11.2 2014 test

Attempt all questions.

1) Show that

[10]

2) Let $x(t)=t^{2}$, $y(t)=\frac{1}{3}t^{3}-t$.

a) Show that $(x(t),y(t))$ lies on the curve $9y^{2}=x^{3}-6x^{2}+9x$.

[2]

b) Calculate the length along the curve from $(x(0),y(0))$ to $(x(3),y(3))$.

[4]

3) State whether each of the following statements is true or false. Briefly (i.e. in about one sentence) justify your answer.

a) (No longer relevant)

b) A function $f(x,y)$ of two variables which satisfies $\frac{\partial^{2}f}{\partial x\partial y}=0$ has no saddle points.

c) (No longer relevant)

[6]

4) Find (but do not classify) the stationary points of the function $f(x,y)=x^{3}-3xy-\frac{1}{2}y^{2}$.

[4]

5) Compute the double integral

[4]

Total: 30