# Practice for the final

You should know how to find the intersection between all kinds of curves. Even when I don't ask you these sorts of questions explicitly, they are usually equivalent to one of the steps of a problem where you had to use solve or fsolve. Find the intersection of the following pairs of curves (either exactly or up to 6 decimal places). You will probably want to graph them too.:
1. $x^2 - y^2+x*y +y = 3$ and $y^2 = x + 2$ in the range $-4 \leq x,y \leq 4$
2. $y^2 = x + 2$ in the range $-4 \leq x,y \leq 4$ and the parametric curve $(3cos(t),sin(t) + cos(2t))$ in the range $0 \leq t \leq 2\pi$
3. the parametric curve $(3cos(t),sin(t) + cos(2t))$ and the curve $(sin(t) + cos(2*t)-1,3*cos(t))$ in the range $0 \leq t \leq 2\pi$.
4. the parametric curve $(sin(t) + cos(2*t)-1,3*cos(t))$ in the range $0 \leq t \leq 2\pi$ and the polar curve $r = abs(1 + 3*sin(theta))$ for $0 \leq \theta \leq 2\pi$
5. the polar curve $r = abs(1 + 3*sin(theta))$ and the polar curve $r = 1 + sin(8*theta)/3$ for $0 \leq \theta \leq 2\pi$
6. the polar curve $r = 1 + sin(8*theta)/3$ for $0 \leq \theta \leq 2\pi$ and the hyperbola $y^2 - x = 2$ in the range $-4 \leq x,y \leq 4$

7. Find the intersection in three dimensions of the parametric curve $(sin(t) + cos(2*t)-1,3*cos(t),sin(t))$ in the range $0 \leq t \leq 2\pi$, and the surface $z = 2x + sin(y)$ for $-4 \leq x,y \leq 4$.

8. Use Lagrange multipliers to find the extrema of the equation $z = 2x + sin(y)$ subject to the constraint $x^2 - y^2+x*y +y = 3$.

9. Write a function which takes a list $L$ of numbers and creates a new list $T$ where the first entries are the same (that is $L_1 = T_1$) and $T_i = L_i + L_{i-1} + T_{i-1}$ for $2 \leq i \leq nops(L)$. For instance if your function is called blah, then you if you calculate blah([1,2,3,4,5,6,7]); should return [1, 4, 9, 16, 25, 36, 49] and blah([7,6,5,4,3,2,1]); should return [7, 20, 31, 40, 47, 52, 55].

10. Find the length and the area inside the polar curve $r = 1 + sin(n \theta/2)/3$ for $1 \leq n \leq 8$.

Plus you should go back to the problems that we solved in assignments and quizzes and you should be able to do calculations similar to those.