Verify Green’s theorem for the given vector field $\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}$ and region D by calculating both $\oint_{\partial D} M d x+N d y$ and $\iint_{D}\left(N_{x}-M_{y}\right) d A.$ $\mathbf{F}=\left(x^{2}-y\right) \mathbf{i}+\left(x+y^{2}\right) \mathbf{j}$ is the rectangle bounded by x = 0, x = 2, y = 0, and y = 1.

Modify the SimpleVector class template presented in this chapter to include the member functions push_back and pop_back. These functions should emulate the STL vector class member functions of the same name. The push_back function should accept an argument and insert its value at the end of the array. The pop_back function should accept no argument and remove the last element from the array. Test the class with a driver program.

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.

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

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?

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

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

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. $f(x)=5 e^{2 x}$

Assuming that numbers is an array of doubles, will the following statement display the contents of the array? cout << numbers << endl;

The differential equation that describes the current in an RLC network is

$$3 \frac{d^{2} i}{d t^{2}}+15 \frac{d i}{d t}+12 i=0$$

Given that i(0) = 0, di(0)/dt = 6 mA/s, obtain i(t).

A rumor spreads through a community of 500 people at the rate $\frac{d N}{d t}=0.02(500-N) N^{1 / 2}$, where N is the number of people who have heard the rumor at time t (in hours). Use Euler’s method with h=0.5 to find the number who have heard the rumor after 3 hours, if only 2 people heard it initially.

Sketch the image of the rectangle with vertices at (0, 0), (1, 0), (1, 2), and (0, 2) under the specified transformation. T is the shear represented by T(x, y) = (x + y, y).

A pie plate is filled up to the brim with pumpkin pie filling. The pie plate is made of Pyrex and its expansion can be neglected. It is a cylinder with an inside depth of 2.10 cm and an inside diameter of 30.0 cm. It is prepared at a room temperature of $68^{\circ} \mathrm{F}$ and placed in an oven at $400^{\circ} \mathrm{F}.$ When it taken out, 151 cc of the pie filling has flowed out and over the rim. Determine the coefficient of volume expansion of the pie filling, assuming it is a fluid.