# 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:

True or False: If and are defined for a function and a point , then defines the tangent line to the curve at .

• AFalse
• BTrue

Q2:

We want to see what a typical tangent plane to the graph of looks like. Fix the point where . This is a point on the graph .

Every plane in that lies on has equation for numbers . For which does this equation not give a plane?

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

Under what conditions on is the plane parallel to the -plane?

• A or
• B
• C
• D
• E

Under what conditions on does the plane contain the line parallel to the -axis through ?

• A
• B
• C
• D
• E

It is not hard to see that if the plane contains any parallel to the -axis, then it must contain the parallel through . Given that this cannot happen for the graph of a differentiable function in the form , we can express the tangent plane at in the form . By considering the section of the graph in the -plane when , find .

• A
• B
• C
• D
• E