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Vector calculus: exam 2

These are exercises for Vector calculus: course.

1. Prove that the function $f(x)=||x||$ is or is not linear.

2. For the linear map $L(x_1,x_2)=(3x_1+x_2,x_1-x_2)$ find the basis of the null space (the kernel).

3. Compute the composition of the functions $L(x_1,x_2)=(3x_1+x_2,x_1-x_2)$ and $f(x_1,x_2)=2x_1+x_2$.

4. Suppose $L:R^2 \rightarrow R^2$ is linear and suppose $L(e_1)=e_1-3e_2, L(e_2)=-2e_1+e_2$. Write $L$ as a $2 \times 2$ matrix.

5. Describe and sketch the parametric curve $f(t)=(t^3-t,1-t^2)$.

6. Represent as a parametric curve ($t \in (- \infty, \infty)$) a 2D spiral approaching $0$, but never reaching it.

7. Find the tangent line to the curve $f(t)=(t,t^2,t^3)$ at the point $(1,1,1)$.

8. Find the function $f:{\bf R} \rightarrow {\bf R}^2$ such that $f'(t)=(1+e^{2t},\sin 3t)$ and $f(0)=(1,1)$.

9. Show that the parametrization of the curve $f(t)=(3\sin t,4\cos t)$ isn't natural. Find the equation for the natural parameter. Do not solve.