Worksheet: Tangent Planes

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In this worksheet, we will practice finding the equation of a tangent plane to a surface at a given point.

Q1:

Find the equation of the tangent plane to the surface 𝑥 + 𝑦 = 4 2 2 at the point √ 3 , 1 , 0 .

  • A 𝑦 = 3 − √ 3 𝑥
  • B 𝑥 − √ 3 = 0
  • C 𝑦 = 4 √ 3 − 𝑥
  • D 𝑦 = 4 − √ 3 𝑥
  • E √ 3 𝑦 = 4 − 𝑥

Q2:

Find the equation of the tangent plane to the surface 𝑥 4 + 𝑦 9 + 𝑧 1 6 = 1 2 2 2 at the point 1 , 2 , 2 √ 1 1 3 .

  • A 1 2 ( 𝑥 − 1 ) + 4 9 ( 𝑦 − 2 ) − √ 1 1 1 2 𝑧 − 2 √ 1 1 3 = 0
  • B 1 4 ( 𝑥 − 1 ) + 1 9 ( 𝑦 − 2 ) + 𝑧 − 2 √ 1 1 3 = 0
  • C 1 2 ( 𝑥 − 1 ) + 1 9 ( 𝑦 − 2 ) − 𝑧 − 2 √ 1 1 3 = 0
  • D 1 2 ( 𝑥 − 1 ) + 4 9 ( 𝑦 − 2 ) + √ 1 1 1 2 𝑧 − 2 √ 1 1 3 = 0
  • E 1 2 ( 𝑥 − 2 ) + 1 9 ( 𝑦 − 1 ) − √ 1 1 2 4 𝑧 − 2 √ 1 1 3 = 0

Q3:

Find the equation of the tangent plane to the surface 𝑥 + 𝑦 − 𝑧 = 0 2 2 2 at the point ( 3 , 4 , 5 ) .

  • A 3 𝑥 + 4 𝑦 − 5 𝑧 − 6 = 0
  • B 3 𝑥 + 5 𝑦 − 4 𝑧 − 9 = 0
  • C 3 𝑥 + 4 𝑦 − 𝑧 = 0
  • D 3 𝑥 + 4 𝑦 − 5 𝑧 = 0
  • E 4 𝑥 + 3 𝑦 − 5 𝑧 = 0

Q4:

Find the equation of the tangent plane to the surface 𝑧 = 𝑥 𝑒 𝑦 at the point ( 1 , 0 , 1 ) .

  • A 𝑥 − 𝑦 − 𝑧 − 2 = 0
  • B 𝑒 𝑥 + 𝑦 − 𝑧 + 1 − 𝑒 = 0
  • C 𝑒 𝑥 + 𝑦 − 𝑧 − 1 − 𝑒 = 0
  • D 𝑥 + 𝑦 − 𝑧 = 0
  • E 𝑥 − 𝑦 − 𝑧 − 1 = 0

Q5:

Find the equation of the tangent plane to the surface 𝑧 = 𝑥 + 2 𝑦 at the point ( 2 , 1 , 4 ) .

  • A 𝑥 + 2 𝑦 − 𝑧 − 8 = 0
  • B 𝑥 − 2 𝑦 − 𝑧 − 1 = 0
  • C 𝑥 − 2 𝑦 − 𝑧 − 9 = 0
  • D 𝑥 + 2 𝑦 − 𝑧 = 0
  • E 𝑥 + 2 𝑦 − 𝑧 − 4 = 0

Q6:

Find the equation of the tangent plane to the surface 𝑧 = 𝑥 𝑦 2 at the point ( − 1 , 1 , 1 ) .

  • A − 2 𝑥 + 𝑦 − 𝑧 − 4 = 0
  • B 2 𝑥 − 𝑦 − 𝑧 − 2 = 0
  • C 2 𝑥 − 𝑦 − 𝑧 − 4 = 0
  • D − 2 𝑥 + 𝑦 − 𝑧 − 2 = 0
  • E − 2 𝑥 + 𝑦 − 𝑧 = 0

Q7:

Find the equation of the tangent plane to the surface 𝑧 = √ 𝑥 + 𝑦 2 2 at the point ( 3 , 4 , 5 ) .

  • A 3 𝑥 + 4 𝑦 − 5 𝑧 − 1 0 = 0
  • B 4 𝑥 + 3 𝑦 − 5 𝑧 + 1 = 0
  • C 4 𝑥 + 3 𝑦 − 5 𝑧 − 4 9 = 0
  • D 3 𝑥 + 4 𝑦 − 5 𝑧 = 0
  • E 3 𝑥 + 4 𝑦 − 5 𝑧 − 2 5 = 0

Q8:

Find the equation of the tangent plane to the surface 𝑥 + 𝑦 + 𝑧 = 9 2 2 2 at the point ( 0 , 0 , 3 ) .

  • A 𝑥 = 0
  • B 𝑦 = 0
  • C 𝑧 − 3 = 2 𝑥 + 2 𝑦
  • D 𝑧 − 3 = 0
  • E 𝑧 + 3 = 2 𝑥 + 3 𝑦

Q9:

Find the equation of the tangent plane to the surface 𝑧 = 𝑥 𝑦 at the point ( 1 , − 1 , − 1 ) .

  • A 𝑦 − 𝑥 − 𝑧 + 3 = 0
  • B 𝑦 − 𝑥 − 𝑧 − 3 = 0
  • C 𝑥 − 𝑦 − 𝑧 + 3 = 0
  • D 𝑦 − 𝑥 − 𝑧 + 1 = 0
  • E 𝑦 − 𝑥 − 𝑧 = 0

Q10:

Find the equation of the tangent plane to the surface 𝑧 = 𝑥 + 𝑦 2 3 at the point ( 1 , 1 , 2 ) .

  • A 2 𝑥 + 6 𝑦 − 𝑧 − 6 = 0
  • B 2 𝑥 + 3 𝑦 − 𝑧 + 1 = 0
  • C 2 𝑥 + 6 𝑦 − 𝑧 − 2 = 0
  • D 2 𝑥 + 3 𝑦 − 𝑧 − 3 = 0
  • E 2 𝑥 + 3 𝑦 − 𝑧 = 0

Q11:

True or False: If 𝐴 = 𝜕 𝑓 𝜕 𝑥 ( 𝑎 , 𝑏 ) and 𝐵 = 𝜕 𝑓 𝜕 𝑦 ( 𝑎 , 𝑏 ) are defined for a function 𝑓 ∶ ℝ → ℝ and a point ( 𝑎 , 𝑏 ) , then 𝐴 ( 𝑥 − 𝑎 ) + 𝐵 ( 𝑦 − 𝑏 ) = 0 defines the tangent line to the curve 𝑓 ( 𝑥 , 𝑦 ) = 0 at ( 𝑎 , 𝑏 ) .

  • AFalse
  • BTrue

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