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Calculus III -- Fall 2017 -- final

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$\bullet$ 1. Sketch the graph of this function: $$f(x,y)=\sqrt{x^2+(y−2)^2−y}.$$

$\bullet$ 2. Find the directional derivative of the function above in the direction of the line $y=x$.

$\bullet$ 3. Suppose $h(t)=(\sin e^t,\cos e^t)$ is a parametric curve. (a) What is its path? (b) Show how the Chain Rule is used to compute its derivative.

$\bullet$ 4. Sketch the vector field given below and estimate its line integral along the boundary of the square oriented counterclockwise (multiple answers are possible): $$\begin{array}{llll} F(0,0)=<1,1>,& F(.5,0)=<0,1>,& F(1,0)=<1,1>,& F(1,.5)=<-1,1>,\\ F(1,1)=<-1,0>,& F(.5,1)=<0,0>,& F(0,1)=<2,1>,& F(0,.5)=<-1,-1>. \end{array}$$

$\bullet$ 5. A mountain ridge has three peaks with two passes between them. Sketch the level curves of the function that represents the terrain.

$\bullet$ 6. Find the best linear approximation of the function $f(x,y)=xe^y$ at the point $(1,1)$.

$\bullet$ 7. Set up, but do not evaluate, an integral that represents the volume of the region bounded by the surface $z=1−x^2$, the $xy$-plane, and the planes $y=0$ and $y=1$.

$\bullet$ 8. In the formula of Green's Theorem shown below, identify all of its parts (such as "$F$ is..."): $$\oint_C F\cdot dP=\iint_D (q_x-p_y)\, dA.$$

$\bullet$ 9. Define the gradient of a function. What does it tell us about the function?

$\bullet$ 10. Define the path-independence property. Does the vector field shown below satisfy the path-independence property? Explain.

Swirly vector field.png