Second Exam

1. 1. Suppose that $z = \sin (x/y)$ , where $x=2t$ and $y=1-t^{2}$ ; find $${\frac{d z}{d t}}$$ .
   2. Suppose that $z = x e^{-y} + y e^{-x}$ , where $x= u \sin v$ and $y=v \cos u$ ; find $${\frac{\partial z}{\partial u}}$$ and $${\frac{\partial z}{\partial v}}$$ .
2. Let $\mathbf{a} = \langle 0 , 2, 1 \rangle$ , $\mathbf{b} = \langle -3, 5, 4 \rangle$ and $\mathbf{z} = \langle 1, -3, -1 \rangle$ .
   1. $2 \, \mathbf{a} + 7 \, \mathbf{b} - 5 \, \mathbf{z}$
   2. $\vert\vert z\vert\vert$
3. A boat is heading due east at $25$ km/hr (relative to the water). The current is moving toward the southwest at $10$ km/hr.
   1. How fast is the boat going, relative to the ground?
   2. By what angle does the current push the boat off its due east course?
4. Find the angle between the planes $5(x-1) + 3 (y+2) + 2 z = 0$ and $x + 3 (y-1) + 2(z+4) = 0$ .
5. For $\mathbf{a} = 3 \, \hat{\mathbf{i}} + \hat{\mathbf{j}} - \hat{\mathbf{k}}$ and $\mathbf{b} = \hat{\mathbf{i}} -4 \, \hat{\mathbf{j}} + 2 \, \hat{\mathbf{k}}$ , find $\mathbf{a} \times \mathbf{b}$ .
6. Find an equation for the plane through the points $(3,4,2)$ , $(-2,1,0)$ and $(0,2,1)$ .
7. Find the directional derivative of the function $f(x,y,z)=xy+z^{3}$ at the point $(1,1,0)$ in the direction of the vector $\mathbf{v} = \hat{\mathbf{j}} + \hat{\mathbf{k}}$ .
8. Let $z=f(x,y)=x^{2} + 3 x y - y^{2}$ .
   1. Find the differential $dz$ .
   2. If $x$ changes from $2$ to $2.05$ and $y$ changes from $3$ to $2.96$ , what is the value of $dz$ ?
9. Find the equation of the tangent plane at the point $(-1,1,2)$ to the surface $x^{2}-xyz=3$ .