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Advanced Calculus II -- Spring 2017 -- final exam

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MATH 528 -- Spring 2017 -- final exam

Name:_________________________ $\qquad$ 6 problems, 60 points total


  • Write the problems in the given order, each problem on a separate page.
  • Show enough work to justify your answers.


$\bullet$ 1. From the definition, prove that l.u.b. of the set $S=\{x=-1/n: n\in {\bf N}\}$ is $0$.

$\bullet$ 2. Suppose $d$ is the standard Euclidean metric of ${\bf R}^2$ while $d_1$ is the taxicab metric: $$d_1\left( (x,y),(u,v) \right)=|u-x|+|v-y|.$$ Do one of the two: (a) prove that if a sequence converges with respect to $d$, it also converges with respect to $d_1$, or (b) prove the converse.

$\bullet$ 3. Prove that a compact set is closed.

$\bullet$ 4. (a) State the definition of a differentiable function $f:{\bf R}^N\to {\bf R}$. (b) Prove that every differentiable function is continuous.

$\bullet$ 5. (a) State the Mean Value Theorem for dimension $N$. (b) Derive from it the Mean Value Theorem for dimension $1$.

$\bullet$ 6. The formula for the Taylor polynomial of $n$th degree of function $f$ at $a\in {\bf R}^N$ is: $$T_n(x)=\sum_{|\alpha|\le n}\frac{1}{\alpha !}D_\alpha f(a)(x-a)^\alpha.$$ (a) Explain the terms in the formula. (b) Compute $T_2$ for $f(x,y,z)=x^2e^yz$ at $0$.