Math Want to have fun with chapter 4? Find the derivative. 1) y = 5x2e3x. 2) y = 2xex - 2ex. 3) y = (x2-2x + 3) ex. 9ex 4) y = 2ex + 1

Transcription

1 Math Want to have fun with chapter 4? Name Find the derivative. 1) y = 52e3 2) y = 2e - 2e 3) y = ( ) e 9e 4) y = 2e + 1 5) y = e e e 6) y = ) y = e3-1 5 Use calculus to find any critical points and inflection points of the given function. Then determine the concavity of the function and the intervals over which it is increasing/decreasing. JUSTIFY ALL ANSWERS! 8) f() = e5 9) f() = e-9 10) f() = 4 - e- Find the indicated tangent line. 11) Find the tangent line to the graph of f() = e5 at the point (0, 1). Solve the problem. 12) The sales in thousands of a new type of product are given by S(t) = e-0.7t, where t represents time in years. Find the rate of change of sales at the time when t = 4. 13) The demand function for a certain book is given by the function = D(p) = 70e-0.003p. Find the marginal demand D'(p). 14) The nationwide attendance per day for a certain motion picture can be approimated using the equation A(t) = 15t2e-t, where A is the attendance per day in thousands of persons and t is the number of months since the release of the film. Find and interpret the rate of change of the daily attendance after 4 months. For the given function, find the requested relative etrema or etreme value. 15) y = 9e- ; maimum value on [0, 2] 16) y = - 6e- 2 ; relative etrema 1

2 Write an equivalent eponential equation. 17) log w Q = 10 Write an equivalent logarithmic equation. 18) FX = A Solve the eponential equation for t. Round your answer to three decimal places if necessary. 19) e-0.06t = 0.6 Find the derivative of the function. 20) y = ln 4 21) y = ln ( - 8) 22) y = ln (7 + 2) 23) y = ln (73-2) 24) y = ln 5 25) f() = ln 4-4 Find the derivative. 26) y = e ln 27) y = e ln 28) f() = ln (e4-4) Solve the problem. 29) A model for advertising response is given by N(a) = ln a, a 1 where N(a) = the number of units sold and a = the amount spent on advertising, in thousands of dollars. How many units are sold after spending $5000 (a = 5) on advertising? 30) A company begins an advertising campaign in a certain city to market a new product. The percentage of the target market that buys the product is a function of the length of the advertising campaign. The company estimates this percentage as 1 - e-0.04t where t = number of days of the campaign. The target market is estimated to be 1,000,000 people and the price per unit is $0.40. The cost of advertising is $4000 per day. Find the length of the advertising campaign that will result in the maimum profit. 2

3 31) The demand function for a certain product is given by D(p) = 800e-0.1p, where p is price per unit. Recall that total revenue is given by R(p) = pd(p). At what price per unit p will the revenue be maimum? 32) The percentage P of doctors who accept a new medicine is given by P(t) = 100(1 - e-0.11t), where t = time in months. Find P'(8). 33) Students in a math class took a final eam. They took equivalent forms of the eam in monthly intervals thereafter. The average score S(t), in percent, after t months was found to be given by S(t) = 71-16ln (t + 1), t 0 What was the average score after 8 months? 34) Students in a math class took a final eam. They took equivalent forms of the eam in monthly intervals thereafter. The average score S(t), in percent, after t months was found to be given by Find S'(t). S(t) = 70-16ln (t + 1), t 0 35) Suppose that the population of a town can be approimately modeled by the formula P = 8 ln 5t + 9 where t is the time in years after 1980 and P is the population of the town in thousands. Find an epression for dp/dt in terms of t. 36) The population of a particular city (in thousands) can be modeled by the function 500 P(t) = e-0.05, where is the number of years after In what year was the growth rate of the population the fastest? Find all relative maima or minima. 37) y = (ln )2 38) y = ln - Solve the problem. 39) Find the eponential function that satisfies the equation dn dt = kn. 40) Find the eponential function that satisfies the equation dy d = 5y. 41) If $2500 is invested in an account that pays interest compounded continuously, how long will it take to grow to $5000 at 6%? 3

4 42) Assume the cost of a gallon of milk is $3.00. With continuous compounding, find the time it would take the cost to be 5 times as much (to the nearest tenth of a year), at an annual inflation rate of 6%. 43) Suppose that P0 is invested in a savings account in which interest is compounded continuously at 5.2% per year. That is, the balance P grows at the rate given by dp dt investment double? = 5.2P. Suppose that $6000 is invested. When will the 44) In 1990, a company's profit was 44.5 million dollars. In 2000, the company's profit was million dollars. Assume that the growth of the company's profit follows the eponential model and use 1990 as the base (t = 0). Estimate what the company's profit will be in ) If a population doubles every 26 years, what is its growth rate to the nearest hundredth of a percent? 46) Initially, a population of rabbits was found to contain 146 rabbits. It was estimated that the population was growing eponentially at the rate of 9% per day. Estimate the population after 47 days. 47) Management at a factory has found that the maimum number of units a worker can produce in a week is given by P(t) = 52 (1 - e- 0.5t), where t is the number of weeks the worker has been on the job. Find the rate of change P (t). 48) A pharmaceutical company introduces a new headache medication on the market. They advertise the product on television and find that the percentage P of people who buy the product after t weeks satisfies the function P(t) = 100% e-0.17t. What percentage buy the product after 16 weeks? 49) A pharmaceutical company introduces a new headache medication on the market. They advertise the product on television and find that the percentage P of people who buy the product after t weeks satisfies the function P(t) = 100% e-0.17t. Find the formula for the rate of change P'(t). 50) The natural resources of an island limit the growth of the population. The population of the island is given by the logistic equation P(t) = e-0.31t where t is the number of years after What is the limiting value of the population? 4

5 51) The number of employees of a company, N(t), who have heard a rumor t days after the rumor is started is given by the logistic equation N(t) = e-0.2t. How many employees have heard the rumor 14 days after it is started? 52) Find the tripling time for an amount invested at a growth rate 6% per year compounded continuously. 53) Following the birth of a child, a parent wants to make an initial investment P0 that will grow to $55,000 by the child's 20th birthday. Interest is compounded continuously at 5.9%. What should the initial investment be? 54) A business estimates that the salvage value V of a piece of machinery after t years is given by V(t) = $35,000e-0.43t. After what amount of time will the salvage value be $7771? For the scatter plot below, determine which, if any, of the following functions might be used as a model for the data. quadratic: f() = a2 + b + c polynomial, not quadratic eponential, f() = bek, k > 0 eponential, f() = be-k, k > 0 logarithmic, f() = a + b ln a logistic, f() = 1 + be-k 55) y

6 56) y Year 57) y Year 58) y Solve the problem. Year 59) A radioactive substance has a decay rate of 5.3% per day. What is its half-life? 60) A radioactive substance has a half-life of 4410 years. What is its decay rate? 61) How old is a skeleton that has lost 18% of its carbon-14? The decay rate, k, of carbon-14 is % per year. 62) An artifact is discovered at a certain site. If it has 69% of the carbon-14 it originally contained, what is the approimate age of the artifact? (carbon-14 decays at the rate of % annually.) (Round to the nearest year.) 6

7 63) A beam of light enters sea water with initial intensity I0. Its intensity at a depth of meters is given by I = I0e-1.4. What percentage of I0 remains at a depth of sea water of 2 meters? 64) The coroner arrives at the scene of a murder at 11 p.m. She takes the temperature of the body and finds it to be 81.4e C. She waits 1 hour, takes the temperature again, and finds it to be 80.1e C. She notes that the room temperature is 69e C. When was the murder committed? 65) The temperature of a hot liquid is 100e C and the room temperature is 75e C. The liquid cools to 96.8e C in 3 minutes. What is the temperature after 13 minutes? Round your answer to the nearest degree. 7

8 Answer Key Testname: CHAPT4 1) 5e3(3 + 2) 2) 2e 3) (2 + 1) e 4) 9e (2e + 1)2 5) -e - 2 e2 6) e ( ) (32 + 7) 2 7) 152e3 e ) Critical points: none Inflection points: none Concavity: concave up for all real numbers Increasing: increasing for all real numbers 9) Critical points: none Inflection points: none Concavity: concave up for all real numbers Decreasing: decreasing for all real numbers 10) Critical points: none Inflection points: none Concavity: concave down for all real numbers Increasing: increasing for all real numbers 11) y = ) 2.6 thousand per year 13) D'(p) = -0.21e-0.003p 14) thousand persons/day œ month; the daily attendance is decreasing. 15) 9 e 16) (0, -6), relative minimum 17) w10 = Q 18) logf A = X 19) ) 1 21) 22) ) ) 1-5ln 6 25) (4-4) 8

9 Answer Key Testname: CHAPT4 26) e ( ln + 1) 27) e ln - e ln2 28) 4e4 e4-4 29) ) 35 days 31) $10 32) 5% 33) 35.8% 34) S'(t) = - 16 t ) dp/dt = 20 5t ) ) (1, 0), relative minimum 38) (1, -1), relative maimum 39) N(t) = N0ekt 40) y = y0e5 41) 11.6 years 42) 26.8 years 43) 0.1 years 44) $328.8 million 45) 2.67 % per year 46) 10,033 47) P (t) = 26 e- 0.5t 48) 27% 49) P'(t) = 714e-0.17t% (1 + 42e-0.17t) 2 50) 4083 people 51) 89 employees 52) 18.3 years 53) $16, ) After 3.5 years 55) Eponential, f() = bek, k > 0 56) Eponential, f() = be-k, k > 0 57) Logarithmic, f() = a + b ln a 58) Logistic, f() = 1 + be-k 59) 13.1 days 60) % per year 61) 1647 years 62) 2969 yr 63) 6.1% 9

10 Answer Key Testname: CHAPT4 64) 3 p.m. 65) 89e C 10