1.  (LM 2.1– 15(a): Continued square roots) Let r be defined by

r “ ffek ` dk ` ck ` bk ` ?k ` ¨ ¨ ¨

where k is any positive integer.

(a)  Calculate r by hand and write down the result as an analytical formula (in terms of k).

(b)  Calculate the value of r numerically for k “ 2, 3, and 4.

2.  (LM 2.1–39(b): Roots of unity)

(a)  Use the approach explained in the problem (the one using Euler’s formula) to find the five distinct solutions of x5  “ ´ 1 in MATLAB.

3.  (Temperature conversion; adapted from LM 2.2–5.)   Write a script which asks for a temperature in Fahrenheit. It should then output the temperature in degrees Celsius, kelvins,

and degree Rankine3. Then run your code with a temperature of 134˝ F.

4.  (Oblate spheroid) An oblate spheroid such as the Earth is obtained by revolving an ellipse about its minor axis as shown in the figure.

Figure 1: An oblate spheroid. Image created using MATLAB.

The Earth’s equatorial radius is about 20km longer than its polar radius.  The surface area of an oblate spheroid is given by

Apr1 , r2 q “ 2π ˜r 1(2) `  log ˆ ˙¸ ,

where r1  is the equatorial radius, r2  is the polar radius, and

γ “ arccos ˆ ˙ .

We assume r2  ă r1 .  Write a script that inputs the equatorial and polar radii and displays both Apr1 , r2 q and the approximation 4πppr1 ` r2 q{2q2 .  Apply the script to the Earth data pr1 , r2 q “ p6378.137, 6356.752q. Use format  long  g to display enough digits.

5.  (3-D coordinate conversion) Recall that Cartesian coordinates px, y, zq in R3  are related to spherical coordinates pρ, ϕ, θq by

x “ ρ sin ϕ cos θ,    y “ ρ sin ϕ sin θ,    z “ ρ cos ϕ,

where ϕ P r0, πs and θ P r0, 2πq. Write a script which takes Cartesian coordinates as inputs and converts them to spherical ones. Be sure to incorporate the following convention correctly.

Convention for special cases.  For points on the z-axis, that is, for points with Cartesian coordinates of the form p0, 0, zq, set r “ |z|, θ “ 0, and