A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field F(x,y,z)=z^2i+3xyj+4y^2k. Use Stokes' Theorem to find the work done.

Use a Numerical approach (i,e., a table of values) to approximate ${\displaystyle \underset{x\to 0}{lim}\frac{e^{2}x-1}{x},}$ Provide support for your analysis (slow the table and explain your resoning).

Use the Divergence Theorem to calculate the surface integral F · dS, that is, calculate the flux of F across S.
${\displaystyle F(x,y,z)=3x{y}^{2}i+x{e}^{z}j+z^{3}k}$,
S is the surface of the solid bounded by the cylinder ${\displaystyle y^2+z^2=9}$ and the planes x = −3 and x = 1.

Let ${\displaystyle z=e^{x^2+3y}+x^3{y}^2}$, where ${\displaystyle x=t\cos r\text{and}y=r{t}^4}$. Use the chain rule fot multivariable functions (Cals III Chain Rule) to find ${\displaystyle \frac{\partial z}{\partial r}}$ and ${\displaystyle \frac{\partial z}{\partial }t)}$. Give your answers in terms of r and t only. Be sure to show all of your work.

Find the complex zeros of the following polynomial function. Write f in factored form.
${\displaystyle f\left(x\right)=x^3-12x^2+49x-58}$
The complex zeros of f are =?

MacDonald Products, Inc., of Clarkson, New York, has the option of (a) proceeding immediately with production of a new top-of-the-line stereo TV that has just completed prototype testing or (b) having the value analysis team complete a study. If Ed Lusk, VP for operations, proceeds with the existing prototype (option a), the firm can expect sales to be 100,000 units at $550 each, with a probability of 6, and a.4 probability of 75,000 at $550. If, however, he uses the value analysis team (option b), the firm expects sales of 75,000 units at $750, with a probability of .7, and a.3 probability of 70,000 units at $750. Value analysis, at a cost of $100,000, is only used in option b. Which option has the highest expected monetary value (EMV)?

3) Answer the following questions considering the complex functions given below.
a) Using the definition of complex derivative, evaluate f(z) expression using derivative operation based on limiting case as ${\displaystyle \underset{\mathrm{△}z\to 0}{lim}}$
a.1 ${\displaystyle f\left(z\right)=\frac{1}{z^2},(z\ne 0)}$

Second derivatives For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step.
${\displaystyle f(x,y)=x^{2}y-x{y}^2\text{ ,where }x=st\text{ and }y=\frac{s}{t}}$