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Worksheet: Tangent Planes

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 𝑧 = π‘₯ 𝑦 at the point ( 1 , βˆ’ 1 , βˆ’ 1 ) .

  • A 𝑦 βˆ’ π‘₯ βˆ’ 𝑧 + 3 = 0
  • B 𝑦 βˆ’ π‘₯ βˆ’ 𝑧 βˆ’ 3 = 0
  • C π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 + 3 = 0
  • D 𝑦 βˆ’ π‘₯ βˆ’ 𝑧 + 1 = 0
  • E 𝑦 βˆ’ π‘₯ βˆ’ 𝑧 = 0

Q2:

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

  • A π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 βˆ’ 2 = 0
  • B 𝑒 π‘₯ + 𝑦 βˆ’ 𝑧 + 1 βˆ’ 𝑒 = 0
  • C 𝑒 π‘₯ + 𝑦 βˆ’ 𝑧 βˆ’ 1 βˆ’ 𝑒 = 0
  • D π‘₯ + 𝑦 βˆ’ 𝑧 = 0
  • E π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 βˆ’ 1 = 0

Q3:

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

  • A π‘₯ + 2 𝑦 βˆ’ 𝑧 βˆ’ 8 = 0
  • B π‘₯ βˆ’ 2 𝑦 βˆ’ 𝑧 βˆ’ 1 = 0
  • C π‘₯ βˆ’ 2 𝑦 βˆ’ 𝑧 βˆ’ 9 = 0
  • D π‘₯ + 2 𝑦 βˆ’ 𝑧 = 0
  • E π‘₯ + 2 𝑦 βˆ’ 𝑧 βˆ’ 4 = 0

Q4:

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

  • A 2 π‘₯ + 6 𝑦 βˆ’ 𝑧 βˆ’ 6 = 0
  • B 2 π‘₯ + 3 𝑦 βˆ’ 𝑧 + 1 = 0
  • C 2 π‘₯ + 6 𝑦 βˆ’ 𝑧 βˆ’ 2 = 0
  • D 2 π‘₯ + 3 𝑦 βˆ’ 𝑧 βˆ’ 3 = 0
  • E 2 π‘₯ + 3 𝑦 βˆ’ 𝑧 = 0

Q5:

Find the equation of the tangent plane to the surface π‘₯ + 𝑦 βˆ’ 𝑧 = 0 2 2 2 at the point ( 3 , 4 , 5 ) .

  • A 3 π‘₯ + 4 𝑦 βˆ’ 5 𝑧 βˆ’ 6 = 0
  • B 3 π‘₯ + 5 𝑦 βˆ’ 4 𝑧 βˆ’ 9 = 0
  • C 3 π‘₯ + 4 𝑦 βˆ’ 𝑧 = 0
  • D 3 π‘₯ + 4 𝑦 βˆ’ 5 𝑧 = 0
  • E 4 π‘₯ + 3 𝑦 βˆ’ 5 𝑧 = 0

Q6:

Find the equation of the tangent plane to the surface 𝑧 = π‘₯ 𝑦 2 at the point ( βˆ’ 1 , 1 , 1 ) .

  • A βˆ’ 2 π‘₯ + 𝑦 βˆ’ 𝑧 βˆ’ 4 = 0
  • B 2 π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 βˆ’ 2 = 0
  • C 2 π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 βˆ’ 4 = 0
  • D βˆ’ 2 π‘₯ + 𝑦 βˆ’ 𝑧 βˆ’ 2 = 0
  • E βˆ’ 2 π‘₯ + 𝑦 βˆ’ 𝑧 = 0

Q7:

Find the equation of the tangent plane to the surface π‘₯ 4 + 𝑦 9 + 𝑧 1 6 = 1 2 2 2 at the point ο€Ώ 1 , 2 , 2 √ 1 1 3  .

  • A 1 2 ( π‘₯ βˆ’ 1 ) + 4 9 ( 𝑦 βˆ’ 2 ) βˆ’ √ 1 1 1 2 ο€Ώ 𝑧 βˆ’ 2 √ 1 1 3  = 0
  • B 1 4 ( π‘₯ βˆ’ 1 ) + 1 9 ( 𝑦 βˆ’ 2 ) + ο€Ώ 𝑧 βˆ’ 2 √ 1 1 3  = 0
  • C 1 2 ( π‘₯ βˆ’ 1 ) + 1 9 ( 𝑦 βˆ’ 2 ) βˆ’ ο€Ώ 𝑧 βˆ’ 2 √ 1 1 3  = 0
  • D 1 2 ( π‘₯ βˆ’ 1 ) + 4 9 ( 𝑦 βˆ’ 2 ) + √ 1 1 1 2 ο€Ώ 𝑧 βˆ’ 2 √ 1 1 3  = 0
  • E 1 2 ( π‘₯ βˆ’ 2 ) + 1 9 ( 𝑦 βˆ’ 1 ) βˆ’ √ 1 1 2 4 ο€Ώ 𝑧 βˆ’ 2 √ 1 1 3  = 0

Q8:

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

  • A 3 π‘₯ + 4 𝑦 βˆ’ 5 𝑧 βˆ’ 1 0 = 0
  • B 4 π‘₯ + 3 𝑦 βˆ’ 5 𝑧 + 1 = 0
  • C 4 π‘₯ + 3 𝑦 βˆ’ 5 𝑧 βˆ’ 4 9 = 0
  • D 3 π‘₯ + 4 𝑦 βˆ’ 5 𝑧 = 0
  • E 3 π‘₯ + 4 𝑦 βˆ’ 5 𝑧 βˆ’ 2 5 = 0

Q9:

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

  • A π‘₯ = 0
  • B 𝑦 = 0
  • C 𝑧 βˆ’ 3 = 2 π‘₯ + 2 𝑦
  • D 𝑧 βˆ’ 3 = 0
  • E 𝑧 + 3 = 2 π‘₯ + 3 𝑦

Q10:

True or False: If 𝐴 = πœ• 𝑓 πœ• π‘₯ ( π‘Ž , 𝑏 ) and 𝐡 = πœ• 𝑓 πœ• 𝑦 ( π‘Ž , 𝑏 ) are defined for a function 𝑓 ∢ ℝ β†’ ℝ  and a point ( π‘Ž , 𝑏 ) , then 𝐴 ( π‘₯ βˆ’ π‘Ž ) + 𝐡 ( 𝑦 βˆ’ 𝑏 ) = 0 defines the tangent line to the curve 𝑓 ( π‘₯ , 𝑦 ) = 0 at ( π‘Ž , 𝑏 ) .

  • AFalse
  • BTrue

Q11:

Find the equation of the tangent plane to the surface π‘₯ + 𝑦 = 4 2 2 at the point ο€» √ 3 , 1 , 0  .

  • A 𝑦 = 3 βˆ’ √ 3 π‘₯
  • B π‘₯ βˆ’ √ 3 = 0
  • C 𝑦 = 4 √ 3 βˆ’ π‘₯
  • D 𝑦 = 4 βˆ’ √ 3 π‘₯
  • E √ 3 𝑦 = 4 βˆ’ π‘₯

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