1105_LW5B_5.3_F11 - MAC1105 Lecture Worksheet'47 Sections...

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Unformatted text preview: MAC1105 Lecture Worksheet '47 Sections 5.3 Part I and 5.3 Part II 1. Each table below shows a constant change in x. That is, Ax remains constant. For each function, determine whether it is linear or exponential. Ifit’s linear, determine its average rate ofchange, initial value, and write its fimction formula. If it’s exponential, deteimine its growth or decay factor, initial value, and write its function formula. 2. In each part below, y is an exponential function of x. You must write the formula. a) in a laboratory setting, a population of fruit flies starts at 200 flies and triples every week. Write an exponential function formula that gives y, the population x weeks from now. Answer: y 2 b) Quantity 3,: starts at 200 milligrams and is halved each day. Write an exponential function formula that gives the y—quantity at days from now. Answer: y = (3) Quantity y starts at 200 milligrams and grows by a factor of 1.065 every hour. Write an exponential function formula that gives the y—quantity 3: hours from now. Answer: y = d) Quantity y starts at 200 milligrams and decays by a factor of 0.558 every hour. Write an exponential function formula that gives the y—quantity 3; hours from now. Answer: y = 3. For each exponential function formula below, determine whether the function represents exponential growth or decay, the initial quantity, the growth or decay factor, and the growth or decayr rate. Sketch its graph, labeling the vertical intercept. Your sketch should show the asymptotic behavior of the graph. and two other points, one with positive input and the other with negative input. a) go) =(1.2><106)(1.17)' b) P(x) =30(0.85)" c) T(n):25(4)’" 4. The four graphs shown correspond to the four function formulas below. Match each graph with its formula by placing the letter of the graph in the blank next to its formula. _ y = on . 1r __ y = C(03)" _ y = car 3! = C (08)" 5. When first taken, a cold medication introduces 200mg of active drug into the blood stream. The amount of active drug decays exponentially according to the formula S (t) = 200(0707)‘ . a) Use the calculator’s table and graph features to estimate how long it will take for there to be only 150 mg of active drug in the blood stream, rounded to the nearest tenth of an hour" day. Answer: 2‘ 2 hours 13) Use the calculator's table and graph features to estimate the half life for this drug, rounded to the nearest tenth of an hour. Answer: t 5 hours c) Use the caiculator’s table and graph features to estimate when only 15% of the initial amount is ' still active in the blood stream, rounded to the nearest tenth of an hour. Answer: if 3 hours d) Sketch the graph and label the points on the graph corresponding to your answers from a)-c). 6. A mutual fund account is currently valued at $2000. a) If the account decreases to $2200 over some time period, what is the percent increase over this time period? AnsWer: % b) If the account decreases to $1700 over this time period, what is the percent decrease over this time period? Answer: _% 7. The value of a condominium is currently $100,000. a) If its value increases by 10% over some time period, find the increase in value and the condo’s value at the end of this time period. Answers: Increase in value: $ Condo’s value: $ b) In part a), by what factor did the condo’s value increase? That is, if we multiply $100,000 by the number we get the condo’s value at the end of this time period. c) If its value decreases by 15% over some time period, find the decrease in value and the condo’s value at the end of this time period. Answers: Decrease in value: $ Condo’s value: $ d) In part a), by what factor did the condo’s value decrease? That is, if we multiply $100,000 by the number we get the condo’s value at the end of this time period. 8. The population of a city grows annually by 1.7%. a) If the population is 40,500 at some particular point in time, find the population one year later. ADSWCI‘ (rounded to the nearest whole number) b) In general, if we multiply the population at any moment by the number , we get the population one year later. c) If the current population is denoted 13;}, write a function formula that gives population t years into the future. Use this function and a graphic approach to determine how many years it will take population to double. ' Calculating Compound Interest - Finite Compounding if P 2 original investment (called principai), r 2 annual (called nominal) interest rate in decimal form, n = the number of times each year that interest is compounded, and t = the number of years over which at . . . . ' r . interest rate is compounded, then account value A is given by A : P(1+—] . The annual interest rate It is often called the nominal rate. When interest is compounded more often than onceeach year (that is, n > 1), account value actually grows faster than the nominal rate suggests. This faster rate is called the i? effective rate, and [6+er —i]x100 gives this effective rate as a percentage. 9. An investment of $4000 cams 7.2% annual (nominal) interest. Round money to the nearest penny, and effective interest rates to the nearest thousandth of a percent. a) If interest is compounded quarterly, find the account value after 10 years and the effective rate. b) If interest is compounded daily, find the account value after 10 years and the effective rate. Natural Exponential Function, Continuous Growth/Decay, and Continuous Compounding As n —-)+oo , the quantity [1+1] w) e a: 231828182846, an irrational number called Euler’s number. H The constant e is the base for the natural exponential function f (x) = e". When the annual (nominal) rate is compounded continuously, account value A is given by A : Pg” , and [er wquo gives the effective rate as a percentage. 10. An investment of $4000 earns 7.2% annual (nominal) interest compounded continuously. Find the account value after 10 years and the effective rate. Round money to the nearest penny, and the effective interest rate to the nearest thousandth of a percent. ll. 12. The amount of drug still active in the blood stream t hours after the drug was first administered can sometimes be modeled by a continuous decay model. Suppose the initial concentration of a particular drug is 2 milligrams per liter of blood, and concentration decays at a continuous rate of 2% per hour. Find a formula of the form AU) = Age“ that gives the drug concentration t hours after it has been administered, and use a graphic approach to estimate how long it will take for concentration to reach 1.5 mg/L. The accident on April 26, 1986, atthe nuclear power plant in Chernobyl, Russia, is considered the ' worst nuclear disaster of the nuclear age. Large amounts of the radioactive substance Cesium-137 were emitted into the air. (http:Ilenwikipedia.org/wiki/Chernobyl disaster) The amount of an initial IOU-milligram sample of Cesium—137 remaining after .1: years is given by the continuous decay model 14h.) = 100 84.02291: . a) What continuous yearly decay rate does the formula suggest? % per year b) The halfulife of a decaying substance is the amount of time for the substance to decay by 50%. Use a graphic approach to estimate the halfwlife of Cesium-137, rounded to the nearest year. Sketch the graph that illustrates your strategy. Haifdife 2 years c) Any natural exponential model of the form A (1) :2 Aogk‘x can be rewritten in the form AU) 3 Cat by letting C : A0 and e" m a . Find base a for AU) =1003'0022951 , round it to four decimal places, and use this rounded base to rewrite the natural exponential model for Cesium-137 in the form A(x) = Ca" . A(x)w ...
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