Sketch or computer plot a graph of the function $y=e^{-x^{2}}$

If $z=x^{2}+2 y^{2}, x=r \cos \theta, y=r \sin \theta$, find the following partial derivatives. $\frac{\partial^{2} z}{\partial y \partial \theta}$ Repeat if $z=r^{2} \tan ^{2} \theta$.

Variation occurs in a formula from photography. In $L=\frac{25 F^{2}}{s t}$, the luminance, L, varies directly as the square of the F-stop, F, and inversely as the product of the film ASA number, s, and the shutter speed, t. (a) What would an appropriate F-stop be for 200 ASA film and a shutter speed of $\frac{1}{250}$ sec when 500 footcandles of light is available? (b) If 125 footcandles of light is available and an F-stop of 2 is used with 200 ASA film, what shutter speed should be used?

Write a 4-by-4 antisymmetric matrix to show that there are 6 different components, not the 4 components of a vector in 4 dimensions.

The work function for zinc is 4.31 eV. (a) Find the cutoff wavelength for zinc. (b) What is the lowest frequency of light incident on zinc that releases photoelectrons from its surface? (c) If photons of energy 5.50 eV are incident on zinc, what is the maximum kinetic energy of the ejected photoelectrons?

Two particles, each with charge 52.0 nC, are located on the y axis at y = 25.0 cm and y = -25.0 cm. (a) Find the vector electric field at a point on the x axis as a function of x. (b) Find the field at x = 36.0 cm. (c) At what location is the field $1.00 \hat{\mathbf{i}} \mathrm{kN} / \mathrm{C}$? You may need a computer to solve this equation. (d) At what location is the field $16.0 \hat{\mathbf{i}} \mathrm{kN} / \mathrm{C}$?

Find the point(s), if any, at which the graph of f has a horizontal tangent line. $f(x)=\frac{x^{2}}{x-1}$

For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint: Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using.

$$\left\{\begin{aligned} 2 x+5 y+z =2 \\ x+y+2 z =1 \\ x +5 z=3 \end{aligned}\right.$$

Complex numbers are used to describe current I, voltage E, and impedance Z (the opposition to current). These three quantities are related by the equation E=IZ, which is known as Ohm's Law. Thus, if any two of these quantities are known, the third can be found. In each exercise, solve the equation E=IZ for the remaining value. $I=5+7 i, Z=6+4 i$

Using the definition of a periodic function, show that a sum of terms corresponding to a fundamental musical tone and its overtones has the period of the fundamental.

Sylvestor Systems borrows $110,000 cash on May 15, 2017, by signing a 60-day, 12% note. 1. On what date does this note mature? 2. Suppose the face value of the note equals$110,000, the principal of the loan. Prepare the journal entries to record (a) issuance of the note and (b) payment of the note at maturity

Find one or more values of each of the following complex expressions and compare with a computer solution. $\left(\frac{1+i}{1-i}\right)^{2718}$

Find two triangles with the given angle measure and side lengths. Round side Lengths To the nearest tenth and angle measures to the nearest degree. $A=39^{\circ}, a=12, b=17$

Find the equation(s) of the tangent line(s) to the graphs of the indicated equations at the point(s) with the given value of x. $x y^{2}-y-2=0 ; x=1$