Find the general solution of the given second-order differential equation.

1. 4?'' + 9? = 15

2. (1/4) ?'' + ?' + ? = ?² − 3x

Solve the differential equation by variation of parameters.

3. ?'' + ? = sin(x)

What are the steps of finding the absolute maximum and absolute minimum?

Problem A) Convert the equation x^2 + y^2 = 4z^2 into cylindrical coordinates.

Problem C) Use cylindrical coordinates to find the volume of the region above the paraboloid z = x^2 + y^2 and inside the sphere x^2 + y^2 + z^2 = 6.

Find integral from (-1)^4 t^3 dt

please show all steps :)

Give an example where the concept of "rate of change" is used in real life.

The World Cereal Organization (WCO) has asked you to redesign cereal boxes to more efficiently use paper. Assume the box must be rectangular with a square base, and that the top and the base are exactly the same. There are a total of 6 sides on the box. Use calculus to find the dimensions of the box that minimizes surface area (paper) while still holding 80 in3 of cereal. Your decision will dictate what the WCO implements, so you must justify your answer.

(a) How do you find the second derivative of a function?

(b) If both  are positive numbers, what does that tell you about the function itself?

(c) If we know that a function is increasing, but at a decreasing rate, what does that say about the first and second derivatives?

(d) Come up with a function where the first and second derivatives are non-zero but the third derivative is zero.

A tree trunk is approximated by a circular cylinder of height 100100 meters and diameter 33 meters. The tree is growing taller at a rate of 11 meters per year and the diameter is increasing at a rate of 55 cm per year. The density of the wood is 50005000 kg per cubic meter.

how quickly is the mass of the tree increasing?

Let F (x, y) = y sin x i – cos x j, where C is the line segment from (π/2,0) to (π, 1). Then C F•dr is

A 1

B 2

C 5/2

D 3

E 7/2

A baseball team plays in a stadium that holds 66000 spectators. With the ticket price at $11 the average attendance has been 27000. When the price dropped to $10, the average attendance rose to 33000. Assume that attendance is linearly related to ticket price.

What ticket price would maximize revenue?

Business: cost from marginal cost. A gourmet popcorn company determines that the marginal cost, in dollars, of the xth bag of gourmet popcorn is given by C' (x) = -0.0004 x + 2.25 C(0)=$0 Find the total cost of producing 1000 bags of popcorn.

Consider the vector function given below.
r(t) =

2t, 3 cos(t), 3 sin(t)

(a) Find the unit tangent and unit normal vectors T(t) and N(t).
T(t)   =   

N(t)   =   

(b) Use this formula to find the curvature.
κ(t) =

Find the maxium number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue R(x) and the cost, C(x) of producing x units are in dollars.

R(x)= 6x, C(x)= 0.01x^2+ 0.3x+7

what is the production level for the maximum profit?

Let

F(x, y, z) = z tan⁻¹(y²)i + z³ ln(x² + 8)j + zk.

Find the flux of F across S, the part of the paraboloid

x² + y² + z = 28

that lies above the plane

z = 3

and is oriented upward.

F · dS =