Roots of a complex number

Find all the solutions to the equation

(a) z^ 4 − 1 = 0

(b) z ^5 + 2^5 i = 0

(c) z^ 6 − 16z^ 3 + 128 = 0

On the standard coordinate grid at initial moment, ship Tiger is at the position (0, 352), ship Lion is at the position (511, 0). Tiger sails along the straight line to the position (724, 0). At the same time, Lion sails along the straight line to the position (0, 565). Lion will reach her destination in one hour, Tiger – in two hours.

       Task:

1. Find the point of intersection of the paths of the ships.
2. At what time from the moment of departure each of the ships will pass the point of intersection of the paths.
3. At what time (in minutes after departure) the distance between Lion and Tiger will be the shortest?
4. What is the shortest distance between Lion and Tiger?
5. What will be the positions of both ships at the moment when the distance between them is the shortest?

Simplify into one fraction:

3/x + 1/7

a) Prove stereographic projection maps circles not containing (0,0,1) from the upper half sphere’s surface into circles on the z = 0 projection plane. b) What happens under stereographic projection to circles containing (0,0,1), or otherwise the North Pole? Draw a sketch showing this.

The number of consumer complaints against the top U.S. airlines in 2013 in given in the following table.

(a) By considering the numbers in the colum labeled "Complaints," caclulate the mean and median number of complaints per airline.

(b) Explain why the averages found in part (a) are not meaningful.

(c) Find the mean and median of the numbers in the column labeled "Complaints per 100,000 Passengers Boarding." Discuss whether these averages are meaningful.

Define a subspace of a vector space V . Take the set of vectors in Rn such that th
coordinates add up to 0. I that a subspace. What about the set whose coordinates add
up to 1. Explain your answers.

An auto dealer's sales numbers are shown in the table below. Find for each month the mean, median, and mode prices of the cars she sold. Round your answers to the nearest dollar.

June

July

Solve 8 cos ( 2 β ) = 8 sin 2 ( β ) + 3 for all solutions 0 ≤ β < 2 π

β=

A doctor administers a drug to a

33-kg

​patient, using a dosage formula of

55

​mg/kg/day. Assume that the drug is available in a

100

mg per 5 mL suspension or in

200

mg tablets. a. How many tablets should a

33-kg

patient take every four​ hours?

b. The suspension with a drop factor of 10​ ggt/mL delivers the drug intravenously to the patient over a​ twelve-hour period. What flow rate should be used in units of​ ggt/hr?

a. The patient should take

nothing

pills every four hours.

​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)

b. The intravenous suspension flow should be set to

nothing

​ggt/hour.

​(Type an integer or decimal rounded to the nearest hundredth as​ needed.)

W=e^x^2+sin(xy) where x=2s-t and y= t-s.

how to write generic chain rule

find when s=, t=1

Find the equation of the tangent plane to the surface,

(4x^2)(y^3) + (5yz) + (2xz^3) = 7

at the point P(-1,1,1). Also nd the parametric equation of the normal

line to that surface at that point . Sketch a picture that illustrates what this

is all about.

1. A 10 ft chain weighs 25 lb and hangs from a ceiling with a 5 lb weight attached to the end. Find the wok done lifting the lower end of the chain and the weight to the ceiling so that they are level with the upper end.

2. Use the method of cylindrical shells to find the volume formula for a sphere with radius r. (in our example we used the disk method. You formula should be the same, but the integral you use to get there should be different.

( NO HAND WRITING PLEASE )

Q1: Suppose that a and b are integers, a ≡ 11 (mod 19), and

b ≡ 3 (mod 19). Find the integer c with 0 ≤ c ≤ 18 such

that

a) c ≡ 13a (mod 19).

b) c ≡ 8b (mod 19).

c) c ≡ a − b (mod 19).

d) c ≡ 7a + 3b (mod 19).

e) c ≡ 2a² + 3b² (mod 19).

Q2:

List all the steps used to search for 10 in the sequence 1,3, 4, 5, 6, 8, 9, 11 using

a) A linear search.

b) A binary search.