Vector calculus: exam 4

Grading

These are exercises for Vector calculus: course.

1. For the following functions $F$ and sets $D$, sketch $F(D)$:

• a. $F(x)=\sin x, D=[0,\frac{\pi}{2}) \cup [{\pi},2{\pi}];$
• b. $F(x,y)=x^2+y^2, D=\{(x,y):|x|{\leq}1,|y|{\leq}1\};$
• c. $F(x)=(2\cos x,2\sin x), D=[0,\frac{\pi}{2}];$
• d. $F(x,y)=(x+y,x-y), D=\{(x,y):0{\leq}x{\leq}1,0{\leq}y{\leq}1\}.$

2. Find the matrix of the total derivative of $f(x,y)=(x\sin y,x-y)$ at $(1,0).$

3. Suppose $f$ is a differentiable function with $f'(0)=3$. Find the derivative of $g(x,y)=f(x^2+x+y^2+y)$ at $(0,0)$.

4. Find the tangent plane to the surface $f(x,y)=\sin (x-y)$ at $(\frac{\pi}{2},0,1)$.

5. Estimate the integral $$\displaystyle\int\int_{B}x^2ydA,$$ where $B={(x,y):0{\leq}x{\leq}2,0{\leq}y{\leq}3}$, by providing a Riemann sum with $6$ squares.

6. Evaluate the integral $$\displaystyle\int\int\int_{B}xyzdV,$$ where $B=\{(x,y):0{\leq}x{\leq}1,0{\leq}y{\leq}1,0{\leq}z{\leq}1 \}.$

7. By using only the properties of the integral, compute $\displaystyle\int\int_{D}f(x,y)dA$, where $D$ is the disk $x^2+y^2{\leq}4$ and $f(x,y)=2$ if $-2{\leq}x{\leq}0$, $f(x,y)=-1$ if $0{\leq}x{\leq}2$, $f(x,y)=55$ for all other values of $(x,y)$.

8. By using polar coordinates, compute the volume of the solid bounded by the cylinder $x^2+y^2=1$ and the planes $z=0$ and $z=1$.