EXERCISES

1. (5) Let $g(x) = \Vert 7x+e \Vert$, where $x,e \in \Re^n$ and $e$ is the vector of ones. Calculate the gradient of $g$, $\nabla g(x)$, when $7x+e \neq 0$.
2. (10) Suppose that $g: \Re^n \rightarrow \Re$, $a,b \in \Re^n$, and $f(t) := g(a-tb)$. Calculate the derivative $f^\prime (t).$

3. Write down the Taylor series of:
   1. (5)
      
      $$f(x) = x^5+x^3, \mbox{~about~} x=1.$$
      
      

   2. (5)
      
      $$f(x) = \log(1+x), \mbox{~about~} x=0.$$
      
      

   3. (5)
      
      $$f(x) = x\cos(x), \mbox{~about~} x=0.$$