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?

1-: ?(?) = ln (3 − √2? + 1)   ? ′(0) =?

2-Passing through the point x = 1 and ? = 2?
Perpendicular to the straight line tangent to 3 + 5? - 2 parabola
what is the equation of the normal?

Find the mass of the solid, moment with respect to yz plane, and the center of mass if the solid region in the first octant is bounded by the coordinate planes and the plane x+y+z=2. The density of the solid is 6x.

Problem: Prove that every polynomial having real coefficients and odd degree has a real root

This is a problem from a chapter 5.4 'applications of connectedness' in a book 'Principles of Topology(by Croom)'

So you should prove by using the connectedness concept in Topology, maybe.

Consider the following planes.

x + y + z = 7,     x + 3y + 3z = 7

(a) Find parametric equations for the line of intersection of the planes. (Use the parameter t.)
(x(t), y(t), z(t)) =

(b) Find the angle between the planes. (Round your answer to one decimal place.)
°

Find the vertex and the x-intercepts (if any) of the parabola. (If an answer does not exist, enter DNE.)

f(x) = 3x² - 8x - 3

Suppose the first and second derivatives of f(x) are: f' (x) = 4x(x^2 − 9) f''(x) = 12(x^2 − 3).

(a) On what interval(s) is f(x) increasing and decreasing?

(b) On what interval(s) is f(x) concave up and concave down?

(c) Where does f(x) have relative maxima? Minima? Inflection points?

Solve differential equation by using undetermined coefficient method

y^''+2y^'+y= 4 sin2x

2-Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.)

f(θ) = 6 cos(θ) + 3 sin²(θ)

3- Consider the following.

f(x) = x⁵ − x³ + 9,    −1 ≤ x ≤ 1

(a) Use a graph to find the absolute maximum and minimum values of the function to two decimal places.

minimum

(b) Use calculus to find the exact maximum and minimum values.

maximum

minimum

A rectangular storage container with an open top is to have a volume of 8 m3. The length of this base is twice the width. Material for the base costs $6 per square meter. Material for the sides costs $10 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

Let S be the set of natural numbers which can be written as a non-empty string of ones followed by a non-empty string of zeroes. For example, 10, 111100 and 11100000 are all in S, but 11 and 1110011 are not in S. Prove that there exists a natural number n∈S, such that 2018 | n.

The business manager of a 90 unit apartment building is trying to determine the rent to be charged. From past experience with similar buildings, when rent is set at $400, all the units are full. For every $20 increase in rent, one additional unit remains vacant. What rent should be charged for maximum total revenue? What is that maximum total revenue?

To help solve the above scenario, perform an internet search for Profit Parabola or Applications of Quadratic Functions. List the URL of one of the applications that you find.

URL ___________________________________________________________________

Go to http://www.purplemath.com/modules/quadprob3.htm to see the process used for determining the quadratic function for revenues R(x) as a function of price hikes x on page 3 with the canoe-rental business problem. Use this process to determine the quadratic function that models the revenues R(x) as a function of price hikes x in the apartment building scenario above. SHOW ALL YOUR WORK!

3. What is the formula for revenues R after   x $20 price hikes in the apartment building?

Graph the function. Clearly label the graph (desmos.com is a great an on-line graphing resource).

Find the maximum revenue (or income) of the apartment building.

What is the rent that coincides with this maximum revenue?

What is the outcome of the rent hike of $20 results in 2 additional vacancies instead of 1 additional vacancy? Recalculate questions 3, 5, 6 for this new scenario.