:

Find the directional derivative of 𝑓(𝑥,𝑦,𝑧)=𝑥𝑒 at the point (1,1,1) in the direction of 𝑣=13,13,13.

  • A2𝑒3
  • B4𝑒
  • C3𝑒3
  • D4𝑒3
  • E𝑒3

Q4:

Find the directional derivative of 𝑓(𝑥,𝑦,𝑧)=𝑥𝑦𝑧sin at the point (1,1,1) in the direction of v=13,13,13.

  • A331cos
  • Bcos13
  • C31cos
  • D3
  • E33

Q5:

Find the directional derivative of 𝑓(𝑥,𝑦)=1𝑥+𝑦 at the point (1,1) in the direction of 𝑣=12,12.

  • A2
  • B22
  • C22
  • D2
  • E322

Q6:

Find the directional derivative of 𝑓(𝑥,𝑦)=𝑥+𝑦+4 at the point (1,1) in the direction of 𝑣=12,12.

  • A233
  • B33
  • C3
  • D322
  • E22

Q7:

Compute the gradient of 𝑓(𝑥,𝑦)=𝑥+𝑦+4.

  • A𝑥𝑥+𝑦+4,𝑦𝑥+𝑦+4
  • B𝑥𝑥+𝑦+4,𝑦𝑥+𝑦+4
  • C2𝑥𝑥+𝑦+4,2𝑦𝑥+𝑦+4
  • D2𝑦𝑥+𝑦+4,2𝑥𝑥+𝑦+4
  • E𝑦𝑥+𝑦+4,𝑥𝑥+𝑦+4

Q8:

The temperature 𝑇 of a solid is given by the function 𝑇(𝑥,𝑦,𝑧)=𝑒+𝑒+𝑒, where 𝑥, 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?

  • A𝑇 decreases the fastest in the direction of 𝑒,2𝑒,4𝑒.
  • B𝑇 decreases the fastest in the direction of 𝑒,2𝑒,4𝑒.
  • C𝑇 decreases the fastest in the direction of 𝑒,2𝑒,4𝑒.
  • D𝑇 decreases the fastest in the direction of 𝑒,2𝑒,4𝑒.

Q9:

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

  • A𝑓 increases the fastest in the direction of 29941,10941 and decreases the fastest in the direction of 29941,10941.
  • B𝑓 increases the fastest in the direction of 29941,10941 and decreases the fastest in the direction of 29941,10941.
  • C𝑓 increases the fastest in the direction of 997,497 and decreases the fastest in the direction of 997,497.
  • D𝑓 increases the fastest in the direction of 997,497 and decreases the fastest in the direction of 997,497.

Q10:

Consider the function 𝑓:, given by 𝑓(𝑥,𝑦)=𝑥𝑦sin. Which of the statements below is incorrect?

  • A𝐷𝑓(𝑥,𝑦)=𝐷𝑓(𝑥,𝑦)=0
  • B𝐷𝑓(𝑥,𝑦)=𝑦sin
  • C𝐷𝑓(𝑥,𝑦)=𝐷𝑓(𝑥,𝑦)
  • D(𝐷𝑓(𝑥,𝑦))+(𝐷𝑓(𝑥,𝑦))=1

Q11:

Compute the gradient of the function 𝑓(𝑥,𝑦)=2𝑥+5𝑦.

  • A15,12
  • B12,15
  • C2𝑥,5𝑦
  • D2,5
  • E5,2

Q12:

Find the gradient of 𝑓(𝑥,𝑦)=𝑥+𝑦1.

  • A𝑥,𝑦
  • B2𝑦1,2𝑥1
  • C2𝑥1,2𝑦1
  • D2𝑦,2𝑥
  • E2𝑥,2𝑦

Q13:

Find the gradient of 𝑓(𝑥,𝑦)=𝑥𝑦.ln

  • A1𝑥𝑦,1𝑥𝑦
  • B1𝑥,1𝑦
  • C(𝑥,𝑦)lnln
  • D(𝑥,𝑦)
  • E1𝑦,1𝑥

Q14:

Compute the gradient of the function 𝑓(𝑥,𝑦)=𝑥𝑒.

  • A2𝑒,𝑥𝑒
  • B2𝑥𝑒,𝑦𝑥𝑒
  • C2𝑥𝑒,𝑥𝑒
  • D𝑥𝑒,2𝑥𝑒
  • E𝑥𝑒,2𝑒

Q15:

Find the gradient of 𝑓(𝑥,𝑦)=1𝑥+𝑦.

  • A2𝑥(𝑥+𝑦),2𝑦(𝑥+𝑦)
  • B2𝑥(𝑥+𝑦),2𝑦(𝑥+𝑦)
  • C𝑥(𝑥+𝑦),𝑦(𝑥+𝑦)
  • D2𝑥(𝑥+𝑦),2𝑦(𝑥+𝑦)
  • E2𝑥(𝑥+𝑦),2𝑦(𝑥+𝑦)

Q16:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧.

  • A2𝑥,2𝑧,2𝑦
  • B2𝑦,2𝑥,2𝑧
  • C2,2,2
  • D𝑥,𝑦,𝑧
  • E2𝑥,2𝑦,2𝑧

Q17:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥𝑒.

  • A2𝑥𝑒,𝑥𝑧𝑒,𝑥𝑦𝑒
  • B𝑥𝑧𝑒,2𝑥𝑒,𝑥𝑦𝑒
  • C𝑥𝑧𝑒,𝑥𝑦𝑒,2𝑥𝑒
  • D𝑥𝑦𝑒,𝑥𝑧𝑒,2𝑥𝑒
  • E2𝑦𝑒,𝑦𝑧𝑒,𝑦𝑒

Q18:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥𝑦𝑧.sin

  • A𝑦𝑧𝑥𝑦𝑧,𝑥𝑧𝑥𝑦𝑧,𝑥𝑦𝑥𝑦𝑧coscoscos
  • B𝑦𝑧,𝑥𝑧,𝑥𝑦
  • C𝑥𝑧,𝑦𝑧,𝑥𝑦
  • D𝑦𝑧𝑥𝑦𝑧,𝑥𝑦𝑥𝑦𝑧,𝑥𝑧𝑥𝑦𝑧coscoscos
  • E𝑥𝑧𝑥𝑦𝑧,𝑦𝑧𝑥𝑦𝑧,𝑥𝑦𝑥𝑦𝑧coscoscos

Q19:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧.

  • A𝑥𝑥+𝑦+𝑧,𝑦𝑥+𝑦+𝑧,𝑧𝑥+𝑦+𝑧
  • B𝑦𝑥+𝑦+𝑧,𝑥𝑥+𝑦+𝑧,𝑧𝑥+𝑦+𝑧
  • C𝑥𝑥+𝑦+𝑧,𝑧𝑥+𝑦+𝑧,𝑦𝑥+𝑦+𝑧
  • D1𝑥+𝑦+𝑧,1𝑥+𝑦+𝑧,1𝑥+𝑦+𝑧
  • E𝑥𝑥+𝑦+𝑧,𝑦𝑥+𝑦+𝑧,𝑧𝑥+𝑦+𝑧

Q20:

Find a function 𝑄(𝑥,𝑦) so that the vector field F(𝑥,𝑦)=𝑥𝑦,𝑄(𝑥,𝑦) is a gradient field.

  • A𝑄(𝑥,𝑦)=𝑥𝑦
  • B𝑄(𝑥,𝑦)=𝑥
  • C𝑄(𝑥,𝑦)=𝑥𝑦2
  • D𝑄(𝑥,𝑦)=𝑥2
  • E𝑄(𝑥,𝑦)=𝑦2

Q21:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝑥+𝑦,𝑥𝑦,𝑥𝑦). Express the gradient 𝑤𝜋,23 (viewed as a 1×2 matrix) in terms of the 1×3 matrix 𝐹(𝑞), where 𝑞=𝜙𝜋,23, and a matrix of partial derivatives of 𝜙.

  • A𝑤𝜋,23=𝐹(𝑞)2𝜋432𝜋4323𝜋
  • B𝑤𝜋,23=𝐹(𝑞)2𝜋432𝜋4323𝜋
  • C𝑤𝜋,23=𝐹(𝑞)2𝜋432𝜋4323𝜋
  • D𝑤𝜋,23=𝐹(𝑞)𝜋23𝜋2323𝜋
  • E𝑤𝜋,23=𝐹(𝑞)𝜋23𝜋2323𝜋

Q22:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝜙,𝜙,𝜙) and that 𝑞=𝜙(𝑝) for a point 𝑝. Express the gradient 𝑤(𝑝) (viewed as a 1×2 matrix) in terms of the 1×3 matrix 𝐹(𝑞) and a matrix of partial derivatives of 𝜙.

  • A𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥
  • B𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦
  • C𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦
  • D𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥
  • E𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦

Q23:

Let 𝑤=𝐹(𝜙(𝑥,𝑦)) where 𝜙=𝑥+𝑦,𝑥𝑦,𝑥𝑦 and 𝐹(𝑝,𝑞,𝑟)=10𝑝+6𝑞16𝑟. Given that there is a line in the 𝑥-𝑦 plane for which 𝑤=0, find the equation of this line.

  • A𝑦=𝑥
  • B𝑦=2𝑥
  • C𝑦=𝑥2
  • D𝑦=2𝑥
  • E𝑦=𝑥

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