a.Express y in terms of x given that dy/dx = (y + 2)(2x + 1) given that y = 2 at x = 0.
b. Solve (x^2 + 1)dy/dx + 3xy = 6x.
c) Obtain a general solution of dy/dx + y/x = sin x.

What path might a basketball travel if you shot from the foul line to a 10 foot hoop?

Steve races down the basketball court, then stops and shoots -- HE SCORES!!! The ball travels through the air and through the basket (10 feet off the ground) according to the function

h = -16t2+ 19.5t + 6.5, where h represents the height of the basketball above the floor, and t represents the number of seconds that the ball was in the air. How long did it take for the ball to pass through the basket?

The next time down the court, Steve took another shot, shooting the ball according to the same function as in part a, but since he was a few feet farther from the basket, the shot was an Air Ball (missing everything completely). How long was the ball in the air before it hit the floor?

Returning to the first part, was there a time, besides when the ball passed through the basket, when it was 10 feet off the floor? Show the results in a graph and a table, and provide at least 1 analytic (algebraic) explanation.

1. x2/3 + 3x1/3 and y = 10. What are the values of X?

A) -2, 5

B) -5, 2

C) -125, 8

D) -8, 125

E) None of the above

2. When a number is decreased by 20% of itself, the result is 144. What is the number?

  36

  72

  180

  900

3. A car rental agency charges $175 per week plus $0.20 per mile to rent a car. How many miles can you travel in one week for $295? ?

  234 miles

  575 miles

  600 miles

  1295 miles

?

1755 miles ?

4. The length of a rectangular room is 6 feet longer than twice the width. If the room's perimeter is 156 feet, what are the room's dimensions? ?

Width = 24 ft; length = 54 ft

Width = 29 ft; length = 64 ft

Width = 36 ft; length = 42 ft

Width = 48 ft; length = 108 ft

Width = 52 ft; length = 48 ft

5. Find the result of

-19

6-25i

6 + 5i

31

None of the above

6. Fine the answer for | 2(x+1) + 4 | =10

  {-6, 0}

  {-6, 4}

  {-6, 8}

  {-8, 0}

  {-8, 2}

7. A landscaping company sells 40-pound bags of top soil. The actual weight x of a bag, however, may differ from the advertised weight by as much as 0.75 pound. Write an inequality involving absolute value that expresses the relationship between the actual weight x of a bag and 40 pounds. Solve the inequality, and express the answer in interval form. ?

|40 - x|? 0.75; [39.25, 40.75]

|40 + x| ? 0.75; [39.25, 40.75]

|x + 0.75| ? 40; [39.25, ?)

|x|- 40 ? 0.75 ; (-?, 40.75]

None of the above

8. Write an equation that passes through (1, -6) with x-intercept = -1.

  y + 6 = - 3(x - 1) or y = - 3(x - 1)

y + 6 = - 3(x - 1) or y = - 3(x + 1)?

  y - 6 = - 3(x + 1) or y = - 3(x + 1)

  y - 6 = - 3(x - 1) or y = - 3(x - 1)

  y - 1 = - 3x or y - 6 = - 3(x + 1)

9. Suppose a life insurance policy costs $20 for the first unit of coverage and then $5 for each additional unit of coverage. Let C(x) be the cost of insurance of x units of coverage. What will 10 units of coverage cost?

  $45

  $55

  $65

  $75

10. An investment is worth $2282 in 2006. By 2010 it has grown to $2986. Let y be the value of the investment in year x, where x =0 represents 2006. Write a linear equation that models the value of the investment in the year x.

11. What is the equation for the line that passes through (4, 3) and perpendicular to the line whose equation is y = 8x + 7.

(x – 3)

(x + 3)

(x – 4)

y (x + 4)

12.   The linear function f(x) = -9.8x + 24 models the percentage of people, f(x), who eats at fast food restaurants each week x years after 2009. What is the slope and what does it mean?

m = 9.8; the percentage of people eating at fast food restaurants each week has decreased at a rate of 9.8% per year after 2009.

m = -9.8; the percentage of people eating at fast food restaurants each week has increased at a rate of -9.8% per year after 2009.

m = 24; the percentage of people eating at fast food restaurants each week has increased at a rate of 24% per year after 2009.

m = 24; the percentage of people eating at fast food restaurants each week has decreased at a rate of 24% per year after 2009.

m = -9.8; the percentage of people eating at fast food restaurants each week has decreased at a rate of -9.8% per year after 2009.

13. The function f(t) = models the U.S population in

millions, ages 65 and older, where t represents year after 1990. The function g(t) = models the total yearly cost of Medicare in billions of dollars, where t represents years after 1990. What does the function represents. Find

Cost per person in thousands of dollars. $0.16 thousand

Cost per person in thousands of dollars. $0.03 thousand

Cost per person in thousands of dollars. $8.64 thousand

Cost per person in thousands of dollars. $34.67 thousand

Cost per person in thousands of dollars. $30.8 thousand

14.   Find the inverse function of f(x) =

15.   Find the center and the radius of the circle

  (4, -3), r = 8

  (-3, 4), r = 8

  (-4, 3), r = 64

  (-3, -4), r = 64

  (-4, -3), r = 64

a)f(u,v) fuction is provide f(6,-2)=2020, f_u(6,-2)=2, f_v(6,-2)=3 equations. g(x,y,z)=f(3yz+x²,2x+2y²-z²) so,

find tangent plane of g(x,y,z)=2020 at the point (0,1,2).

b)Find the tangent line ,which is parallel to question a) tangent plane, of r(t)=<t²+1,2t+7,4t-t²>(-∞<t<∞)

1.) Let f′(x) = 3x^2 − 8x. Find a particular solution that satisfies the differential equation and the initial condition f(1) = 12.

2.) An object moving on a line has a velocity given by v(t) = 3t^2 −4t+6. At time t = 1 the object’s

position is s(1) = 2. Find s(t), the object’s position at any time t.

(a) Determine the inverse Laplace transform of F(s) =(2s−1)/s^2 −4s + 6
(b) Solve the initial value problem using the method of Laplace transform. d^2y/dx^2 −7dy/dx + 10y = 0, y(0) = 0, dy/dx(0) = −3.
(c) Solve the initial value problem:
1/4(d^2y/dx^2)+dy/dx+4y = 0, y(0) = −1/2,dy/dx(0) = −1.

using these axioms ( finite affine plane ) :

AA1 : there exists at least 4 distinct points , no there of which are collinear .

AA2 : there exists at least one line with n (n>1) points on it .

AA3 : Given two distinct points , there is exactly one line incident with both of them .

AA4 : Given a line l and a point p not on l , there is exactly one line through p that does not intersect l .

prove : in an affine plane of order n , each line contains exactly n points .

Find the solutions of the equation

4x^2 + 3 = 2x

Among all triangles with a perimeter of 2s=9 units, find the dimension of the triangle with the maximum area. (Hint: Heron's Formula for the area of a triangle may be useful - A=Sqrt(s(s-a)(s-b)(s-c).  

Use the Laplace Transform method to solve the following differential equation problem: y 00(t) − y(t) = t + sin(t), y(0) = 0, y0 (0) = 1

Please show partial fraction steps to calculate coeffiecients

Consider the following differential equations:

?³?/??³ + 9 ?²?/??² + 20 ??/?? + 12? = 15

?(0) = ?̇(0) = ?̈(0) = 0

a) Find the solutions to the equation using Laplace transform.
b) Use the Final Value Theorem to determine ?(?) as ?→∞ from?(?).

Note: Dots denote differentiation with respect to time.

Given that h(x) = x.sinx . Find the root of the function h(x) = 1, where x is between [0, 2] using substitution method.

Two towns had a population of 12,000 in 1990. By 2000, the population of town A had increased by  13⁢ % while the population of town B had decreased by  13⁢ %. Assume these growth and decay rates continued.

a. Write two exponential population models A(T) and B(T) for towns A and B, respectively, where T is the number of decades since 1990.

A(T)=

12000*e0.12*T

B(T)=

12000*e−0.14*T

b. Write two new exponential models a(t) and b(t) for towns A and B, where t is the number of years since 1990.

Give the answers in the form C⋅at. Round the growth factor to four decimal places.

a(t)=

12000*1.13t

b(t)=

12000*0.87t

c. Find A(2), B(2), a(20), and b(20) and explain what you have found.

Round your answers to the nearest integer.

A(2)=

B(2)=

a(20)=

b(20)=

Each of these values represent the population

2 years after 199020 decades after 19902 decades after 1990

Q­: In the questions below, nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. In how many ways can this be done if:

1. Ben is to be in the picture?
2. Both Ed and Gail are in the picture?
3. Neither Ed nor Fran are in the picture?
4. Dot is on the left end and Ed is on the right end?
5. Hal or Ida (but not both) are in the picture.
6. Ed and Gail are in the picture, standing next to each other?
7. Ann and Ben are in the picture, but not standing next to each other?

Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $25,000 at a rate of 8%/year compounded monthly. Her bank is now charging 11.3%/year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of 3 yr for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank? (Round your answers to the nearest cent.)

interest paid to manufacturer?

interest paid to bank?

savings?