question 1:

The total revenue received from the sale of x units of a product is given by

TRx= -3x^5 + 3/2 x^4+ x/4+ 12square x+ 6y

Find the

1. 1. Average revenue AR.
2. 2. Marginal revenue MR.
3. 3. MR at x = 4.      
4. 4. The actual revenue from selling10^th item.
5. 5. Write down different rule of derivative.

question 2 :

1. Calculate the principle amount borrowed from the bank for 20 months at a rate of 7 3/4 % per annum with interest of 2250 RO. Also find the total amount to be paid back to the bank .

2. Find the rate of interest required for an investment of 2000 RO to earn 320 RO interest over a period of 18 months.

3. A family spends 30 % of their income on mortgage installments of 500 RO per month. Calculate the monthly income of the family.

4. Find the sale price of a car if its cost price is 5500 RO and % loss is 15%.

5. The price of a PC decreases from 60 RO to 55 RO. Express this decrease as a % decrease.

Determine whether or not W is a subspace of V. Justify your answer.

W = {p(x) ∈ P(R)|p(1) = −p(−1)}, V = P(R)

Prove the following test: Let {x_n} be a sequence and lim |X_n| ^1/n = L

1. If L< 1 then {x_n} is convergent to zero

2. If L> 1 then {x_n} is divergent

Solve the differential equation. x''(t)+2x'(t)+5x(t) = 2

Find the power series solution for the equation y'' + (sinx)y = x; y(0) = 0; y'(0) = 1

Provide the recurrence relation for the coefficients and derive at least 3 non-zero terms of the solution.

Let

f(x, y) = x²y(2 − x + y²)⁵ − 4x²(1 + x + y)⁷ + x³y²(1 − 3x − y)⁸.

Find the coefficient of x⁵y³ in the expansion of f(x, y)

A mass of 50 g stretches a spring 3.828125 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 10 cm/s, and if there is no damping, determine the position u of the mass at any time t.

Enclose arguments of functions in parentheses. For example, sin(2x).

Assume g=9.8 ms2. Enter an exact answer.

Prove that rect(x)*rect(x) = tri(x)

Notes 2.7 Using CRT notation, show what is going on for all the combinations you considered in Notes 2.6. Explain why gcd(s + t, 35) sometimes gave you a factor, and it sometimes did not

Notes 2.6 is:

Notes 2.6 Consider all the possible sets of two square roots s, t of 1 (mod 35) where s ≢ t (mod 35) (there are six of them, since addition is commutative (mod 35).

For all possible combinations, compute gcd(s + t, 35). Which combinations give you a single prime factor of 35?

1.) Prove that Z+, the set of positive integers, can be expressed as a countably infinite union of disjoint countably infinite sets.

2.) Let A and B be two sets. Suppose that A and B are both countably infinite sets. Prove that there is a one-to-one correspondence between A and B.

Please show all steps. Thank you!

(I rate all answered questions)

A person buying a personal computer system is offered a choice of three models of the basic unit, four models of keyboard, and three models of printer. How many distinct systems can be purchased?

  systems

1) Show that if A is an open set in R and k ∈ R \ {0}, then the set kA = {ka | a ∈ A} is open.

1. Find the 100th and the nth term for each of the following sequences.

a. 50,90,130,...

b. 1,4,16,...

c. 7, 7⁴,7⁷,7¹⁰,...

d. 197+7x3²⁷, 197+8x3²⁷, 197+9x3²⁷,...

2. Find the first five terms in sequences with the following nth terms.

a. 10ⁿ-5

b.2n-1

3. How many terms are there in each of the following sequences?

a. 11,15,19,23,...331

b. 59,60,61,62,...459

find all primes p such that 6 is a square mod p?

please with clear hand writing