Use the Laplace transform to solve the following initial value problem,

y′′ − y′ − 30y  =  δ(t − 7),y(0)  =  0,  y′(0)  =  0.

The solution is of the form ?[g(t)] h(t).

(a) Enter the function g(t) into the answer box below.

(b) Enter the function h(t) into the answer box below.

Joan wants to buy a rug for a room that is 13 feet by 16 feet. She wants to leave a uniform strip of floor around the rug. She can afford 130 square feet of carpeting. What dimensions should the rug​ have? Let x represent the width of the uniform strip of floor around the rug. Set up an equation that can be used to solve for x.

Consider the set of numbers {1...200}

Denote by A the set of multiples of 2 in this set.

Denote by B the set of multiples of 5 in this set.

Denote by C the set of multiples of 11 in this set.

How many numbers in {1...200} are divisible by at least one of 2,5 and 11?

How many numbers in {1...200} are relatively prime to 110 ?

(Relatively prime means they do not share any factors.)

How many numbers in {1...200} are divisible by 2 and 11 but not by 5?

How many numbers in {1...200} are divisible by 2 or 11 but not by 5?

Solve the following problem using algebraic techniques. A manufacturer produces 40-inch, 46-inch and 52-inch television sets that require assembly, testing and packing. Each 40-inch set requires 45 minutes to assemble, 30 minutes to test and 10 minutes to package. Each 46-inch set requires 1 hour to assemble, 45 minutes to test and 15 minutes to package. Each 52-inch set requires 1.5 hours to assemble, 1 hour to test and 15 minutes to package. If the assembly line operates for 17.75 hours per day, the test facility is used for 12.5 hours per day and the packing equipment is used for 3.75 hours per day, how many of each type of set can be produced?

(1) Define your variables

(2) Set up a system of equations

(3) Solve the system

(4) Write a complete sentence to indicate your solution

a) Find the intervals over which f is increasing or decreasing.

b) Find the local maximum and minimum values ​​of f.

c) Find the concavity intervals and the inflection points.

a. ?(?) = 2x 3 + 3? 2 − 26?

b. ?(?) = 4x 3 + 3? 2 − 6? + 4

c. ?(?) = ? 4 − 2? 2 + 1

Provide a detailed explanation of empirical research on corporate in relation to disclosure on environmental and/or climate change information.

2. For the function : f(x) = x2 − 30x − 2

a) State where f is increasing and where f is decreasing

b) Identify any local maximum or local minimum values. c) Describe where f is concave up or concave down d) Identify any points of inflection (in coordinate form)

3. For the function f (x)= x4 − 50 2

a) Find the intervals where f is increasing and where f is decreasing.

b) Find any local extrema and state whether they are a local max or a local min.

4. For the function: f(x) = x4 + 2x 3 − 12x 2 − 24x

a) State the intervals where f is concave up and where it is concave down

b) Find any points of inflection (in coordinate form).

5. For the function: f(x) = x4 + 5x 2

a) State the intervals where f is concave up and where it is concave down

b) Find any points of inflection (in coordinate form).

*please show work*

(1) z=ln(x^2+y^2), y=e^x. find ∂z/∂x and dz/dx.

(2) f(x1, x2, x3) = x1^2*x2+3sqrt(x3), x1 = sqrt(x3), x2 = lnx3. find ∂f/∂x3, and df/dx3.

Determine the centroid, C(x̅, y̅, z̅), of the solid formed in the first octant bounded by z + y − 16 = 0 and 2x² − 32 + 2y = 0.

Solve y'''-4y'=32xe^2x-24x² , y(0)=0 y'(0)=0 y''(0)=-1