In this problem, p is in dollars and x is the number of units.

Find the producer's surplus for a product if its demand function is

p = 144 − x² and its supply function is p = x² + 12x + 130.

(Round your answer to two decimal places.)

In this problem, p is in dollars and x is the number of units.

The demand function for a certain product is

p = 123 − 2x²

and the supply function is

p = x² + 33x + 36.

Find the producer's surplus at the equilibrium point. (Round x and p to two decimal places. Round your answer to the nearest cent.)
$

Differential Calculus - [Related Rates]

—

At Noon, ship A is 200 km east of ship B and ship A is sailing north at 30 km/h. ten mins later, ship B starts to sail south at 35 km/h.

a) What is the distance between the two ships at 3pm?

b) How fast (in km/h) are the ships moving apart at 3pm?

—

Source Material:

Stewart, J. (2016). Single variable calculus: early transcendentals. [Chapter 3.9]

Use Newton's method to find all real roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

8/x = 1 + x^3

Calculate k(t) when r(t) = <4t^-1,-6,6t>

Thank you!

A manufacturing firm has a monthly production of units modeled by?(?, ?) = 50?0.7?0.3 where x is the number of units of labor and y is the number of units of capital.
The company has a budget of $75,000 per month for labor and capital. If each unit of labor costs $30 and each unit of capital costs $25, how much should the company spend on labor and materials in order to maximize their production?

Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim x → ∞ 3x − 1 / 2x + 5

A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing $2 per running foot and on one side by a brick wall costing $6 per running foot. Let x be the length of the brick wall side. Which of the following represents the total cost of the material?

2. Solve the second-order linear differential equation y′′ − 2y′ − 3y = −32e−x using the method of variation of parameters.

Solve the following boundary value problem by Laplace Transform. (If you solve with another
method you will NOT get credit. There is nothing wrong about the conditions.)
d2y
dt2 + y = cos(2t); y0(0) = 0; y0(

2
) = ?1:

The U.S. census questionnaire defines kitchens with complete facilities as those having a sink with piped water, a range, and a refrigerator. Homes that lack complete kitchen facilities have been rare in the United States for many years. The first census for which data were tabulated on this subject was in 1970. The table shows the percentage of housing units lacking complete kitchen facilities in the western United States.

Percent of Western U.S. Homes with Incomplete
Kitchens

(a) Use the method of least squares to find the multivariable function f with inputs a and b for the best-fitting line

y = ax + b,

where x is years since 1970.
f(a, b) =

(b) Calculate the minimum value of

f(a, b).

Explain what this minimum value indicates about the relationship between and the best-fitting line.The minimum value of

f(a, b)

is  , which indicates that the line with parameters a =  and b =  passes through each data point.
(c) Write function of the linear model that best fits the data to give the percentage of homes with incomplete kitchens in the Western United States, where x is years since 1970, with data from

0 ≤ x ≤ 20.

h(x) =

(d) In what year does the best-fitting line predict that no housing units will lack complete kitchen facilities?

Integration question: Fluid flows into a tank for 10 minutes. The tank is initially empty. After t minutes, fluid flows in at a rate of 1 + t/2 litres per minute. How much fluid, in litres, flows into the tank?

Prove that:

a) |sinx|<= |x|

b) x = sin x has only one solution in real number using mean value theorem

Find the mass of the solid bounded by the ??-plane, ??-plane, ??-plane, and the plane (?/2)+(?/4)+(?/8)=1, if the density of the solid is given by ?(?,?,?)=?+3?.

(a) Find the limit of the following functions:

-lim as x approaches 0 (1-cos³(x)/x)

-lim as x approaches 0 (sin(x)/2x)

-lim as theta approaches 0 (tan (5theta)/theta)

(b) Find the derivative of the following functions:

-f(x) = cos (3x²-2x)

-f(x) = cos³ (x²/1-x)

(c) Determine the period of the following functions:

-f(x) = 3 cos(x/2)

-f(x)= 21+ 7 sin(2x+3)

6. (a) let f : R → R be a function defined by
f(x) =



x + 4 if x ≤ 1
ax + b if 1 < x ≤ 3
3x x 8 if x > 3
Find the values of a and b that makes f(x) continuous on R. [10 marks]
(b) Find the derivative of f(x) = tann 1
1 ∞x
1 + x

. [15 marks]
(c) Find f
0
(x) using logarithmic differentiation, where f(x) = e
e 3x
√
2x x 5
(6 65x)
4
. [15 marks]
(d) Evaluate the integral Z
(x
3 + 1)1/3x
5
dx.