Lesson: Tangent Planes and Linear approximation

Mathematics

In this lesson, we will learn how to find the equation of a tangent plane to a surface at a point and how to use tangent planes for linear approximation of functions of two variables.

Worksheet

Q1:

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).

Q2:

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?

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

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)?

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.

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