Using the methods of this section: (a) Find the first few terms of the Maclaurin series for each of the following functions. (b) Find the general term and write the series in summation form. (c) Check your results in (a) by computer. (d) Use a computer to plot the function and several approximating partial sums of the series. $\ln (x+\sqrt{1+x^{2}})=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}}$

A metal rod of mass m slides without friction along two parallel horizontal rails, separated by a distance $\ell$ and connected by a resistor R. A uniform vertical magnetic field of magnitude B is applied perpendicular to the plane of the paper. The applied force acts only for a moment, to give the rod a speed v. In terms of m, $\ell$, R, B, and v, find the distance the rod will then slide as it coasts to a stop.

Identify each of the following elements. Give the number of protons and neutrons in each nucleus. $^{173}_{70} \mathrm{X}$

For each given set of basis vectors, use the Gram-Schmidt method to find an orthonormal set. (a) A = (0, 2, 0, 0), B = (3,−4, 0, 0), C = (1, 2, 3, 4). (b) A = (0, 0, 0, 7), B = (2, 0, 0, 5), C = (3, 1, 1, 4). (c) A = (6, 0, 0, 0), B = (1, 0, 2, 0), C = (4, 1, 9, 2).

For these problems, the most important sketch is the projection in the plane of integration, which is easy to do by hand. However, you might like to use your computer to plot the corresponding 3 dimensional picture. The part of the plane $x + y + z = 1$ which is in the first octant is a triangular area (sketch it). Find the area and its centroid by integration. You might like to check your work by geometry.

Find the solutions of the following differential equations in terms of Bessel function. $4 x y^{\prime \prime}+y=0$

Identify each of the following costs as either direct materials (DM), direct labor (DL), or factory overhead (FO). The company manufactures tennis balls.

$$\begin{matrix} \text{\_\_\_\_\_\_\_ 1. Rubber used to form the cores}\\ \text{\_\_\_\_\_\_\_ 2. Factory maintenance}\\ \text{\_\_\_\_\_\_\_ 3. Wages paid to assembly workers}\\ \text{\_\_\_\_\_\_\_ 4. Glue used in binding rubber cores to felt covers}\\ \text{\_\_\_\_\_\_\_ 5. Depreciation—Factory equipment}\\ \text{\_\_\_\_\_\_\_ 6. Cans to package the balls}\\ \end{matrix}$$

Write an equation for a sinusoidal sound wave of amplitude 1 and frequency 440 hertz (1 hertz means 1 cycle per second). (Take the velocity of sound to be 350 m/sec.)

Solve the following differential equations by the methods discussed above and compare computer solutions. $y^{\prime \prime}+9 y=0$

The values of the following integrals are known and can be found in integral tables or by computer. Your goal in evaluating them is to learn about contour integration by applying the methods discussed in the examples above. Then check your answers by computer. $\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)}$

Two circular coils of radius R, each with N turns, are perpendicular to a common axis. The coil centers are a distance R apart. Each coil carries a steady current I in the same direction. (a) Show that the magnetic field on the axis at a distance x from the center of one coil is

$$B=\frac{N \mu_{0} I R^{2}}{2}\left[\frac{1}{\left(R^{2}+x^{2}\right)^{3 / 2}}+\frac{1}{\left(2 R^{2}+x^{2}-2 R x\right)^{3 / 2}}\right]$$

(b) Show that dB/dx and $d^{2} B / d x^{2}$ are both zero at the point midway between the coils. We may then conclude that the magnetic field in the region midway between the coils is uniform. Coils in this configuration are called Helmholtz coils.

Determine whether the following series converge. Justify your answers. $\sum_{k=1}^{\infty} \tan ^{-1} \frac{1}{\sqrt{k}}$

Using both the binomial distribution and the normal approximation. Consider a biased coin with probability 1/3 of heads and 2/3 of tails and suppose it is tossed 450 times. (a) Find the probability of getting exactly 320 tails. (b) Find the probability of getting between 300 and 320 tails.

Write seven sentences to tell about what kind of clothes you like to wear. Use the words in the box to help you.

$$\begin{array}{l} \text{la ganga} & \text{la liquidación} & \text{el mercado} & \text{algodón} & \text{de cuero}\\ \text{está hecho(a) de} & \text{claro(a)} & \text{oscuro(a)} & \text{vivo(a)} & \text{apretado(a)}\\ \text{flojo(a)} & \text{mediano(a)} & \text{la talla} & \text{el dinero} & \text{el regalo}\\ \text{el precio} & \text{los pantalones} & \text{la blusa} & \text{la falda} & \text{la chaqueta}\\ \text{tela sintética} & \text{ } & \text{ } & \text{ } & \text{ }\\ \end{array}$$