Name: Calculus III
Final Exam Problems
Top Sekrit!

Instructions: Show all work to receive full or partial credit. You may use a scientific calculator/graphing calculator. All University rules and guidelines for student conduct are applicable.

1.

Find the angle between vectors $\langle0, 1, -2 \rangle$ and $\langle-4, 1, 1 \rangle$.

2.

If

$$f(x,y) = e^x \cos(xy),$$

calculate $\nabla f$, $\frac{\partial f^2}{\partial x^2}$ and $\frac{\partial f^2}{\partial x \partial y}$.

3.

Find the equation of the plane passing through the points (0,0,1), (3,2,1) and (6,0,1).

4.

What is the area of the triangle with vertices (0,1,2), (3,-1,2), (3,3,-1)?

5.

Find the maximum and minimum value of

f(x,y) = x² y + 3 x² y² + 2

for $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$.

6.

Find the equation of the tangent plane to the surface

$$x^2 + \frac{y^2}{9} + z^2 = 1$$

at the point $(\sqrt{2},3,1)$.

7.

Calculate the directional derivative of

f(x,y) = e^x (x + y²)

at (0,1) in the direction $3 \widehat{i} + 4 \widehat{j}$.

8.

If

$$\v f (x,y,z) = xy \widehat{i} + e^{yz} \widehat{j} + \ln(1+x^2 + y^2 + z^2) \widehat{k}$$

find $\nabla \cdot \v f$ and $\nabla \times \v f$.

9.

Find the volume under the surface

f(x,y) = 4-x²-y²

over the region $R = \{(x,y) \vert x^2+y^2 \leq 4\}$.

10.

Find the centroid of the object sketched below assuming that it has uniform density.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{wing.eps}}$

11.

Find the flux of

$$\v f (x,y,z) = x^2 \widehat{i} + z \widehat{j} + e^z \widehat{k}$$

on the cube shown below.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{cube.eps}}$

12.

Find $\oint_C \v f \cdot d\v x$ where

$$\v f(x,y) = (\cos(x) + y^2) \widehat{i} + 2 x y \widehat{j}$$

and C is the path shown below leading from (1,-5) to (-1,-5).

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{tree2.eps}}$

13.

Find the mass and centroid of the object sketched below assuming the material has uniform density $\rho = 1$.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{cone.eps}}$

14.

Calculate

$$\iiint_R f(x,y,z) dx dy dz$$

where

f(x,y,z) = xyz

and R is the region bounded by the xy-plane, xz-plane, yz-plane, the plane y=2 and the plane

x+z = 1.

15.

Calculate $\int_C \v f \cdot d\v x$ where

$$\v f(x,y) = (\cos(\pi x) + y) \widehat{i} + \left(x + \frac{1}{1+y}\right) \widehat{j}$$

over the path C shown below.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{path.eps}}$

16.

Calculate $\int_C \v f \cdot d\v x$ where

$$\v f (x,y) = xy \widehat{i} + (x^2 + z) \widehat{j} + x \widehat{k}$$

where C is the line segment connecting (0,0,0) to (1,2,3).

17.

Find

$$\oint_C \v f \cdot d\v x$$

where

$$\v f(x,y,z) = (x+y) \widehat{i} + (x+z) \widehat{j} + 6x \widehat{k}$$

over the path C.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{path2.eps}}$

18.

Calculate the center of mass of this 3/4 section of a washer.

$\parbox{1.5in}{\epsfxsize=1.5in \epsfbox{washer.eps}}$

19.

Calculate

$$\iint_R z dS$$

where R is the upper surface of a sphere of radius 2 centered at the origin.

20.

Find a potential for the following vector field.

$$\v F = \sin y \widehat{i} + (x \cos y + y^2) \widehat{j} + e^{3z} \widehat{k}$$