Farmers wishing to avoid the use of purchased seeds are increasingly concerned about inadvertently growing hybrid plants as a result of pollen drifting from nearby farms. Assuming that these farmers raise their own​ seeds, the fractional portion of their crop that remains free of hybrid plants t years later can be approximated by ​P(t)=(0.93)^ t.

​a) Using this​ model, predict the fractional portion of the crop that will be free of hybrid plants 10 yr after a neighboring farm begins to use purchased seeds.

​b) Find P' (10) and explain its meaning.

​c) When will half of the crop be hybrid​ plants?

Summarize the pertinent information obtained by applying the graphing

strategy and sketch the graph of f(x)=-2x/(x-1)^2

Part 1: Find the​ x-intercepts of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.The​ x-intercept(s) is/are at x=____. ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

B. There are no​ x-intercepts.

Part 2. Find the​ y-intercepts of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.The​ y-intercept(s) is/are at y=_____. ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

B. There are no​ y-intercepts.

Part 3. Find any horizontal asymptotes of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer​box(es) to complete your choice.

A. The function has one horizontal​ asymptote, ____ ​(Type an​ equation.)

B. The function has two horizontal asymptotes. The top asymptote is ____ and the bottom asymptote is___. ​(Type equations.)

C. There are no horizontal asymptotes.

Part 4. Find any vertical asymptotes of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer​box(es) to complete your choice.

A. The function has one vertical​ asymptote,____. ​(Type an​ equation.)

B. The function has two vertical asymptotes. The leftmost asymptote is_____ and the rightmost asymptote is ______. ​(Type equations.)

C. There are no vertical asymptotes.

Part 5. Find the intervals where​ f(x) is increasing or decreasing. Select the correct choice below and fill in the answer​box(es) to complete your choice.

A. The function is increasing on____. It is never decreasing. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)

B. The function is increasing on______ It is decreasing on______. ​(Type your answers in interval notation. Use a comma to separate answers as​ needed.)

C. The function is decreasing on_____. It is never increasing.

Part 6. Find the location of any local extrema of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer​box(es) to complete your choice.

A. There is a local maximum at x=______. and there is a local minimum at x=_____. ​(Type integers or decimals. Use a comma to separate answers as​needed.)

B. There is a local maximum at x=____. There is no local minimum. ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

C.There is a minimum at x=____. There is no local maximum. ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

D. There are no local extrema. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)

Part 7. Find the intervals where​ f(x) is concave upward or downward. Select the correct choice below and fill in the answer​box(es) to complete your choice.

A. The function is concave upward on____ It is never concave downward. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)

B. The function is concave upward on_____It is concave downward on_____. ​(Type your answers in interval notation. Use a comma to separate answers as​ needed.)

C. The function is concave downward on_____ It is never concave upward. ​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)

Part 8. Find the location of any inflection points of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.There​ is/are (a) inflection​ point(s) at x=_____. ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

B. There are no inflection points.

Mr. Smith is purchasing a $ 160000 house. The down payment is 20 % of the price of the house.
He is given the choice of two mortgages:

a) a 20-year mortgage at a rate of 10 %.
Find
(i) the monthly payment: $

equation editor

Equation Editor

(ii) the total amount of interest paid: $

equation editor

Equation Editor

b) a 15-year mortgage at a rate of 10 %.
Find
(i) The monthly payment: $

equation editor

Equation Editor

(ii) the total amount of interest paid: $

The profit P, in thousands of dollars, that a manufacturer makes is a function of the number N, in thousands, of widgets produced in a year, and the formula is P(N)= 10N-N²-6.34. The formula is valid up to a level of 7,000 widgets produced. Express in functional notation the profit at a production level of 4,500 widgets. Calculate the value and explain the result in practical terms. What are the fixed costs? Determine the break-even point(s) for this manufacturer. What is the maximum profit?

Let the natural number n have the decimal numeral 123,461,46d, where d is the units digit. Use divisibility tests to give all of the choices of d by which n is divisible. Complete parts (a) through (h) below. (a) For what value(s) of d is n divisible by 2? (Use a comma to separate answers as needed.) (b) For what value(s) of d is n divisible by 3? (Use a comma to separate answers as needed.) (c) For what value(s) of d is n divisible by 4? (Use a comma to separate answers as needed.) (d) For what value(s) of d is n divisible by 5? (Use a comma to separate answers as needed.) (e) For what value(s) of d is n divisible by 6? (Use a comma to separate answers as needed.) (f) For what value(s) of d is n divisible by 8? (Use a comma to separate answers as needed.) (g) For what value(s) of d is n divisible by 9?

(h) For what value(s) of d is n divisible by 10?

The velocity function of a particle moving along a line is given by the equation v(t) = t² - 2t -3. The particle has initial position s(0) = 4.

a. Find the displacement function

b. Find the displacement traveled between t = 2 and t = 4

c. Find when the particle is moving forwards and when it moves backwards

d. Find the total distance traveled between t = 2 and t = 4

e. Find the acceleration function, and use it to find the acceleration of the particle at t = 3

Let F(x)=f(f(x)) and G(x)=(F(x))^2 . You also know that f(6)=14,f(14)=3,f′(14)=4,f′(6)=3.

Find F′(6)= and G′(6)= .

Determine the tangent line approximation for ?(?) = 2sin(? ? ) at the point (0, 2???1)

Take the Laplace transform of the following initial value and solve for X(s)=L{x(t)}X(s)=L{x(t)}:

x′′+16x={sin(πt),0}

0≤t<1

1≤t

x(0)=0

x′(0)=0.

a) X(s)=  

Now find the inverse transform to find

b) x(t)=

Use u(t−a) for the Heaviside function shifted a units horizontaly.

Farmer S. Unkist has a fruit grove consisting of lemons, bananas, and watermelons along a straight moat. To prevent thieves from stealing his fruit while at the same time make sure the fruit do not mix before they are processed, he plans to fence in each fruit plot using identical rectangular enclosures. The side along the moat needs no fence because the moat is infested with man eating crocodiles as shown below. If he has 1200 yards of fence, what should be the dimensions of each enclosure if the total area of the grove is to be maximized?

A cup of coffee is made with boiling water at 100 and stands in a room where the temperature is 25. The change in temperature, H in degrees , with respect to time t, in minutes, is given by the following differential equation. (dH)/(dt)=-k(H-25) Solve this differential equation. If the coffee cools to 90 in 3 minutes, how long will it take to cool to 60 degrees? Round your answer to the nearest integer.

Suppose ?(?) represents the number of bacteria in a dish t hours after the start of a lab experiment. a) Interpret the meaning of ?(20) = 100

1. 100 is the average rate of change in the number of bacteria after 20 hours.

2. 100 is the rate of change in the number of bacteria after 20 hours.

3. After 20 hours, there are 100 bacteria in the dish.

4. 100 is the total change in the number of bacteria between the first hour and 20 hours later.

b) ) Interpret the meaning of ?′(20) = 100

1. 100 is the average rate of change in the number of bacteria after 20 hours.

2. 100 is the rate of change in the number of bacteria after 20 hours.

3. After 20 hours, there are 100 bacteria in the dish.

4. 100 is the total change in the number of bacteria between the first hour and 20 hours later.

Solve the initial–value problem by using the Laplace transform.

y′′ +2y′ +10y = δ(t−π), y(0) = 0, y′(0) = 4

Solve the IVP: y’’’ – y ’= 2sinx

where: y(0)=0, y’(0)=0, y”(0)=1

Use an annihilator method, please.

3. Let g(θ) = 2 cos(θ) + sin(2θ) . Find the absolute maximum and minimum values of g on the interval [0, π/2]