Maximize each of the following utility functions, with the cost of each commodity and total amount available to spend given. $f(x, y)=x^{3} y^{4}$, cost of a unit of is $3, cost of a unit of is$3, and $42 is available.

Give a geometric description of the linear transformation defined by the matrix product.

$$\left[\begin{array}{rr} {0} & {12} \\ {1} & {0} \end{array}\right]=\left[\begin{array}{rr} {12} & {0} \\ {0} & {1} \end{array}\right]\left[\begin{array}{ll} {0} & {1} \\ {1} & {0} \end{array}\right]$$

T F If the starting address of a C-string is passed into a pointer parameter, it can be assumed that all the characters, from that address up to the byte that holds the null terminator, are part of the string.

Use Euler’s method to approximate the indicated function value to 3 decimal places, using h=0.1. Next, solve the differential equation and find the indicated function value to 3 decimal places. Compare the result with the approximation. $\frac{d y}{d x}=2 x y$; y(1)=1; find y(1.6)

Two identical strings on different cellos are tuned to the 440-Hz A note. The peg holding one of the strings slips, so its tension is decreased by 1.5%. What is the beat frequency heard when the strings are then played together?

Find the Taylor series for the functions defined as follows. Give the interval of convergence for each series. $f(x)=\frac{7 x}{1+2 x}$

Is it possible to have more than one constructor? Is it possible to have more than one destructor?

For which type of thermodynamic process is the change in entropy equal to zero: (a) isothermal, (b) isobaric, (c) isometric, or (d) none of the preceding?

The autoignition temperature of a fuel is defined as the temperature at which a fuel-air mixture would self-explode and ignite. Thus, it sets an upper limit on the temperature of the hot reservoir in an automobile engine. The autoignition temperatures for commonly available gasoline and diesel fuel are about $495^{\circ} \mathrm{F}$ and $600^{\circ} \mathrm{F},$ respectively. What are the maximum Carnot efficiencies of a gasoline engine and a diesel engine if the cold reservoir temperature is $40^{\circ} \mathrm{C} ?$

Evaluating the integral

$$\int_{-\infty}^{\infty} \frac{10 \delta(\omega)}{4+\omega^{2}} d \omega$$

results in: (a) 0 (b) 2 (c) 2.5 (d) $\infty$

Find Taylor polynomials of degree 4 at 0 for the functions defined as follows. $f(x)=5 e^{2 x}$

The time (in years) until a certain machine requires repairs is a random variable t with probability density function defined by $f(t)=\frac{5}{112}\left(1-t^{-3 / 2}\right)$ for t in [1, 25]. a. Find the probability that no repairs are required in the first three years by finding the probability that a repair will be needed in years 4 through 25. b.Find the expected value for the number of years before the machine requires repairs. c. Find the standard deviation.

The “Transylvania hypothesis” claims that the full moon has an effect on health-related behavior. A study investigating this effect found a significant relationship between the phase of the moon and the number of general practice consultations nationwide, given by $y=100+1.8 \cos \left[\frac{(t-6) \pi}{14.77}\right]$, where y is the number of consultations as a percentage of the daily mean and t is the days since the last full moon. a. What is the period of this function? What is the significance of this period. b.There was a full moon on October 11, 2011. On what day in October 2011 does this formula predict the maximum number of consultations? What percent increase would be predicted for that day? c. What does the formula predict for October 28, 2011?

Find the following function values without using a calculator. $\cot -\frac{3 \pi}{4}$