M11 Short Course In Calculus V. J. Motto Spring 01 Applications of Derivatives Worksheet 1. A tomato is thrown from the top of a tomato cart its distance from the ground in feet is modeled by the equation d( t) 16t 5 5.4 where t is measured in seconds and the initial height of the cart is 5.4 feet. (A) At what time is the tomato at its maimum height? (B) What is the maimum height? (C) What is the initial velocity of the tomato (at t = 0)?. A landlord rents an apartment building with 50 apartments. The monthly profit in dollars can be modeled by P( ) 8 00 80000 where is the number of apartments rented. How many apartments should be rented to maimize profit?. 4. The population of a town is approimated by the function 90000(1.0)t the number of years since 1980. Find P'(0). Round to the nearest whole number. The value of a car is falling at 10% per year so that if C0 is the purchase price of the car in V ( t) C dollars its value after t years is given by 0(0.9) t. How fast is the car depreciating after 4 years? 5. Find the equation of the tangent line to the curve y e which passes through the origin. 6. The population of Ghostport has been declining since the beginning of 1800. The 0.005t population in thousands is modeled by P( t) 0e measured in years. At what rate was the population declining at the beginning of 000? 7. 8. Consider the function g( ). Give the equation of the tangent line at =. Round the coefficients to decimal places. P P With a yearly inflation rate of % prices are described by 0 (1.0)t where P0 is the price in dollars when t = 0 and t is time in years. If P 0 = 1. how many cents per year are prices rising when t = 1? Round to the nearest tenth of a cent. - 1 -
9. The equation of the tangent line to the curve y-ais is y = +. g( ) e at the point where it touches the 10 If $100 is invested at r % interest per year compounded yearly then the yield after 15 years 15 r df F 100 1 is given by 100. Find dr r 5. 11 1. 1 0.07t The population of Meico in millions is described by the formula the number of years after 1980. In the year 015 the population will be increasing at the rate of million people per year. Round to decimal places. 0.07t The population of Meico in millions is described by the formula the number of years after 1980. How many years will it take for the population to double? Round to decimal places. 0.07t The population of Meico in millions is described by the formula the number of years after 1980. How many years will it take before the population is increasing at a rate of 8 million people per year? Round to decimal places. 14 The price in dollars of a house during a period of mild inflation is described by the formula 0.05t P( t) (80000) e the number of years after 1990. By how many dollars per year will the value of the house be increasing in the year 010? Round to the nearest dollar. 15 The price in dollars of a house during a period of mild inflation is described by the formula 0.05t P( t) (100000) e the number of years after 1990. How many years will it take for the house to triple in value? Round to the nearest year. 16. 17. 0.848 The cost to produce q aircraft in South Africa is given by the function C( q).5q with C in millions of rands. At a production level of 50 aircraft how many million rands will it cost to produce one more aircraft? Round to decimal places. 0.5t A drug has a concentration in the body given in ng/ml by the function f ( t) 15e where t is the number of hours after it was administered. By how many ng/ml per hour is the amount of the drug in the body changing after 7 hours? Round to decimal places. - -
18. The following table gives values for two functions f and g and their derivatives. What is d [ f ( ) g ( )] d 0? -1 0 1 f 1 0 1 g 1.5 4 f ' - - -1.5-1 1 g'.5 19. The quantity q of tickets sold for a certain flight is a function of the selling price p. Thus q f ( p). You are given the information that f (50) 180 and f '(50) 1. Revenue is given by R pq. When tickets are being sold at a price of $50 an increase of $1 in the sales price will cause revenue to go down by how much? 0. g The concentration in ml of a drug introduced gradually into the body can be modeled 4t Ct () by 0.01t minutes. At what time does the concentration reach its maimum? 1. The following figure is a graph of a derivative function f '. Indicate on the graph the - values that are critical points and label each as a local maimum a local minimum or neither.. For which interval(s) is the function f ( ) 1 decreasing?. Find all of the critical points of f ( ) 1. List them from smallest to largest separated by commas. 4. Suppose f has a continuous derivative whose values are given in the following table. -4 - - -1 0 1 4 f '( ) - -1 1-1 - 1 Is =.5 a potential local minimum local maimum or neither? - -
5. Sketch a graph of a function such that 0 when > 1. f ( ) 0 at = 1 f ( ) < 0 when < 1 f ( ) < 5 y 4 1-5 -4 - - -1 1 4 5-1 - - -4-5 6. A stone is thrown vertically upward so that its height measured in feet after t seconds is given by s( t) 88t 16t. What is the time of flight of the stone (in seconds)? 7. The quantity of a medication in the bloodstream t hours after it is ingested is given in mg by q( t) 00te t. What is the maimum quantity of the medication in the bloodstream? A) 110 mg B) 00 mg C) 815 mg D) 150 mg 8. Daily production levels in a plant can be modeled by the function G( t) t 1t 15 which gives units produced at t the number of hours since the factory opened at 8 am. Factory productivity is at a maimum at am. - 4 -
9. The distance s traveled by a runner in a 0 mile race is given in the following figure where time t is in hours. At which of the following values of t is the runner's speed is the slowest? A) 4 B) C) 1 D) 0. The following table shows cost and revenue for a product (in dollars). A. What is the price of the product? B. At what value of q is profit is maimized? q 0 1000 000 000 4000 5000 R(q) 0 500 1000 1500 000 500 C(q) 100 50 400 800 1400 000 1. Total cost and revenue are approimated by the functions C 1600.9q and both in dollars. R 7q A. What is the marginal cost per item? B. What is the profit function?. A factory produces a product that sells for $11. They currently produce 400 items per month at an average cost of $5 per item. The marginal cost at this level is $4. Assume that the factory can sell all the items that it produces. A. What is the profit at this production level? B. Would increasing production increase or decrease average cost? - 5 -