Worksheet: Tangent Planes and Linear approximation

In this worksheet, we will practice finding the equation of a tangent plane to a surface at a point and using tangent planes for linear approximation of functions of two variables.

Q1:

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

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

Q2:

Find the equation of the tangent plane to the surface 𝑥4+𝑦9+𝑧16=1 at the point 1,2,2113.

  • 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 ) + 4 9 ( 𝑦 2 ) 1 1 1 2 𝑧 2 1 1 3 = 0
  • D 1 2 ( 𝑥 2 ) + 1 9 ( 𝑦 1 ) 1 1 2 4 𝑧 2 1 1 3 = 0
  • E 1 2 ( 𝑥 1 ) + 1 9 ( 𝑦 2 ) 𝑧 2 1 1 3 = 0

Q3:

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

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

Q4:

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

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

Q5:

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

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

Q6:

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

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

Q7:

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

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

Q8:

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

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

Q9:

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

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

Q10:

Find the equation of the tangent plane to the surface 𝑧=𝑥+𝑦 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 + 3 𝑦 𝑧 = 0
  • D 2 𝑥 + 3 𝑦 𝑧 3 = 0
  • E 2 𝑥 + 6 𝑦 𝑧 2 = 0

Q11:

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

  • AFalse
  • BTrue

Q12:

We want to see what a typical tangent plane to the graph of 𝑧=𝑓(𝑥,𝑦) looks like. Fix the point (𝑎,𝑏,𝑐) where 𝑐=𝑓(𝑎,𝑏). This is a point on the graph 𝑧=𝑓(𝑥,𝑦).

Every plane in 𝑅 that lies on (𝑎,𝑏,𝑐) has equation 𝑅(𝑥𝑎)+𝑆(𝑦𝑏)+𝑇(𝑧𝑐)=0 for numbers 𝑅,𝑆,𝑇. For which (𝑅,𝑆,𝑇) does this equation not give a plane?

  • AWhen one of 𝑅,𝑆,𝑇 is zero
  • BWhen 𝑅=0
  • CWhen all of 𝑅,𝑆,𝑇 are zero
  • DWhen 𝑇=0
  • ENever, this will always give a plane.

Under what conditions on 𝑅,𝑆,𝑇 is the plane 𝑅(𝑥𝑎)+𝑆(𝑦𝑏)+𝑇(𝑧𝑐)=0 parallel to the 𝑥𝑦-plane?

  • A 𝑅 0 or 𝑆0
  • B 𝑅 = 0
  • C 𝑇 = 0
  • D 𝑅 = 𝑆 = 0
  • E 𝑆 = 0

Under what conditions on 𝑅,𝑆,𝑇 does the plane 𝑅(𝑥𝑎)+𝑆(𝑦𝑏)+𝑇(𝑧𝑐)=0 contain the line parallel to the 𝑧-axis through (𝑎,𝑏,𝑐)?

  • A 𝑅 = 𝑆 = 0
  • B 𝑅 = 0
  • C 𝑅 + 𝑆 + 𝑇 = 0
  • D 𝑆 = 0
  • E 𝑇 = 0

It is not hard to see that if the plane 𝑅(𝑥𝑎)+𝑆(𝑦𝑏)+𝑇(𝑧𝑐)=0 contains any parallel to the 𝑧-axis, then it must contain the parallel through (𝑎,𝑏,𝑐). Given that this cannot happen for the graph of a differentiable function in the form 𝑧=𝑓(𝑥,𝑦), we can express the tangent plane at (𝑎,𝑏,𝑐) in the form 𝐴(𝑥𝑎)+𝐵(𝑦𝑏)+(𝑧𝑐)=0. By considering the section of the graph in the 𝑥𝑧-plane when 𝑦=𝑏, find 𝐴.

  • A 𝜕 𝑓 𝜕 𝑦 ( 𝑎 , 𝑏 ) 𝜕 𝑓 𝜕 𝑥 ( 𝑎 , 𝑏 )
  • B 𝜕 𝑓 𝜕 𝑦 ( 𝑎 , 𝑏 )
  • C 𝜕 𝑓 𝜕 𝑦 ( 𝑎 , 𝑏 )
  • D 𝜕 𝑓 𝜕 𝑥 ( 𝑎 , 𝑏 )
  • E 𝜕 𝑓 𝜕 𝑥 ( 𝑎 , 𝑏 )

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