- For each angle below, draw two vectors (label them $\vec F$ and $\vec d$) with the given angle between them. Then draw the projection of $\vec F$ onto $\vec d$. End by stating whether the work done by $\vec F$ as an object moves through the displacement $\vec d$ is positive, negative, or zero.
- 0, 60, 90, 135, 180 degrees.
- Find the point on the curve $y=\frac{4}{x^2}$ that is closest the origin. (State $f$, $g$, $\vec \nabla f$, $\vec \nabla g$, and the three equations you will need to solve using Lagrange multipliers. Then solve the system. You should get $x=\pm \sqrt[3]{2}$.
- Find the work done by $\vec F = (2x+2y, -2x)$ on an object that moves around a circle of radius 3 in the counter-clockwise direction.
- For the curve $\vec r(t) = (\cos t,\sin t)$ for $0\leq t\leq \pi$, and for the function $f(x,y) = x+y$, compute the integral $\int_Cf ds = \int_a^b f(\vec r(t))\left|\frac{d\vec r}{dt}\right|dt$.
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