Lecture 11, Monday, 9/22/08: Section 12.4:

Topics

Differential of a function of several variables
Linear approximation to a function of several variables
Tangent planes and normal lines (for functions z = f(x,y))
Applications: Approximation of multivariable functions, error estimates

Read

Section 12.4. You can skip the last example (Ex. 4.6) and skim through the parts involving Delta (p. 966, Theorem 4.2, Example 4.4, and Definition 4.1). (However, you do need to know the definition of a differential given at the bottom of p. 967.)

Homework

Section 12.4: 1, 9, 13, 17, 29.
12.Review (p. 1026): 31. In addition to finding L(x,y) for this problem, also do the following: (a) find the differential dz at the given point, (b) use the differential to estimate the effect on z that increasing x from -2 to -1.9 and decreasing y from 5 to 4.9 has.
Solutions.

Notes

Tangent planes will come up again in 12.6 in a more general context, namely attached to surfaces of the form F(x,y,z)=c, where c is a constant. (The latter equation may look familiar; in fact, it is the equation of a level surface for the function F(x,y,z), a topic that came up two lectures ago.)

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Last modified Thu 25 Sep 2008 11:23:01 AM CDT