Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 4(x2 + y2)ey2 − x2

local maximum value(s)=?

local minimum value(s) =?

saddle point(s) (x, y, f) =?

3. Congratulations, you just won the lottery! In one option presented to you, you will be paid one

million dollars a year for the next 25 years. You can deposit this money in an account that will earn

5% each year.

(a) Let M(t) be the amount of money in the account (measured in millions of dollars) at time

t (measured in years). Set up a differential equation that describes the rate of change in the

amount of money in the account. Two factors cause the amount to grow – first, you are

depositing one millon dollars per year and second, you are earning 5% interest.

(b) If there is no amount of money in the account when you open it, how much money will you

have in the account after 25 years?

(c) The second option presented to you is to take a lump sum of 10 million dollars, which you

will deposit into a similar account. Set up a new initial value problem (that is, differential

equation with initial condition) to model this situation.

(d) How much money will you have in the account after 25 years with in this case?

(e) Do you prefer the first or second option? Explain your thinking.

(f) At what time does the amount of money in the account under the first option overtake the

amount of money in the account under the second option?

Steve and Elsie are camping in the desert, but have decided to part ways. Steve heads north, at 8 AM, and walks steadily at 2 miles per hour. Elsie sleeps in, and starts walking west at 2.5 miles per hour starting at 10 AM.

When will the distance between them be 25 miles? (Round your answer to the nearest minute.)

If exactly 186 people sign up for a charter flight, Leisure World Travel Agency charges $302/person. However, if more than 186 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Determine how many passengers will result in a maximum revenue for the travel agency. Hint: Let x denote the number of passengers above 186. Show that the revenue function R is given by

R(x) = (186 + x)(302 − x).

-------- passengers

What is the maximum revenue?

$ ----

What would be the fare per passenger in this case?

  ---- dollars per passenger

Sketch the region and find the area completely enclosed by the functions f(x)=x²-1 and g(x)=1-x

7. The Sorry State Lottery requires you to select five different numbers from 0 through 69. (Order is not important.) You are a Big Winner if the five numbers you select agree with those in the drawing, and you are a Small-Fry Winner if four of your five numbers agree with those in the drawing. (Enter your answers as exact answers.)

What is the probability of being a Big Winner?


What is the probability of being a Small-Fry Winner?


What is the probability that you are either a Big Winner or a Small-Fry Winner?

8. In a New York State daily lottery game, a sequence of two digits (not necessarily different) in the range 0-9 are selected at random. Find the probability that all two are different.

Consider the function f(x) = x^3 / x − 2

1. Give the intervals of increase and the intervals of decrease for f(x).

2. Give the location of any local maximums and local minimums on the graph of y = f(x). Give your answer(s) as a point, listing both the x-coordinates and y-coordinates.

3. List the intervals of concavity for f(x).

4. Identify any inflection points on the graph of y = f(x). Give your answer(s) as a point, listing both the x-coordinates and y-coordinates.

5. Sketch the graph of y = f(x) on the axes below. Be sure your graph reflects all of the information above.

Consider the differential equation dy/dx = y^2 + y - 2 (1) Sketch its phase portrait and classify the critical points. (2) Find the explicit solution of the DE.

When you cough, your windpipe contracts. The speed, v(r), with which you expel air through your windpipe, depends in the radius, r, of your windpipe. If a is normal (rest) radius of your windpipe, then for 0<r<a. The speed is given by: v(r)=k(a-r)r^2 where k is a positive constant. For an individual, assume a to be constant. a. Find all critical points of v(r). b. what value of r minimizes the speed? Explain briefly using the second derivative test. c. What value of r maximizes the speed, and what is this maximum speed? explain briefly using the first derivative test. d. Consider two individuals, Jay and Dee. Jay's value of a is larger than Dee's value of a. How would Jay's maximum windpipe velocity compare to that of Dee? Explain briefly.

given the 3rd order differential equation: y''' - 3y'' + 2y' = e^x / (1 + e^-x)

i) set u = y' to reduce the order of the equation to order 2

ii) solve the reduced equation using variation of parameters

iii) find the solution of the original differential equation

Suppose the function  g(x) has a domain of all real numbers except

x = −2

. The second derivative of g(x) is shown below.

g ''(x) =

(a) Give the intervals where  

g(x)

is concave down. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

(b) Give the intervals where  

g(x)

is concave up. (Enter your answer using interval notation. If an answer does not exist, enter DNE.)

(c) Find the x-coordinates of the inflection points for

g(x)

. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

1) Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 7y cos(x),    0 ≤ x ≤ 2π

2)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 4 − x⁴ + 2x² − y²

You have three sheets of 2 × 4 m cardboard. You cut squares of side x from each of the twelve corners. You fold up the resulting flaps on the three sheets to make 3 open-topped boxes, and you use the twelve squares to form two cubes. Maximize the total volume