1. For the function ?(?)=−3(?−1)^2+5, choose a domain restriction to make ?(?) a one-to-one function, and then find the inverseof ?(?).

2. For the following function, find the horizontal and vertical asymptote(s):?(?)=3?^2+3?−18 /?^2+5?+6

3. Find the inverse function for ?(?)=4? / 5−3?

4.Find all zeroes (including complex ones) for the following function without using a graph:?(?)=?^4+5?^3+4?^2−7?−3

1- Find the solution of the following equations. For each equation, 2- determine the type of the category that the equation belongs to. Separable equation, Homogenous equation, Linear equation, or Bernolli equation?

1. ydx − x ln xdy = 0

2. y ′ = (1+y^2) / (xy(1+x2))

3. xy′ + (1 + y^2 ) tan^−1 y = 0

4. y ′ = (y) / (x+ √xy)

5. y ′ = (y−x) / (y+x)

6. tan x dy/dx + y = 3x sec x

7. (2xy^5 − y)dx + 2xdy = 0

The volume of a large cube is 64 cubic inches. A new shape is formed by removing a smaller cube from one corner of the larger cube. What is the surface area of the new shape?

A young couple buying their first home borrow $65,000 for 30 years at 7.4%, compounded monthly, and make payments of $450.05. After 5 years, they are able to make a one-time payment of $2000 along with their 60th payment.

(a) Find the unpaid balance immediately after they pay the extra $2000 and their 60th payment. (Round your answer to the nearest cent.) $

(b) How many regular payments of $450.05 will amortize the unpaid balance from part (a)? (Round your answer to the nearest whole number.) -this answer is in payments not $

(c) How much will the couple save over the life of the loan by paying the extra $2000? (Use your answer from part (b). Round your answer to the nearest cent.) the answer is not  2395.

For examples 1, you will consider a standard 4-year car loan of $10,000, with 6% APR compounded monthly. Payments are $234.85 per month, assuming $0 down payment. All of our expected values are from the perspective of the bank offering the loan.

1)     [Scenarios] Consider people who pay for 2 years, then stop…..

a)      Assume that the bank can recover an average of $2000 from the repossession process after 2 years.   How much $ does the bank get back from these people total (in payments and repo $ combined)?

b)     If 10% of purchasers default after 2 years, and the rest pay in full, what is the expected value of the loan for the bank?

c)      If 5% of car buyers make no payments at all, 10% default after 2 years, and the rest pay in full, what is the expected value of the loan for the bank?

Suppose that y′=0.162sin²(ty)+1. Plot y(t) from t=0 to t=4 with y(0)=1.286 using Euler's method with a step size of 0.4.

A rock is thrown upward from a bridge that is 83 feet above a road. The rock reaches its maximum height above the road 0.61 seconds after it is thrown and contacts the road 3.87 seconds after it was thrown.

Write a function f that determines the rock's height above the road (in feet) in terms of the number of seconds tt since the rock was thrown. THE FUNCTION MUST BE IN THE FORMAT f(t)=c⋅(t−t_1)(t−t_2) for fixed numbers c, t_1, and t_2. NOT f(x)= -16t^2+40.47297t+83 I already know how to do that.

2. Find all eigenvalues and corresponding linearly independent eigenvectors of A = [2 0 3 4] (Its a 2x2 matrix)

4. Find all eigenvalues and corresponding linearly independent eigenvectors of A = [1 0 1 0 2 3 0 0 3] (Its's a 3x3 matrix)

6. Find all eigenvalues and corresponding eigenvectors of A =    1 2 3 0 1 2 0 0 1    .(Its a 3x3 matrix)

Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the vectors in the order in which they are given. B = {(1, 3, 0), (0, 0, 3), (1, 1, 1)}

u₁₌

u₂₌

u₃₌

find Fourier series of periodic function ?(?) = { −?, −1 ≤ ? ≤ 0 with the period 2

{ ?, 0 ≤ ? ≤ 1

let G = D2n = {e,r,r^2,...,r^n-1,s,sr,sr²,..,sr^n-1} a diedergroup of order 2n, where n >=3
(a) prove that [G,G] = <r²>
(b) prove that G/[G,G] consists of two elements if n is uneven and 4 elements if n is even

Homework #3

a) Find the point of intersection of the line x = 2-3t, y = 3 + t, z = 5t and the plane 3x-2y + z = 5

b)Find the equation of the plane that passes through (1,2,3) and is parallel to the plane 2x-3y + z = 1

c)Find the equation of the plane that contains the line x = 2-3t, y = 3 + t, z = 5t and goes through the point (1,2, 3)

1)The total profit​ P(x) (in thousands of​ dollars) from a sale of x thousand units of a new product is given by

​P(x)= ln (-x³+9x²+48x+1) (0≤x≤​10).

​a) Find the number of units that should be sold in order to maximize the total profit.

​b) What is the maximum​ profit?

2)Suppose that the cost function for a product is given by C(x)=0.003x3+9x+9,610.

Find the production level​ (i.e., value of​ x) that will produce the minimum average cost per unit C(x).

a)The production level that produces the minimum average cost per unit is

x=__