Final Exam Review - Business Calculus - Spring x x

Transcription

1 Final Exam Review - Business Calculus - Spring 2016 Name: 1. (a) Find limit lim x 1 x 1 x 1 (b) Find limit lim x 0 x x 1

2 2. Use the definition of derivative: dy dx = lim f(x + h) f(x) h 0 h Given y = f(x) = x 2 + x find dy using the definition dx 3. Find the derivative of y = x 1 x x + 1 x 2

3 4. Find the derivative of y = e x sin x cos x 5. Find the derivative of y = e x3 +9x 2 3

4 6. Implicitly find dx dy for 4x2 + 3y 2 = You invest $20,000 in a new business venture. Your monthly revenue is $1,500 and monthly cost of doing business is $1,000; how long would it take to break even in this business at this rate? 4

5 8. Demand and Supply: For a consumer, quantity demanded decreases as price increases. But for a supplier, it is just the opposite; the quantity supplied increases as price increases: (a) Find the equilibrium price, the price at which the demand meets the supply, for an item in a free market system given that the demand curve for the item is x = p and the supply curve is x = p (b) Sketch the general shape of the demand and supply curve 5

6 9. An object is thrown straight up with initial velocity of 160 feet per second from the ground. Find the follwing for this object: (a) Find x(t), the position of the object as a function of time (b) Find the velocity of the object as a function of time (c) Find the acceleration of the object as a function of time (d) Find the highest point the object reaches (e) How long after the object is thrown, does it return to the ground? 6

7 10. Marginal Profit: Profit for a cell phone accessories business owner is given by P (x) = 0.05x 2 + 5x. Find the marginal profit for this business when quantity sold is For the function f(x) = x 4 + 3x 3 5 do the following: (a) Find the critical points and label it in a rectangular coordinate system (b) Find the local maximum and minimum values for this function (c) Find the regions where the function is increasing or decreasing 7

8 (d) Find the points of inflection for this function (e) Determine concavity for this function at different regions (f) Use the second derivative test to determine local extreme values 12. A rectangular paper will have 25 square inches of writing space on it with top and bottom margins 1 inch each, and right and left margins 1 inch each. Find the dimension of the page that will minimize paper consumption. 8

9 13. If advertisement cost for a business is given by f(x) = 1 12 x3 + 5x , at what point does the does the law of diminishing return start to take effect for an advertisement dollar? 14. Find the extreme values of the function f(x) = x 3 x 2 5x on the interval [ 3, 3] 9

10 15. For a linear demand function, we noticed that when price is $50, quanitiy desired is 500 units per year and when price is $60, quantity desired is 400 per year. Cost function is given by C(x) = 20x Find the number of units that needs to be sold to maximize profit. 16. Find the value/s of x in the demand function p(x) = x for which price elasticity is 1,that is unit elasticity and demand is elastic, and demand is inelastic. 10

11 17. $10,000 was invested for 10 years at 5% yearly interest rate. Find the amount of money in the account if interest is compounded daily and continuously. 18. The value of a home doubles roughly every 17 years. If a home is bought now for $300,000 then in how many years would the price of the home is expected to be about $800,000? 11

12 19. At 5% yearly rate what is the present value of $10, 000 to be received in 5 years if interest is compunded continuously? 20. A business depreciates a top end electronic equipment using exponential model. After 2 years the value of the item is $111,000 and after 4 years the value of the item is $78,000. Find the purchase price of the item and its value in 7 years. 12

13 21. Integrate (5x 3 3x 2 + 2)dx 22. Marginal cost for a business is given by dc = x. Find the cost function for dx this business given that the production cost for an item is $ Integrate (3x 2 + 2) x 3 + 2xdx 24. Integrate 3x x3 + 2x dx 13

14 25. Integrate xe x2 dx 26. Integrate 2x + 5 x 2 + 5x dx 27. Integrate 3 1 e x 1 + e x dx 28. Find the average value of the function f(x) = x 2 x over the interval [ 1, 1]. Use the 1 b formula a b a f(x)dx 14

15 29. Amount of annuity is given by A(t) = e rt T 0 c(t)e rt dt. If you start saving $500 per month when you are 25 and continue to do so until you retire at 65. You invest the money in S&P 500 that returns about 5% per year. How much money would you have in your account when you retire? 30. Producer and consumer surpluses: Demand p = 0.36x + 9 and supply p = 0.14x + 2. When price is $3.96 find the consumer and producer surpluses. 31. Integrate xe x dx 15

16 32. Integrate x 2 ln xdx 33. Integrate ln xdx 34. You will work for the next 30 years making on the average $100,000 per year. Use integral to find out how much money you will make over the 30 years and also find the present value of this cash flow assuming 3% yearly interest rate. Use present value formula T 0 c(t)e rt dt 16