# Lesson Worksheet: Tangent Planes and Linear approximation Mathematics

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