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.

  • A𝑦=33𝑥
  • B𝑥3=0
  • C𝑦=43𝑥
  • D3𝑦=4𝑥
  • E𝑦=43𝑥

Q2:

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

  • A12(𝑥1)+49(𝑦2)+1112𝑧2113=0
  • B14(𝑥1)+19(𝑦2)+𝑧2113=0
  • C12(𝑥1)+49(𝑦2)1112𝑧2113=0
  • D12(𝑥2)+19(𝑦1)1124𝑧2113=0
  • E12(𝑥1)+19(𝑦2)𝑧2113=0

Q3:

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

  • A3𝑥+5𝑦4𝑧9=0
  • B4𝑥+3𝑦5𝑧=0
  • C3𝑥+4𝑦𝑧=0
  • D3𝑥+4𝑦5𝑧6=0
  • E3𝑥+4𝑦5𝑧=0

Q4:

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

  • A𝑥+𝑦𝑧=0
  • B𝑒𝑥+𝑦𝑧1𝑒=0
  • C𝑥𝑦𝑧2=0
  • D𝑥𝑦𝑧1=0
  • E𝑒𝑥+𝑦𝑧+1𝑒=0

Q5:

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

  • A𝑥+2𝑦𝑧=0
  • B𝑥+2𝑦𝑧8=0
  • C𝑥2𝑦𝑧1=0
  • D𝑥+2𝑦𝑧4=0
  • E𝑥2𝑦𝑧9=0

Q6:

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

  • A2𝑥𝑦𝑧2=0
  • B2𝑥𝑦𝑧4=0
  • C2𝑥+𝑦𝑧=0
  • D2𝑥+𝑦𝑧4=0
  • E2𝑥+𝑦𝑧2=0

Q7:

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

  • A3𝑥+4𝑦5𝑧=0
  • B3𝑥+4𝑦5𝑧10=0
  • C4𝑥+3𝑦5𝑧+1=0
  • D3𝑥+4𝑦5𝑧25=0
  • E4𝑥+3𝑦5𝑧49=0

Q8:

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

  • A𝑧+3=2𝑥+3𝑦
  • B𝑦=0
  • C𝑧3=0
  • D𝑥=0
  • E𝑧3=2𝑥+2𝑦

Q9:

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

  • A𝑦𝑥𝑧+1=0
  • B𝑦𝑥𝑧=0
  • C𝑦𝑥𝑧3=0
  • D𝑦𝑥𝑧+3=0
  • E𝑥𝑦𝑧+3=0

Q10:

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

  • A2𝑥+6𝑦𝑧6=0
  • B2𝑥+3𝑦𝑧+1=0
  • C2𝑥+3𝑦𝑧=0
  • D2𝑥+3𝑦𝑧3=0
  • E2𝑥+6𝑦𝑧2=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?

  • A𝑅0 or 𝑆0
  • B𝑅=0
  • C𝑇=0
  • D𝑅=𝑆=0
  • E𝑆=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)?

  • A𝑅=𝑆=0
  • B𝑅=0
  • C𝑅+𝑆+𝑇=0
  • D𝑆=0
  • E𝑇=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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