Math Answers

1.    y''' - 9y''+ 27y'-27y=0        ans, y=c₁e^3x+c₂xe^3x+c₃x²e^3x

2.    y''' + 2y''- 13y' + 10y =0    ans, y= c₁e^-5x+c₂e^2x+c₃e^x

1. y'' - 2y' = 0                ans, y=c₁+c₂e^2x
2. 4y'' - 4y' + y = 0       ans, y=c₁e^1/2x+c₂xe^1/2x

1. y'' + 4y' + 7y = 0      ans, y= e^-2x(c₁cosx+c₂sinx)
2. 2y'' - 7y' + 3y = 0     ans, y=c₁e^3x+c₂e^1/2x

An electric refrigerator is built in a form of rectangular parallelepiped. The inside dimensions are 3 ft. by 2.6 ft. by 1.8 ft. A freezing unit (1.1 ft. by 0.8 ft. by 0.7 ft.) subtracts from the storage room of the box. Find the capacity of the refrigerator.

a) Define Bijective function and Surjective function.

b) Find the limiting value of i) lim

𝑥→7

𝑥

2+2𝑥−63

𝑥−7

ii) lim

𝑥→∞

5𝑥−1

5𝑥+1

.

c) Test the continuity of following functions at x= -2 and x=3

𝑓(𝑥) = {

7𝑥 − 1 𝑖𝑓 𝑥 > 3

𝑥

2 − 8 𝑖𝑓 − 2 ≤ 𝑥 ≤ 3

8𝑥 + 3 𝑖𝑓 𝑥 < −2

Assume z is a normal random variable with mean 0 and standard deviation 1.

An investor is considering a $25,000 investment in a start-up company. She estimates that she

has

probability 0.05 of a $15,000 loss, probability 0.15 of a $20,000 loss, probability 0.35 of a $35,000 profit,

and probability 0.45 of breaking even (a profit of $0). What is the expected value of the profit?

Scores of an achievement test show that if follows a normal distribution. Its mean is 78 with a standard deviation of 8. Find the interval wherein the middle 80% of the scores lie?

If A and Bare vector fields, prove the following:

nabla(A* B)=(B* nabla)A+(A* nabla)B+B*( nabla* A)+A*( nabla* B) .

[SADT10] Let r=x hat i +y hat j +z hat k and r = ||r||

Show that:

nabla(lnr)= r r^ 2 .

and

nabla*(r^ n r)=0 .