Worksheet: Directional Derivatives and Gradient

Download

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+𝑦−1 at the point (1,1) in the direction of 𝑣=1√2,1√2.

  • A√2
  • B3√2
  • C4√2
  • D4
  • E2√2

Q2:

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

  • A2𝑒√2
  • B2√2𝑒
  • C2𝑒√2
  • D3𝑒√2
  • E3𝑒√2

Q3:

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

  • A2𝑒√3
  • B4𝑒
  • C3𝑒√3
  • D4𝑒√3
  • E𝑒√3

Q4:

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

  • A3√31cos
  • Bcos1√3
  • C√31cos
  • D√3
  • E3√3

Q5:

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

  • A√2
  • B√22
  • C−√22
  • D−√2
  • E−3√22

Q6:

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

  • A2√33
  • B√33
  • C√3
  • D3√22
  • E√22

Q7:

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

  • A𝑥√𝑥+𝑦+4,𝑦√𝑥+𝑦+4
  • B𝑥√𝑥+𝑦+4,𝑦√𝑥+𝑦+4
  • C2𝑥√𝑥+𝑦+4,2𝑦√𝑥+𝑦+4
  • D2𝑦√𝑥+𝑦+4,2𝑥√𝑥+𝑦+4
  • E𝑦√𝑥+𝑦+4,𝑥√𝑥+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 29√941,10√941 and decreases the fastest in the direction of −29√941,−10√941.
  • B𝑓 increases the fastest in the direction of −29√941,−10√941 and decreases the fastest in the direction of 29√941,10√941.
  • C𝑓 increases the fastest in the direction of 9√97,4√97 and decreases the fastest in the direction of −9√97,−4√97.
  • D𝑓 increases the fastest in the direction of −9√97,−4√97 and decreases the fastest in the direction of 9√97,4√97.

Q10:

Consider the function 𝑓ℝ→ℝ:, 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
  • C⟨2𝑥,5𝑦⟩
  • D⟨2,5⟩
  • E⟨5,2⟩

Q12:

Find the gradient of 𝑓(𝑥,𝑦)=𝑥+𝑦−1.

  • A⟨𝑥,𝑦⟩
  • B⟨2𝑦−1,2𝑥−1⟩
  • C⟨2𝑥−1,2𝑦−1⟩
  • D⟨2𝑦,2𝑥⟩
  • E⟨2𝑥,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.

  • A⟨2𝑒,𝑥𝑒⟩
  • B2𝑥𝑒,𝑦𝑥𝑒
  • C2𝑥𝑒,𝑥𝑒
  • D𝑥𝑒,2𝑥𝑒
  • E⟨𝑥𝑒,2𝑒⟩

Q15:

Find the gradient of 𝑓(𝑥,𝑦)=1𝑥+𝑦.

  • A2𝑥(𝑥+𝑦),2𝑦(𝑥+𝑦)
  • B−2𝑥(𝑥+𝑦),−2𝑦(𝑥+𝑦)
  • C−𝑥(𝑥+𝑦),−𝑦(𝑥+𝑦)
  • D2𝑥(𝑥+𝑦),−2𝑦(𝑥+𝑦)
  • E−2𝑥(𝑥+𝑦),2𝑦(𝑥+𝑦)

Q16:

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

  • A⟨2𝑥,2𝑧,2𝑦⟩
  • B⟨2𝑦,2𝑥,2𝑧⟩
  • C⟨2,2,2⟩
  • D⟨𝑥,𝑦,𝑧⟩
  • E⟨2𝑥,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

  • A⟨𝑦𝑧𝑥𝑦𝑧,𝑥𝑧𝑥𝑦𝑧,𝑥𝑦𝑥𝑦𝑧⟩coscoscos
  • B⟨𝑦𝑧,𝑥𝑧,𝑥𝑦⟩
  • C⟨𝑥𝑧,𝑦𝑧,𝑥𝑦⟩
  • D⟨𝑦𝑧𝑥𝑦𝑧,𝑥𝑦𝑥𝑦𝑧,𝑥𝑧𝑥𝑦𝑧⟩coscoscos
  • E⟨𝑥𝑧𝑥𝑦𝑧,𝑦𝑧𝑥𝑦𝑧,𝑥𝑦𝑥𝑦𝑧⟩coscoscos

Q19:

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

  • A𝑥𝑥+𝑦+𝑧,𝑦𝑥+𝑦+𝑧,𝑧𝑥+𝑦+𝑧
  • B𝑦√𝑥+𝑦+𝑧,𝑥√𝑥+𝑦+𝑧,𝑧√𝑥+𝑦+𝑧
  • C𝑥√𝑥+𝑦+𝑧,𝑧√𝑥+𝑦+𝑧,𝑦√𝑥+𝑦+𝑧
  • D1√𝑥+𝑦+𝑧,1√𝑥+𝑦+𝑧,1√𝑥+𝑦+𝑧
  • E𝑥√𝑥+𝑦+𝑧,𝑦√𝑥+𝑦+𝑧,𝑧√𝑥+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,𝑥−𝑦,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.

  • A∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣2𝜋432𝜋−43−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • B∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣2𝜋−432𝜋−43−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • C∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣2𝜋−432𝜋43−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • D∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣𝜋−23𝜋23−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • E∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣𝜋−23𝜋−23−23𝜋⎤⎥⎥⎥⎥⎥⎦

Q22:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝜙,𝜙,𝜙) and that 𝑞=𝜙(𝑝) for a point 𝑝∈ℝ. 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.

  • A∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎣𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥⎤⎥⎥⎥⎦
  • B∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎣𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦⎤⎥⎥⎥⎦
  • C∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦⎤⎥⎥⎥⎥⎥⎥⎦
  • D∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥⎤⎥⎥⎥⎥⎥⎥⎦
  • E∇𝑤(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,𝑥−𝑦,𝑥𝑦 and 𝐹(𝑝,𝑞,𝑟)=10𝑝+6𝑞−16𝑟. 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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.