In Yahtzee, a “large straight” is a roll that leads to five dice displaying 1-2-3-4-5 or 2-3-4-5-6. What is the probability that a player will score a large straight with a single roll of all six dice?

Find the integrals of the following:

1) 6sin²xcos³x dx

2) 9xcos²x dx

3) 3cos²xtan³x dx

4) 7cot⁵xsin⁴x dx

5) 3tan²xsecx dx

1:

Given that f(4) = 6 and f'(x) = 2/x²+9 for all x.

a) Use a linear approximation or differentials to estimate f(4.04)

b) Is your estimate in part (a) too large or too small? Explain.

2:

a) Given f(x) = (x + 3)^sinx, find f'(π) using logarithmic differentiation.

b) Find the value of h'(0) if h(x)+xsin(h(x))= x²+4x-π/2

Assume we roll a fair four-sided die marked with 1, 2, 3 and 4.
(a) Find the probability that the outcome 1 is first observed after 5 rolls.
(b) Find the expected number of rolls until outcomes 1 and 2 are both observed.
(c) Find the expected number of rolls until the outcome 3 is observed three times.
(d) Find the probability that the outcome 3 is observed exactly three times in 10 rolls
given that it is first observed after 5 rolls.
(e) Find the probability that the outcome 3 is first observed after 5 rolls given that
it is observed exactly three times in 10 rolls

1. What is dollarization?

2. What is currency union?

3. What is asymmetric information?

4. What was the name of the first bankers?

part 1)

Let f(x) = x^4 − 2x^2 + 3. Find the intervals of concavity of f and determine its inflection point(s).

part 2)

Find the absolute extrema of f(x) = x^4 + 4x^3 − 8x^2 + 3 on [−1, 2].

f(x)= 2x^4 - 4x^2 + 1

a. Indicate where the function is increasing or decreasing.

b. List the coordinates of where extrema occur.

c. State where the graph is concave up or concave down.

d. List the coordinates of where points of inflection occur.

4) Consider total cost and total revenue given in the following table:

Quantity 0 1 2 3 4 5 6 7 7

Total Cost $120 150 160 170 190 230 300 410 610

Total Revenue $0 70 140 210 280 350 420 490 560

A. Create a table that includes the fixed cost at each level of output as well as the variable cost

at each level of output. If capital is fixed and the firm uses 10 units of capital, then what is

the price of capital? If labor is variable and the firm uses 5 units of labor to produce 3 units

of output, then what is the price of labor?

B. Create a table that includes the marginal cost, average variable cost, and average total cost

associated with each level of production. Graph these three cost measures on a single graph

with quantity on the horizontal axis. Explain how you know the firm exhibits the law of

diminishing returns.

C. Calculate profit (revenue minus costs) for each quantity. Calculate marginal revenue for

each quantity. Verify that the profit maximizing firm will choose the highest level of profit

at a point where MR = MC.

D. This firm is in a perfectly competitive industry? Explain how we know this is true. What

do you expect to happen in the long run in this industry? Will firms enter or leave the

market in the long run? Will the market price increase or decrease in the long run?

A set of data has the following coordinates

a) Find the least-squares fit to this data by a linear function of t (that is, find constants c1,c0 so that y(t) = c1t + c0 is the best linear fit to this set of data).

b) Find the equation of the best quadratic fit to the same set of data. Then find the equation of the polynomial of smallest degree that fits the data exactly.