Rapid Recall
Consider the function $f(x,y)=x^3+3x^2+y^2+10y$ at the point $P=(-1,2)$.
- Write an equation for the line tangent to the level curve through $P$.
Solution
Another day
- Write a vector or scalar equation for the line normal to the level curve through $P$.
Solution
Another day
- Find the slope of the surface $z=f(x,y)$ at the point $P$ in the direction $\vec u = \left< 3,-4\right>$.
Solution
Another day
Group problems
- A box lies inside the rectangle $ [2,10]\times [1,5] $ (so $2\leq x\leq 10$ and $1\leq y \leq 5$ ).
- Compute the integral formula $\ds\frac{\int_2^{10}\int_1^5 x dydx}{\int_2^{10}\int_1^5 1 dydx}.$
- Compute the integral formula $\ds\frac{\int_2^{10}\int_1^5 y dydx}{\int_2^{10}\int_1^5 1 dydx}.$
- What physical quantities do the two integrals above compute?
- Compute the integral formula $\ds\frac{\int_1^5 \int_2^{10}y dxdy}{\int_1^5 \int_2^{10}1 dxdy}.$
- Draw the region described the bounds of each integral.
- $\ds\int_{0}^{2}\int_{2x}^{4}dydx$
- $\ds\int_{0}^{4}\int_{0}^{y/2}dxdy$
- $\ds\int_{0}^{3\pi/2}\int_{0}^{2+2\cos\theta}rdrd\theta$
- $\ds\int_{-3}^{3}\int_{0}^{9-x^2}\int_{0}^{5}dzdydx$
- $\ds\int_{0}^{1}\int_{0}^{1-z}\int_{0}^{\sqrt{1-x^2}}dydxdz$
- Set up an integral formula to compute each of the following:
- The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
- The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
- The $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies under the plane $2x+3y+6z=6$.
- The $y$-coordinate of the center-of-mass (so $\bar y$) of the same object.
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