Find the eigenvalues

λ_n

and eigenfunctions

y_n(x)

for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.)

y'' + λy = 0,  y(0) = 0,  y(π/6) = 0

Explain in your own words, what can happen if D = 0 (Hessian Determinant).
Given in particular an example where we have a maximum, where we have
a minimum, or where we have neither a maximum nor a minimum.

A shelf in the Metro Department Store contains 95 colored ink cartridges for a popular ink-jet printer. six of the cartridges are defective. (a) If a customer selects 2 cartridges at random from the shelf, what is the probability that they are all defective? (Round your answer to five decimal places.) (b) If a customer selects 2 cartridges at random from the shelf, what is the probability that at least 1 is defective? (Round your answer to three decimal places.)

Find the absolute maximum and minimum values of f on the set D. Also note the point(s) where these absolute maximum and minimum values are located. f(x, y) = 9x^2 + 36x^2 y - 4y - 1 D is the region described as follows: D = { (x,y) | -2 < x < 3; -1 < y < 4}

Solve for​ Y(s), the Laplace transform of the solution​ y(t) to the initial value problem below.

y"-6y'+9y=cos 2t- sin 2t , y(0)=6, y'(0)=3

For the function ?(?) = 2? 3 + 9? 2 − 108? + 30 find the following (round to the nearest thousandth if needed):

a. Interval(s) where the function is increasing

b. Interval(s) where the function is decreasing

c. Relative maximum and minimum points (x and y values)

d. Inflection point(s) (x and y values)

e. Interval(s) where the function is concave up

f. Interval(s) where the function is concave down

Suppose a ball has a maximum height of 24 feet after 11 seconds of travel. This same ball will have a height of 8 feet after 20 seconds of travel. Find

Upper H left parenthesis Upper T right parenthesis equals AT squared plus BT plus Upper C commaH(T)=AT2+BT+C,

where​ H(T) is the height of the ball in inches after T seconds. Answers in Exact Form.

One year, an accountant prepared individual income tax returns for 439 clients. For these individual tax returns, completion of the following schedules was required: 298 required schedule A (itemized deductions), 256 required schedule B (interest and ordinary dividends), 167 required schedule C (business income), 212 required schedules A and B, 87 required schedules A and C, 61 required schedules B and C, and 54 required schedules A, B and C. Draw a Venn diagram to illustrate this situation. How many individual tax returns required:

(a) only schedule A?

(b) only schedule B?

(c) only schedule C?

(d) schedules A or C but not B?

(e) A and C but not B?

(f) none of these three schedules?

Find f.

f ''(x) = x⁻²,    x > 0,    f(1) = 0,    f(4) = 0

f(x)=

Suppose that A is a triangle with integer sides and integer area. Prove that the semiperimeter of A cannot be a prime number.

(Hint: Suppose that a natrual number x is a perfect square, and suppose that p is a prime number that divides x. Explain why it must be the case that p divides x an even number of times)

(2) A matrix A is given. Find, if possible, an invertible matrix P and a diagonal matrix D such that P −1AP = D. Otherwise, explain why A is not diagonalizable.

(a) A =   −3 0 −5

                0 2 0

                2 0 3

(b) A =   2 0 −1

             1 3 −1

             2 0 5

(c) A = 1 −1 2

             −1 1 2

              2 2 2

Find the relative maximum and minimum values.

a. f(x,y)=x^3-6xy+y^2+6x+3y-1/5

Relative minimum: ________ at ________

Relative maximum: ________ at ________

b. f(x,y)= 3x-6y-x^2-y^2

Relative minimum: ________ at ________

Relative maximum: ________ at ________

1.

You are given the graph of a function f defined on the interval (−1, ∞). Find the absolute maximum and absolute minimum values of f (if they exist) and where they are attained. (If an answer does not exist, enter DNE.)

The x y-coordinate plane is given. A curve, a horizontal dashed line, and a vertical dashed line are graphed.

• A vertical dashed line crosses the x-axis at x = −1.
• A horizontal dashed line crosses the y-axis at y = 1.
• The curve enters the window in the third quadrant just to the right of x = −1, goes up and right becoming less steep, becomes nearly horizontal at the origin, goes up and right becoming more steep, passes through the approximate point (1, 0.5), goes up and right becoming less steep, and exits the window just below y = 1.

absolute maximum

(x, y)=

absolute minimum(x, y)=

2.

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

g(x) = −x² + 4x + 9

maximum =

minimum=

3.

We first note that the function

f(x) = −x² + 2x + 8

is continuous and defined on the closed interval

[3, 6].

Recall that to find the absolute extrema of the given function we must first find any critical numbers of the function that lie in interval

[3, 6].

In other words, we need to find any values of x for which

f '(x) = 0,

or

f '(x)

does not exist.

Therefore, we must first find

f '(x).

4.

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) = x² − x − 3 on [0, 3]

maximum=

minimum=

5.

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) = 4x −9/X on [9, 11]

maximum=

minimum=

6.

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

f(x) =

on [−2, 1]

maximum=

minimum=

7.

Average Speed of a Vehicle

The average speed of a vehicle on a stretch of Route 134 between 6 A.M. and 10 A.M. on a typical weekday is approximated by the function

f(t) = 27t − 54

+ 64    (0 ≤ t ≤ 4)

where f (t) is measured in miles per hour, and t is measured in hours, with t = 0 corresponding to 6 A.M. At what time of the morning commute is the traffic moving at the slowest rate?

=A.M.

What is the average speed of a vehicle at that time?

=mph

8.

Maximizing Profits

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, produced by Phonola Media, is related to the price per compact disc. The equation

p = −0.00054x + 6    (0 ≤ x ≤ 12,000)

where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by

C(x) = 600 + 2x − 0.00003x²    (0 ≤ x ≤ 20,000).

To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is

R(x) = px,

and the profit is

P(x) = R(x) − C(x).

(Round your answer to the nearest whole number.)

= discs/month

9.

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

g(x) =

x² − 16

on [0, 36]

maximum=

minimum=

10.

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

g(t) =

on [4, 6]

maximum=

minimum=

11.

Enclosing the Largest Area

The owner of the Rancho Grande has 3,044 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river, what are the dimensions (in yd) of the largest area he can enclose?

A rectangular piece of land has been enclosed along a straight portion of a river. The enclosure is bordered by the river (a long side), another long side of fence, and two short sides of fence.

shorter side = yd

longer side= yd

What is this area (in yd²)?

= yd²

^12.

Packaging

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 16 in. long and 6 in. wide, find the dimensions (in inches) of the box that will yield the maximum volume. (Round your answers to two decimal places if necessary.)

smallest value = in

= in

largest value = in

13.

Minimizing Packaging Costs

A rectangular box is to have a square base and a volume of 72 ft³. If the material for the base costs $0.49/ft², the material for the sides costs $0.12/ft², and the material for the top costs $0.15/ft², determine the dimensions (in ft) of the box that can be constructed at minimum cost. (Refer to the figure below.)

A closed rectangular box has a length of x, a width of x, and a height of y.

x= ft

y= ft

14.

Charter Revenue

The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $568/person/day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 100) for the cruise, then each fare is reduced by $4 for each additional passenger.

Assuming at least 20 people sign up for the cruise, determine how many passengers will result in the maximum revenue for the owner of the yacht.

= passengers

What is the maximum revenue?

$ =

What would be the fare/passenger in this case? (Round your answer to the nearest dollar.)

= dollars per passenger

15.

Minimizing Packaging Costs

If an open box has a square base and a volume of 103 in.³ and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. (Round your answers to two decimal places.)

height = in

length = in

width= in

How far down (from a vertex) along each diagonal do the diagonals of a parallelogram intersect?