Use Bisection method to determine the drage coefficient needed so that an 80-kg bungee jumper has a velocity of 36 m/s after 4 s of free fall. Note: The acceleration of gravity is 9.81 m/s^2. Start with initial guesses of xl = 0.1 and xu = 0.2 and iterate until the approximate relative error falls below 2%.

Show that there are only two distinct groups with four elements, as follows. Call the elements of the group e, a. b,c.

Let a denote a nonidentity element whose square is the identity. The row and column labeled by e are known. Show that the row labeled by a is determined by the requirement that each group element must appear exactly once in each row and column; similarly, the column labeled by a is determined. There are now four table entries left to determine. Show that there are exactly two possible ways to complete the multiplication table that are consistent with the constraints on multiplication tables. Show that these two ways of completing the table yield the multiplication tables of the two groups with four elements that we have already encountered.

Let K = { s+t * 2^(1/2), such that s, t are Rational}. Show that K is a Field

Determine if ~w = (−4, 6, 1) is a linear combination of ~u = (1, 0, −1) and
~v = (1, −11, 3) . If so, then express ~w as a linear combination of ~u and ~v .

Let ~u = (1, 1, −1) and ~v = (2, 1, 3). Determine if ~w = (7, 6, 3) is a linear
combination of ~u and ~v. If so, express ~w as a linear combination of ~u and ~v.

Let
~x1 = (2, −1, 3, 1), ~x2 = (1, 0, −1, 1), ~x3 = (0, 1, 4, 2).
(i) Determine if ~x1, ~x2, and ~x3 are linearly independent. Justify your answer.
(ii) Determine if ~v = (2, −1, 3, 1) is a linear combination of ~x1, ~x2, and ~x3.
If so, express ~v as a linear combination of ~x1, ~x2, and ~x3. If not, justify
your answer.
(iii) Determine if ~u = (1, 0, 0, 1) is a linear combination of ~x1, ~x2, and ~x3. If
so, express ~u as a linear combination of ~x1, ~x2, and ~x3. If not, justify
your answer.

1). Consider the quadratic equation
x^2+ 100 x + 1 = 0
(i) Compute approximate roots by solving
x^2 -100 x = 0
(ii) Use the quadratic formula to compute the roots of equation
(iii) Repeat the computation of the roots but use 3 digit precision.
(iv) Compute the relative absolute errors in the two 3 digit precision root approximations in (iii).
(v) With x1 =1/2a (-b + sqrt b^2 - 4ac and x2 = 1/2a (-b + sqrt b^2 - 4a, show that x1x2 =c/a and  Discuss
the conditions under which one of the zeros can be trusted and the other zero not.
(vi) Using your 3 digit precision computations, recompute the second zero by rearranging x1x2 = c/a appromately

Let G a graph of order 8 with V (G) = {v1, v2, . . . , v8} such that deg vi = i for 1 ≤ i ≤ 7. What is deg v8? Justify your answer.

Please show all steps thank you

Brainstorm an example of a skewed data distribution relating to something in real life.

For your example would the median be a better description of the center of the distribution, why or why not? What could be the ramifications of the MEAN being communicated to the audience instead of the median? Can you think of any situations where the data may be skewed and the mean communicated not the median?

Textbook : Garfunkel, S. (2016). For all practical purposes: Mathematical literacy in today's world (10th ed).

An agricultural products manufacturer's research and development division is testing a newly developed wheat fertilizer.

Wheat is planted in several test areas (plots), which are in a controlled environment. Each area is randomly assigned a specific quantity of the fertilizer: either no fertilizer, a small amount, or a large amount of fertilizer.

The amount of water and the temperature are carefully controlled at specified levels and monitored for each test area: each area is maintained at either 78 °F or 71 °F for the duration of the test. Each area is administered either 2, 4 or 8 litres of water per week.

After three months, the yield (i.e. the amount of wheat produced) for each group is measured and the results are compared

a)This testing procedure is an example of a:

A. completely randomized design experiment
B. randomized block design experiment
C. matched pairs experiment

b)The unit of this experiment is the:

A. fertilizer
B. wheat

c)The factors of this experiment are:

A. season
B. application of fertilizer
C. sunlight
D. temperature
E. water
F. time period

d)Identify the response in the experiment:

A. amount of fertilizer
B. yield

Use a relative error in p_n (relative error= abs(p_n- p_n-1)/abs(p_n)) of 0:0001 to find the root of f(x) = x + exp(x) = 0 using both the Bisection method, the Fixed Point method and Newton's method. How many iterations are required for each method

Homework problems: Nested quantifiers (1.9-1.10)

1. Determine the truth value of each expression below if the domain is the set of all real numbers.

   1. ∃x∀y (xy = 0) (If true, give an example.)

2. ∀x∀y∃z (z = (x - y)/3) (If false, give a counterexample.)

3. ∀x∀y (xy = yx) (If false, give a counterexample.)

4. ∃x∃y∃z (x2 + y2 = z2) (If true, give an example.)

2. Redo the above (problem 1), with the domain of positive integers.

3. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all integers.

   1. There are two numbers whose sum is equal to their product.
      

2. The product of every two positive integers is positive.
   

3. Every positive integer can be expressed as the sum of the squares of four integers.

4. There is a positive integer that is smaller than all other positive integers.

4. The domain of discourse is the members of a chess club. The predicate B(x, y) means that person x has beaten person y at some point in time. Give a logical expression equivalent to the following English statements.

   1. No one has ever beat Nancy.

2. Everyone has been beaten before.

3. Everyone has won at least one game.

4. No one has beaten both Ingrid and Dominic.

5. There are two members who have never been beaten.

5. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all real numbers.

   1. The reciprocal of every positive number is positive.

2. There is no smallest number.

3. There are two numbers whose ratio is less than 1.

6. Write the negation of each of the following logical expressions so that all negations immediately precede predicates.

   1. ∀x ∃y ∃z P(y, x, z)

2. ∃x ∃y P(x, y) ∧ ∀x ∀y Q(x, y)

3. ∃x ∀y ( P(x, y) ↔ P(y, x) )

4. ∃x ∀y ( P(x, y) → Q(x, y) )

Homework problems: Logical reasoning (1.11-1.13)

7. Use a truth table to prove the conclusion from the hypotheses. The hypotheses are:

   • If I drive on the freeway, I will see the fire.

   • I will either drive on the freeway or take surface streets.

   • I am not going to take surface streets.

Conclude that I will see the fire.

Use the following variable names:

• p: I drive on the freeway

• r: I take surface streets

• q: I see the fire

8. Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. Use the set of all students as the domain of discourse. The hypotheses are:

   • Larry and Hubert are taking Boolean Logic.

   • Any student who takes Boolean Logic can take Algorithms.

Conclude that Larry and Hubert can take Algorithms.

9. Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. Use the set of all people as the domain of discourse. The hypotheses are:

   • Everyone who practices hard is a good musician.

   • There is a member of the orchestra who practices hard.

Conclude that someone in the orchestra is a good musician.

10. Which of the following arguments are valid? Explain your reasoning.

    • I have a student in my class who is getting an A. Therefore, John, a student in my class is getting an A.
      
      
      
      

    • Every girl scout who sells at least 50 boxes of cookies will get a prize. Suzy, a girl scout, got a prize. Therefore Suzy sold 50 boxes of cookies.

11. Use the laws of logic to show that ∀x(P(x) ∧ Q(x)) implies that ∀x Q(x) ∧ ∀x P(x).

Solve the ordinary differential equation analytically:

y''-4y-+3y = 5cos(x) + e^(2x)

y(0)=1, y'(0)=0

20 pairwise distinct positive integers are all smaller than 70. Prove that among their pairwise differences there are at least 4 equal numbers

1. A supplier of gasoline is considering whether to build a new station. The annual returns will depend on both the size of the station and marketing factors such as the oil industry and local demand of gasoline. The following data is available for decision making:

1. Develop an appropriate decision table for this case.
2. What is the maximax decision?
3. What is the maximin decision?
4. What is the equally likely decision?
5. What is the criterion of realism decision? Use α = 0.8.
6. Develop an opportunity loss table?
7. What is the minimax regret decision?