weebly.com/ Core Mathematics 3 Exponentials and Natural Logarithms

Transcription

1 weebly.com/ Core Mathematics 3 Exponentials and Natural Logarithms Core Maths 3 Exponentials and natural Logarithms Page 1

2 Ln and Exponentials C3 Content By the end of this unit you should have knowledge of: the function e x and its graph (including y = e ax+b ). the function lnx and its graph; lnx as the inverse function of e x. solutions to equations of the type n = e ax+b and ln(ax+b) = r Rules 1. e x =k, x=ln(k) 2. ln(x)=k, x= e k 3. ln(x) + ln(y)= ln(xy) 4. ln(x) ln(y) = ln(x/y) 5. nln(x) = ln(x n ) 6. ln(e x ) = x 7. e ln(x) = x Core Maths 3 Exponentials and natural Logarithms Page 2

3 The exponential function e x The graphs below display y = 2 x and y = 3 x and the tangents to the curves at the point where x = 0. 4 y=2 x 4 y=3 x For the curve y = 2 x, the gradient at the point where x = 0 is and for y = 3 x, the gradient is 1.1. There must be a number whose exponential graph has a gradient of 1 at x = 0. That number is e. e = The function f(x) = e x has a domain x and range f(x) > 0 The exponential function e x is the only function that differentiates to itself (more later). Core Maths 3 Exponentials and natural Logarithms Page 3

4 The logarithmic Function lnx. Lnx is the inverse of e x and its graph is displayed below. The function lnx has domain x > 0 and range f (x) What do you notice? Recap of rules from C2 One of the most important rules you should have learnt in C2 was the interchangeability of the following statement: a c = b logab = c This idea can be applied to e x where e is the base: e x = y logey = x logey is written as lny. This reinforces the idea that ln is the inverse of e. Core Maths 3 Exponentials and natural Logarithms Page 4

5 Rules of logarithms were introduced in C2 and obviously they all apply to logs in base e. ln7 + ln8 = ln56 ln8 - ln2 = ln4 n ln7 = ln7 n lne = 1 ln1 = 0 lna log c a lnc Most of the exam questions on the C3 papers will bring together other areas of the specification but we will start with more simple ones. Example 1 Given that e -x = 5, find the value of x to 4 decimal places. Taking ln of both sides: -x = ln 5 x = Core Maths 3 Exponentials and natural Logarithms Page 5

6 Example 2 Given that ln(y + 4) lny = 1.2, find the value of y, correct to 3 decimal places. Using rules of logarithms y + 4 ln = 1.2 y Therefore by using the interchangeable statement: lna = b e b = a e 1.2 y + 4 y 1.2 ye y ye y y(e 1) 4 y Core Maths 3 Exponentials and natural Logarithms Page 6

7 Example 3 Find the exact solution to the equation e x + 25e -x = 10 Multiplying by e x gives: e 2x + 25 = 10e x We now have an exponential quadratic Replacing e x with y gives: y 2 10y + 25 = 0 Therefore (y 5)(y 5) = 0 y = 5 and hence e x = 5 So finally x = ln5 The fact that e x is the inverse of lnx is used time and again in C3 questions Core Maths 3 Exponentials and natural Logarithms Page 7

8 The following question deals with exponential heat loss. Remember that one of the first things we discussed in C1 and C2 was that rate of change is the fundamental principle behind differentiation. Therefore if you are asked to find a rate of change then you must differentiate. Example 5 A heated metal ball is dropped into a liquid. As the ball cools, its temperature, TºC, t minutes after it enters the liquid, is given by T = 400e -0.04t + 25, t 0 a) Find the temperature of the ball as is enters the liquid. b) Find the value of t for which T = 400, giving your answer to 3 sig fig. c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50, giving your answer in ºC per minute to 3 sig fig. d) Explain why the temperature of the ball can not fall below 10ºC. a) Find the temperature when t = 0 T = 400e -0.04t + 25 T = 400e = 425 b) Find the value of t for which T = 400, giving your answer to 3 sig fig. 400 = 400e -0.04t e t Core Maths 3 Exponentials and natural Logarithms Page 8

9 Taking ln of both sides 375 ln 0.04t 400 t 1.61 c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50, giving your answer in ºC per minute to 3 sig fig. The question is asking for dt dt when t = t T = 400e + 25 dt dt 16e -0.04t t 50 rate = C/min d) Explain why the temperature of the ball can not fall below 10ºC. As t tends to infinity, e -0.04t tends to 0, therefore the temperature can not fall below 25ºC. The final question is one of the more tricky types of rates questions but still give it a go! Core Maths 3 Exponentials and natural Logarithms Page 9

10 Example 6 The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula 1 t x De where x is the amount of the blood in the blood stream in milligrams and D is the dose given in milligrams. A dose of 7 milligrams is given. a) Find the amount of the drug in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 7mg of the drug is given after 4 hours. b) Show that the amount of the drug in the bloodstream 1 hour after the second dose is mg to 3 decimal places. No more doses of the drug are given. At time T hours after the second dose is given, the amount of the drug in the bloodstream is 5mg. c) Find the value of T. a) From the question D = 7 and t = 4 x De 1 t 4 x 7e x 5.016mg Core Maths 3 Exponentials and natural Logarithms Page 10

11 b) One hour after the second dose is given the first dose has been at work for five hours. Therefore we have to do the question in two parts. For dose one D = 7 and t = 5 For dose two D = 7 and t = 1 x De 1 t x De 1 t 5 x 7e 1 x 7e x mg x mg required. These two amounts add up to give mg to 3 decimal places as c) The thing to be careful with here is the variable T is the time after the second dose therefore the formula to calculate the amount of the first dose still present is given by: 1 ( T 4) x 7e and for the second dose it is T x 7e 1 these two should add together to give 5mg. Therefore: ( T 4) T e 7e By using rules of indices the first part becomes T T e e 7e Core Maths 3 Exponentials and natural Logarithms Page 11

12 Factorising gives: Therefore 1 4 T 5 7 e e 7 e T e 4 1 Taking ln of both sides and multiplying by - gives: T = 10.5hrs This is a tough question and the algebraic manipulation is challenging but you should be able to score some marks in part (c). Core Maths 3 Exponentials and natural Logarithms Page

13 C3 Exponentials and Logarithms past paper questions 1. A particular species of orchid is being studied. The population p at time t years after the study started is assumed to be 0.2t 2800ae p =, where a is a constant. 0.2t 1 ae Given that there were 300 orchids when the study started, (a) show that a = 0., (b) use the equation with a = 0. to predict the number of years before the population of orchids reaches (c) Show that p =. 0.2t 0. e (d) Hence show that the population cannot exceed (Q7, June 2005) 2. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, T C, t minutes after it enters the liquid, is given by T = 400e 0.05t + 25, t 0. (a) Find the temperature of the ball as it enters the liquid. (b) Find the value of t for which T = 300, giving your answer to 3 significant figures. (c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50. Give your answer in C per minute to 3 significant figures. (d) From the equation for temperature T in terms of t, given above, explain why the temperature of the ball can never fall to 20 C. (Q4, June 2006) 3. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. (Q1, June 2007) Core Maths 3 Exponentials and natural Logarithms Page 13

14 4. The amount of a certain type of drug in the bloodstream t hours after it has been taken is given by the formula x = De 81, where x is the amount of the drug in the bloodstream in milligrams and D is the dose given in milligrams. A dose of 10 mg of the drug is given. (a) Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours. (b) Show that the amount of the drug in the bloodstream 1 hour after the second dose is mg to 3 decimal places. No more doses of the drug are given. At time T hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg. (c) Find the value of T. (Q8, June 2007) 5. The radioactive decay of a substance is given by R = 1000e ct, t 0. where R is the number of atoms at time t years and c is a positive constant. (a) Find the number of atoms when the substance started to decay. It takes 5730 years for half of the substance to decay. (b) Find the value of c to 3 significant figures. (c) Calculate the number of atoms that will be left when t = (d) Sketch the graph of R against t. (Q5, Jan 2008) 6. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation 1 t 5 P = 80 e, t R, t 0. (a) Write down the number of rabbits that were introduced to the island. (b) Find the number of years it would take for the number of rabbits to first exceed d P (c) Find. dt dp (d) Find P when = 50. dt (Q3, June 2009) - t Core Maths 3 Exponentials and natural Logarithms Page 14

15 7. (a) Simplify fully 2 2x 9x 5. 2 x 2x 15 Given that ln (2x 2 + 9x 5) = 1 + ln (x 2 + 2x 15), x 5, (b) find x in terms of e. (Q8, June 2010) 8. Joan brings a cup of hot tea into a room and places the cup on a table. At time t minutes after Joan places the cup on the table, the temperature, C, of the tea is modelled by the equation = 20 + Ae kt, where A and k are positive constants Given that the initial temperature of the tea was 90 C, (a) find the value of A. The tea takes 5 minutes to decrease in temperature from 90 C to 55 C. (b) Show that k = 5 1 ln 2. (c) Find the rate at which the temperature of the tea is decreasing at the instant when t = 10. Give your answer, in C per minute, to 3 decimal places. (Q4, Jan 2011) 9. The mass, m grams, of a leaf t days after it has been picked from a tree is given by m = pe kt, where k and p are positive constants. When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams. (a) Write down the value of p. (b) Show that k = 4 1 ln 3. (c) Find the value of t when dm dt = 0 6 ln 3. (6) (Q5, June 2011) Core Maths 3 Exponentials and natural Logarithms Page 15

16 10. The area, A mm 2, of a bacterial culture growing in milk, t hours after midday, is given by A = 20e 1.5t, t 0. (a) Write down the area of the culture at midday. (b) Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute. (5) (Q5, Jan 20) 11. The value of Bob s car can be calculated from the formula V = 17000e 0.25t e 0.5t where V is the value of the car in pounds ( ) and t is the age in years. (a) Find the value of the car when t = 0. (b) Calculate the exact value of t when V = (c) Find the rate at which the value of the car is decreasing at the instant when t = 8. Give your answer in pounds per year to the nearest pound. (Q8, Jan 2013). Find algebraically the exact solutions to the equations (a) ln(4 2x) + ln(9 3x) = 2ln(x + 1), 1 < x < 2 (5) (b) 2 x e 3x+1 = 10 Give your answer to (b) in the form a ln b c ln d where a, b, c and d are integers. (5) (Q6, June 2013) Core Maths 3 Exponentials and natural Logarithms Page 16

17 13. Figure 3 The population of a town is being studied. The population P, at time t years from the start of the study, is assumed to be where k is a positive constant P, t 0 1 7e kt The graph of P against t is shown in Figure 3. Use the given equation to (a) find the population at the start of the study, (b) find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study, (c) calculate the value of k to 3 decimal places. (5) Using this value for k, (d) find the population at 10 years from the start of the study, giving your answer to 3 significant figures. (e) Find, using d P, the rate at which the population is growing at 10 years from the dt start of the study. (Q8, June 2013_R) Core Maths 3 Exponentials and natural Logarithms Page 17

18 14. Find the exact solutions, in their simplest form, to the equations (a) 2 ln (2x + 1) 10 = 0 (b) 3 x e 4x = e 7 (Q2, June 2014) 15. A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation 0.1t 800e P, t 0, t 0.1t 1 3e (a) Calculate the number of primroses at the start of the study. (b) Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers. (c) Find the exact value of d P dt when t =10. Give your answer in its simplest form. (d) Explain why the population of primroses can never be 270. (Q8, June 2014) 16. Water is being heated in an electric kettle. The temperature, C, of the water t seconds after the kettle is switched on, is modelled by the equation = 0 100e t, 0 t T. (a) State the value of when t = 0. Given that the temperature of the water in the kettle is 70 C when t = 40, (b) find the exact value of, giving your answer in the form integers. ln a, where a and b are b When t = T, the temperature of the water reaches 100 C and the kettle switches off. (c) Calculate the value of T to the nearest whole number. (Q4, June 2015) Core Maths 3 Exponentials and natural Logarithms Page 18

19 17. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula x = De 0.2t, where x is the amount of the antibiotic in the bloodstream in milligrams, D is the dose given in milligrams and t is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given. (a) Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose, (b) show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is mg to 3 decimal places. No more doses of the antibiotic are given. At time T hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg. (c) Show that T = a ln b b, where a and b are integers to be determined. e (Q9, June 2016) 18. Find the exact solutions, in their simplest form, to the equations (a) e 3x 9 = 8 (b) ln(2y + 5) = 2 + ln(4 y) (Q1, June 2017) Core Maths 3 Exponentials and natural Logarithms Page 19

20 P P T O T Figure 3 t The number of rabbits on an island is modelled by the equation 100e P 1 3e 0.1t 0.9t 40, t R, t 0 where P is the number of rabbits, t years after they were introduced onto the island. A sketch of the graph of P against t is shown in Figure 3. (a) Calculate the number of rabbits that were introduced onto the island. (b) Find dp dt The number of rabbits initially increases, reaching a maximum value PT when t = T (c) Using your answer from part (b), calculate (i) the value of T to 2 decimal places, (ii) the value of PT to the nearest integer. (Solutions based entirely on graphical or numerical methods are not acceptable.) For t > T, the number of rabbits decreases, as shown in Figure 3, but never falls below k, where k is a positive constant. (d) Use the model to state the maximum value of k. (Q8, June 2017) Core Maths 3 Exponentials and natural Logarithms Page 20

21 20. The value of Lin s car is modelled by the formula V = e 0.2t e 0.1t , t 0 where the value of the car is V pounds when the age of the car is t years. A sketch of t against V is shown in Figure 1. (a) State the range of V. According to this model, (b) find the rate at which the value of the car is decreasing when t = 10. Give your answer in pounds per year. (c) Calculate the exact value of t when V = (Q8, IAL Jan 2015) Core Maths 3 Exponentials and natural Logarithms Page 21

22 21. The mass, m grams, of a radioactive substance t years after first being observed, is modelled by the equation where k is a positive constant. m = 25e 1 kt (a) State the value of m when the radioactive substance was first observed. Given that the mass is 50 grams, 10 years after first being observed, (b) show that k = 1 ln 1 e 10 2 (c) Find the value of t when m = 20, giving your answer to the nearest year. (Q6, IAL June 2015) 22. A hot piece of metal is dropped into a cool liquid. As the metal cools, its temperature T degrees Celsius, t minutes after it enters the liquid, is modelled by T = 300e 0.04t + 20, t 0 (a) Find the temperature of the piece of metal as it enters the liquid. (b) Find the value of t for which T = 180, giving your answer to 3 significant figures. (Solutions based entirely on graphical or numerical methods are not acceptable.) (c) Show, by differentiation, that the rate, in degrees Celsius per minute, at which the temperature of the metal is changing, is given by the expression 20 T 25 (Q6, IAL Jan 2016) Core Maths 3 Exponentials and natural Logarithms Page 22

23 23. The population of a species of animal is being studied. The population P, at time t years from the start of the study, is assumed to be where k is a positive constant. kt 9000e P, t kt 0 3e 7 A sketch of the graph of P against t is shown in Figure 2. Use the given equation to (a) find the population at the start of the study, (b) find the value for the upper limit of the population. Given that P = 2500 when t = 4 (c) calculate the value of k, giving your answer to 3 decimal places. (5) Using this value for k, (d) find, using d P, the rate at which the population is increasing when t = 10 dt Give your answer to the nearest integer. (Q9, IAL June 2016) Core Maths 3 Exponentials and natural Logarithms Page 23

24 DO NOT WRITE IN THIS 24. A population of insects is being studied. The number of insects, N, in the population, is modelled by the equation N = 300 t R, t 0-0.2t 3+17e where t is the time, in weeks, from the start of the study. Using the model, (a) find the number of insects at the start of the study, (b) find the number of insects when t = 10, (c) find the time from the start of the study when there are 82 insects. (Solutions based entirely on graphical or numerical methods are not acceptable.) (d) Find, by differentiating, the rate, measured in insects per week, at which the number of insects is increasing when t = 5. Give your answer to the nearest whole number. (Q10, IAL Jan 2017) P blank O T t Figure 5 A colony of ants is being studied. The number of ants in the colony is modelled by the equation P = e 15 e 0.6t 0.8t t R, t 0 where P is the number of ants, measured in thousands, t years after the study started. A sketch of the graph of P against t is shown in Figure 5. (a) Calculate the number of ants in the colony at the start of the study. Core Maths 3 Exponentials and natural Logarithms Page 24

25 (b) Find dp dt The population of ants initially decreases, reaching a minimum value after T years, as shown in Figure 5. (c) Using your answer to part (b), calculate the value of T to 2 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.) (Q13, IAL June 2017) Core Maths 3 Exponentials and natural Logarithms Page 25