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Vector calculus: exam 3

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These are exercises for Vector calculus: course.


1. Let $$f(x,y)= \sqrt{x^2+(y-2)^2-4}+1.$$

  • a. Find the domain of $f$.
  • b. Sketch the graph of $f$.
  • c. What is it?

2. Using the $\epsilon - \delta$ definition, prove that the function $f(x)=||x||,x \in R^n$, is continuous.

3. Suppose $A$ and $B$ are path-connected subsets of $R^n$ and suppose $a \in A \cap B$. Prove that $A \cup B$ is path-connected.

4. Give an example of a function $z=f(x,y)$ such that $\frac{\partial f}{\partial x}(0,0)$ exists but $\frac{\partial f}{\partial y}(0,0)$ does not.

5. From the definition, compute the directional derivative of the function $$f(x,y)=xy+y^2$$ at the point $a=(2,1)$ in the direction $v=(1,1)$.

6. From the definition, prove that $T(x)=-2x_2-1$ is the best affine approximation of the function $$f(x)=x_1^4+x_2^2,$$ where $x=(x_1,x_2)$, at the point $a=(0,-1)$.

7. Represent the function $$f(t)=(\sin e^{t},\cos e^{t})$$ as the composition of two functions and then use the Chain Rule to compute its derivative at $t_0=0$.

8. Find all critical points of the function $$f(x,y,z)=xz+5y^2.$$

9. Classify these subsets of ${\bf R}^3$ as closed, open, bounded, path-connected or not (no proof required):

  • a. $\{(x,y,z):x^2+y^2+z^2=1$ and $0<x<5\},$
  • b. the complement of the union of the three axes,
  • c. $\{(x,y,z):x^2+y^2=1$ and $z^2=1\}$.