Exam 13: Mathematical Problem Set: Calculus, Algebra, and Optimization

A manufacturer with exclusive rights to a sophisticated new industrial machine is planning to sell a limited number of the machines to both foreign and domestic firms.The price the manufacturer can expect to receive for the machines will depend on the number of machines made available.(For example,if only a few of the machines are placed on the market,competitive bidding among prospective purchasers will tend to drive the price up.)It is estimated that if the manufacturer supplies x machines to the domestic market and y machines to the foreign market,the machines will sell for $60 - \frac { x } { 5 } + \frac { y } { 20 }$ thousand dollars apiece at home and for $50 - \frac { y } { 10 } + \frac { x } { 20 }$ thousand dollars apiece abroad.If the manufacturer can produce the machines at a cost of $38,000 apiece,how many should be supplied to each market to generate the largest possible profit?

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

The given series converges. $\sum _ { k = 1 } ^ { \infty } k ^ { - 2.5 }$

Find the points of intersection (if any)of the given pair of curves. y = x + 3 and y = 2x + 4

Use Lagrange multipliers to find the maximum value of f (x,y,z)= 6xyz subject to 2x + 5y + 2z = 60.

Evaluate the given expression. cos $\left( - \frac { 2 \pi } { 3 } \right)$

Consider the random experiment of rolling a fair die two times,and let X denote the random variable that counts the number of times 6 appears.Find $P ( X = 2 )$

A factory produces MP3 players.During regular inspections,2% of the MP3 players are found to be defective.On any particular day,what is the probability that the twentieth MP3 player inspected is the first defective player found that day? Round your answer to four decimal places.