Due date: Before you take the final exam
Last updated Apr 18 06:20 EDT 2014
Do all of the problems below. You will not be asked to hand any in as homework, but the material they cover is fair game for the final exam.
- A. Rosenlicht Chap. IX (pp. 212-214)/ 1, 11, 12. For #1 and harder problems of the same type (more variables, more complicated formula, etc.), the following may be useful.
Recall that when a function F : Rn → R has continuous first partial derivatives at a point p, F is automatically differentiable at p. When F does not have continuous first partials at p, or if it’s hard to tell
whether F has continuous first partials at p, here is one strategy for deciding whether a function F is differentiable at p. (This strategy applies to more general F : V → W, where V and W are finite-dimensional vector spaces.)- Compute all directional derivatives (DpF)(v). If any fails to exist, F is not differentiable at p. If all exist, go to step 2.
- Determine whether the map T defined by T(v) = (DpF)(v) is linear. If T is not linear, then F is not differentiable at p. If T is linear, then T is the only candidate for the derivative of F at p; go to step 3.
- Determine whether limv→ 0 (|| F(p+v) – F(p) – T(v)|| / ||v||) = 0. (There are too many sub-strategies to list for this step. But for functions R2 → R, the methods you learned in Calculus 3 generally suffice. There is no substitute for familiarity with inequalities, familiarity with the behavior of various simple functions, and the ability to recognize relevant, simple sub-expressions in a more complicated expression.) If this limit is 0, then F is differentiable at p; if this limit does not exist, or exists but is not 0, then F is not differentiable at p.
- B. Read the first two and half pages of the handout Matrices, Power Series, and Functions of Matrices and do the following problems in that handout: 1, 2, 3, 5, 6, 7(a)-(d). Click here for additional non-book problems. Problems B2 and B3 relate to older material (nothing beyond Chapter VI is needed other than the definition of an infinite series). B3, from part (b) on, is very straightforward; you should all be able to do it and arrive at a fun fact to impress your friends with.