5.25. Solve the following initial value problem. ty' - 2y = t³ cost, y(л) = 2

8. The manager of an appliance manufacturing firm determines that when toasters are priced at p dollars apiece, the number sold each month can be modeled by 300 A(p) = P The manager estimates that t months from now, the unit price of the toasters will be p(t) = 0.04ť² + 18.5 dollars. At what rate will the monthly demand for toasters, A(p), be changing two years from now? Will it be increasing or decreasing at this time?

5. The accumulated average price in millions of dollars of a two-bedroom apartment in downtown New York City from 1990 to 2010 could be approximated by P(t) = = 0.63e0.25t, where t is time in years since 1990. a) Determine P(6) and interpret your answer in context of the problem. b) Determine P'(6) and interpret your answer in context of the problem c) Determine P"(6) and interpret your answer in context of the problem.

4. Let C(x) represent the cost, R(x) the revenue, and (x) the total profit, in dollars of producing x units of a widget. If C'(400) = 18 and R'(400) = 15, what is the approximate profit or loss by the 401st widget? Show your computations that lead to your answer.

21. Suppose the number of users for a new social media app increases according to y = Yoe0.1t where y is measured in millions and t is measured in weeks. How long will it take for the number of users to triple? (a) In(30) weeks (b) 10 In(3) weeks (c) 10 In(3yo) weeks (d) 0.1 In(3) weeks (e) 3 In(1.1) weeks

26. A population P(t) of small rodents has birth rate ß = (0.001)P (births per month per rodent) and constant death rate 8. If P(0) = 100 and P'(0) = 8, how long (in months) will it take this population to double to 200 rodents? (Sug- gestion: First find the value of 8.)

1. Using differentials, solve for the approximate change in the hypotenuse of a right triangle if the lengths of its legs change from 8 meters and 15 meters to 8.1 meters and 14.75 meters, respectively. Express your answer in four (4) decimal places. Hint: Set up the equation of the hypothenuse in terms of its legs.

2.4ax'y = 4 find y %3D

1. Using differentials, solve for the approximate change in the hypotenuse of a right triangle if the lengths of its legs change from 8 meters and 15 meters to 8.1 meters and 14.75 meters, respectively. Express your answer in four (4) decimal places.

6. The average cost of a product changes at the rate C '(x) = −4x−2 +  1 4 and the average cost of 4 units is $8.00. (a) Find the average cost function. C(x) =        (b) Find the average cost of 13 units. (Round your answer to the nearest cent.)