Math 141 Final Exam Review

Transcription

1 Math 141 Final Eam Review Problems appearing on your in-class final will be similar to those here but will have numbers and functions changed. 1. (.1) Use the graph below to find the following: a) lim 4 f() = f) lim f() = b) lim 4 + f() = g) lim + f() = c) lim 4 f() = h) lim f() = d) f(4) = i) f() = e) Is f continuous at = 4? j) Is f continuous at =? Math 141 Final Review, Dec. 18, 015

2 . (.1)Given that lim 1 f() = 5 and lim 1 g() = 4, find the limits below. a) lim 1 ( 5f()) b) lim 1 (g() + 6) c) lim 1 f() + g(). (.1) If 6 1, if 1 4, if 1 Find the following: a) lim 1 f() = b) lim 1 + f() = c) lim 1 f() = d) f( 1) = e) Is f continuous at = 1? Math 141 Final Review, Dec. 18, 015

3 4. (.1) If f() = 4 5 +, then find the limit of the following difference quotient: lim h 0 f(6 + h) f(6). h 5. (.) Find all vertical and horizontal asymptotes of the function f( ) (.) Solve the inequality given below. Use both intervals notations and inequality notations for your answer (.4) If an object moves along a line so that it is at y = f() = 4 at time (in seconds), find the instantaneous velocity function v = f () and find the velocity at times = 1,, and 5 seconds (y is measured in feet). 8. (.5) Find f () if f() = (.6) Find dy and Δy for y = ( 8) when = and d = Δ = (.7) The total cost and the total revenue (in dollars) for the production of alarm clock radios are given by the following functions: C() = 5 +,500 and R() = a) Find the profit function P(). b) Find the marginal profit at a production level of 150 alarm clock radios. c) Find the average profit per unit at a production level of 150 alarm clock radios. d) Find the marginal average profit at a production level of 150 alarm clock radios. e) Use the results from part c) and d) to estimate the average profit per unit at a production level of 151 alarm clock radios. Math 141 Final Review, Dec. 18, 015

4 11. (.7) The total cost (in dollars) of producing food processors is C( ), (a) Find the eact cost of producing the 1 st food processor. (b) Use marginal cost to approimate the cost of producing the 1 st food processor. 1. (.6) A sphere with a radius of 5 centimeters is coated with ice 0.1 centimeter thick. Use 4 differentials to estimate the volume of the ice. [Recall that V r. ] 1. (.1) A couple paid $10,000 for a house in They sold the house in 015 for $450,000. If interest is compounded continuously, what annual nominal rate of interest did the original $10,000 earn? 14. (.) The model T ln p 7.91 p relates the temperature T, in F at which water boils at pressure p in pounds per square inch. Find the rate of change of temperature when pressure is 70 pounds per square inch. 15. (.1,.,.4) The drug concentration in the bloodstream t hours after injection is given approimately by C( t) 5 0.t e, where () Ct is concentration in milligrams per millimeter. a. What is the rate change of concentration after 5 hours? b. When is the rate of change of concentration 0.8 mg per milliliter per hour? c. When is the drug concentration 1.5 mg per milliliter? 16. (.,.,.4) Find dy for the following functions. d 1 a. y 6e ln log b. y e ln(5 ) 4 c. y 5 Math 141 Final Review, Dec. 18, 015

5 d. y ln e. 9 y (.4) Ecologists estimate that the average level of carbon monoide in the air above a city is given by L where L is in parts per million and is the population of the city in thousands. The population is estimated as a function of t years from the present by 75 5t 0.5t a. Find dl d b. Find dl dt c. How fast is the carbon monoide level changing at time t years? 18. (.5) Consider the equation ye y 0 a. Use implicit derivative and find y b. Find the equation of the tangent line at the point 0,1 19. (.7) Given the demand equation f ( p) 8,100 9p, p f ( p) a. Find E( p ), the elasticity of demand. Recall: E( p). f( p) b. Determine whether the demand is elastic, inelastic or has unit elasticity at p (.) The function R ( ) 80, models the amount of plastic carry-out bags recycled, in tons, in the United States where is the number of years since a. Find the amount of plastic carry-out bags recycled in 015 b. Find the rate of growth in recycled plastic carry-out bags (in tons per year) in 015 Math 141 Final Review, Dec. 18, 015

6 1. (.) When a company produces and sells thousands units per week, its total weekly 00 profit in thousands of dollars is given by P. 100 a. Find (6) P and P 6 b. How fast is the total weekly profit of the company changing when 6 thousand units?. (.6) A 41 foot long ladder is leaning against a vertical wall of a house. The bottom of the ladder is pulled away from the house wall at a constant rate of 1.6 feet per second. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 9 feet from the wall?. (.6)Suppose that for a company manufacturing calculators, the cost, revenue and profits equations are given by C90,000 0, R00 and P R C where the production 0 output in one week is calculators. If production is increasing at the rate of 00 calculators per week when the production output is 4,000 calculators, find the rate of increase (or decrease) in a. the cost, b. the revenue and c. the profit. 4. (.7) Given the demand equation p a. Epress the demand as a function of a price p and find its domain. b. Epress the revenue R as a function of the price p and find its domain. c. Find the elasticity of demand E( p ). Recall: d. For which values of p is demand inelastic? e. For which values of p is revenue decreasing? p f ( p) E( p) f( p) Math 141 Final Review, Dec. 18, 015

7 5. (4.1) Use the given graph of y f ( ) to find the intervals on which f is increasing, the intervals on which f is decreasing, and the coordinates of the local etrema of f. Sketch a possible graph of y f ( ). 6. (4.1, 4.) Use the given information to sketch the graph of f. Assume that f is continuous on its domain and that all intercepts are included in the information given. 7. (4.) A company estimates that it will sell N units of a product after spending $ thousand on advertising, as given by 4 N , 000, 9 4 When is the rate of change of sales increasing and when is it decreasing? What is the point of diminishing returns and the maimum rate of change of sales? Graph N and N on the same coordinate system. Math 141 Final Review, Dec. 18, 015

8 8. (4.) Find each limit. Note that L Hôpital s rule does not apply to every problem, and some problems will require more than one application of L Hôpital s rule. e (a) lim (b) lim 0 6 ln(1 ) ln(1 ) (c) lim (d)lim e e e (e) lim (f) lim ln (g) lim (h) lim 5 0 ln(1 6 ) ln(1 6 ) (i) lim (j) lim ln(1 ) 0 ln(1 ) 9. (4.1, 4., 4.4) Use the given information to sketch the graph of f. Assume that f is continuous on its domain and that all intercepts are included in the information given. 0. (4.5, 4.6) A company manufactures and sells e-book readers per month. The monthly cost and price-demand equations are, respectively, C( ) 50 50,000 p , 0 0,000 (a) Find the maimum revenue. (b) How many readers should the company manufacture each month to maimize its profit? What is the maimum monthly profit? How much should the company charge for each reader? Math 141 Final Review, Dec. 18, 015

9 (c) If the government decides to ta the company $0 for each reader it produces, how many readers should the company manufacture each month to maimize monthly profit? How much should the company charge for each reader? 1. (4.5, 4.6) A fence is to be built to enclose a rectangular area. The fence along three sides is to be made of material that costs $5 per foot. The material for the fourth side costs $15 per foot. (a) If the area is 5,000 square feet, find the dimensions of the rectangle that will allow for the most economical fence. (b) If $,000 is available for the fencing, find the dimensions of the rectangle that will enclose the most area.. (4.5, 4.6) A 00-room hotel in Reno is filled to capacity every night at a rate of $40 per room. For each $1 increase in the nightly rate, 4 fewer rooms are rented. If each rented room costs $8 a day to service, how much should the management charge per room in order to maimize gross profit? What is the maimum gross profit?. (4.5, 4.6) The price-demand equation for a GPS device is p( ) 1, e where is the monthly demand and p is the price in dollars. Find the production level and price per unit that produce the maimum revenue. What is the maimum revenue? 4. (4.1 and 4.4) Nicole owns a company that makes luurious velvet robes. Her total cost to make robes can be modeled by the function C( ) 1500, 0. (a) Find the average cost function. (b) How many robes must be produced for the average cost to be minimized? (c) What is the minimum average cost? Math 141 Final Review, Dec. 18, 015

10 5. (5.1) Find each indefinite integrals. d dw e d w (a) 4 (b) (c) t t (d) 1 d (e) d (f) dt t 6. (5.1) The marginal profit from the sales of items is given by P P if P(100) 500. Find 7. (5.1) The rate of change of the monthly sales of a newly released football game is given by S t t S 1/4 500, 0 0 where t is the number of months since the game was released and St is the number of games sold each month. Find S t. When will monthly sales reach 0,000 games? 8. (5.) Find each indefinite integral and check the result by differentiating (a) 6 1 d (b) d (c) 10 1 d (d) 7 e d (e) 1 e d (f) d 1 (g) d (h) 1 d (i) d t 4 7 Math 141 Final Review, Dec. 18, 015

11 9. (5.) The market research department for an automobile company estimates that sales (in millions of dollars) of a new electric car will increase at the monthly rate of 0.08 t S t 4e 0 t 4 t months after the introduction of the car. What will be the total sales St t months after the car is introduced if we assume that there were 0 sales at the time the car entered the marketplace? What are the estimated total sales during the first 1 months after the introduction of the car? How long will it take for the total sales to each $40 million? 40. (5.) The area of a healing skin wound changes at a rate given approimately by da dt 5t 1 t 5 where t is time in days and wound in 5 days? A1 5 square centimeters. What will be the area of the 41. (5.) The marginal price dp at units of supply per day is proportional to the price p. d There is no weekly demand at a price of $00 per unit [ demand of 10 units at a price of $50 per unit [ p10 50 ]. p0 00 ], and there is a weekly (a) Find the price-demand equation. (b) At a demand of 0 units per week, what is the price? (c) Graph the price-demand equation for (5.5) The total cost (in dollars) of making music boes is given by C 1, (a) Find the average cost per unit if 00 music boes are produced. (b) Find the average value of the cost function on the interval [0, 00]. (c) Eplain the difference in the meaning of the values found in part (a) and (b). Math 141 Final Review, Dec. 18, 015

12 4. (5.5) A company produces a printer that also scans documents. The research department produced the marginal cost function C 00 where 5 C is the total cost (in dollars) and is the number of printers produced in a month. Compute the increase in cost going from a production level of 100 printers per month to 500 printers per month. Set up a definite integral and evaluate. 44. (5.5) The total accumulated costs Ct and revenues respectively, for a coal mine satisfy 0.1t and 0 C t R t e Rt (in thousands of dollars), where t is the number of years that the mine has been in operation. Find (a) the useful life of the mine, to the nearest year, recall the value of t for witch C( t) R( t) is called the useful life. (b) What is the total profit accumulated during the useful life of mine? Answer Keys: 1. a) -1. b) -1. c) -1. d) -1. e) yes. f) -5. g) -. h) does not eist. i) -5. j) no.. a) -5. b) -6. c) 1.. a) -6. b) 4. c) does not eist. d) 4. e) no Vertical asymptotes: The line 1 and the line 4. The horizontal asymptote is the line: y. 6. Interval notations:(, 1] (0, ). Inequality notations: 1 or ft/sec; ft/sec; 8ft/sec Math 141 Final Review, Dec. 18, 015

13 dy = 4; y = a) ,500 b) $10 per alarm clock radio c) $1.67 per alarm clock radio d) The average profit is increasing at the rate of $0.06 per alarm clock radio e) About $1.7 per alarm clock radio. 11. (a) $9.50 (b) $ cm % F per pound per square inch 15. a. 0.5 mg/ml b..10 hours c. 4.01hours 16. a. 6e 1 1 (ln10) b. e e ln 5 c d e a b t 0.001t 5 t per million per year c. increasing at the rate of14.9 parts 18. a. ye y e b. y a. p 900 p b. elastic 0. a.1,180,645.5 tons b. 167,65.8 tons/year 1. a. P(6) 5.16 and P(6) 0.17 b. Decreasing at the rate of 1 cents per unit Math 141 Final Review, Dec. 18, 015

14 . 0.6 ft/s. a. $9,000 /week b. $10,000 /week c. $1,000 /week 4. a p, 0 p 5 b., 0 p 5 R( ) 500 p 100 p p c. E( p) 5 p d. 0,1.5 e. 1.5,5 5. Increasing on (, )and (1, ) ; decreasing on (,1) ; local maimum at ; local minimum at (a) (b) -1/5 (c) (d) 0 (e) (f) 1 (g) 0 (h) 0 (i) 1 (j) Math 141 Final Review, Dec. 18, 015

15 9. 0. (a) Ma R( ) R(10,000) $,500,000.(b) Maimum profit is $175,000 when,000 readers are manufactured and sold for $45 each. (c) Maimum profit is $119,000 when,600 readers are manufactured and sold for $45 each. 1. (a) The epensive side is 50 ft; the other side is 100 ft. (b) The epensive side is 75 ft; the other side is 150 ft.. $49; $6,74.. A maimum revenue of $18,94 is realized at a production level of 50 units at $67.88 each (a) C( ) 1500, (b) robes, (c) $ (a) 4 t t C C (b) ln w C (c) 6. P , 450 e C (d) 4 C (e) 4 ln C (f) 9 7. S( t) 400 5/4 t, 4/5 50 mo (a) 4 1 C (b) ln C (g) 1 t 4 C C (c) 6 C (d) (h) e 5/ / C 5 C(e) 1 e C (f) Math 141 Final Review, Dec. 18, 015

16 / 1/ C (i) t S t e e 40. 1cm ; $1million; ln 0. / mo. 41. (a) p e (b) $09.0 (c) The graph of p ( ) in the window [0, 50] by [-10, 10] 4. (a) $100 (b) $16,000 (c) The average cost function yields the average cost per music bo. The average value of the cost function yields the average total cost of each production level between 0 and d 5 ; $56, (a) Useful life = 0 10 ln 19 years; (b) Total profit = e or $11,086. Math 141 Final Review, Dec. 18, 015