Exponential Functions PDF

Exponential functions  L What Are Exponential Functions? The variable is in the exponent, the base is a constant 𝑦 = 2𝑥  base = 2 𝑓 𝑥 = 𝑒 𝑥  base = e It shows a growth or decay is proportional to the function's current value....

Exponential functions  L What Are Exponential Functions? The variable is in the exponent, the base is a constant 𝑦 = 2𝑥  base = 2 𝑓 𝑥 = 𝑒 𝑥  base = e It shows a growth or decay is proportional to the function's current value. Meaning that if the current amount is large, so is the rate of change, if it’s small, the rate of change is small too. E.g. if 1000 bacteria double in 1 hour, the increase is 1000, but if 500000 bacteria double, the increase is 500000 -> the greater the starting value, the greater the change Graph of exponentials Values: -4 1/16 -3 1/8 -2 ¼ -1 ½ 0 1 1 2 2 4 3 8 4 16 etc. Exponential Functions If x increases by 1, y gets multiplied by the base. A basic exponential function looks like: y = 𝑦0 × 𝑏 𝑥  Exponential growth The number of bacteria in a Petri dish at t=0 is 5 × 104. Their doubling time is T=8 h. How many bacteria will there be at t=64 hours? Exponential growth y = 𝑦0 × 𝑏 𝑥 We know 𝑦0 = 5 × 104 , the starting value, since it’s given. b is 2, because we’re talking about doubling, so the amount gets multiplied by 2 every time. But what is x?  Exponential growth x shows the number of times the original value gets multiplied by the base (doubled in this case). How many times the bacteria can double in that time. our bacteria double every 8 hours, so in 64 hours they double 64/8 times. 64 y = 5 × 104 × 2 (they will double 64/8 = 8 times) 8 𝐲 = 𝟓 × 𝟏𝟎𝟒 × 𝟐𝟖 = 𝟓 × 𝟏𝟎𝟒 × 𝟐𝟓𝟔 = 𝟏. 𝟐𝟖 × 𝟏𝟎𝟕  L Exponential decay (negative exponential) When the exponent is negative, y decreases as x increases example: y=100 × 2-x x y 0 100 (=100/1) 1 50 (=100/2) 2 25 (=100/4) 3 12.5 (=100/8) 4 6.25 (=100/16) 5 3.125 (=100/32) Exponential decay The half-life of a radioactive sample is 80 days. We have 106 radioactive nuclei. How many will we have left in 200 days? Exponential decay We use base 2 with a negative exponent because it gets halved. y = 𝑦0 × 2−𝑥 𝑡 200 𝑑𝑎𝑦𝑠 𝑥= = = 2.5 times will be halved 𝑇 80 𝑑𝑎𝑦𝑠 𝑦0 = 106 200 𝑑𝑎𝑦𝑠 − y = 106 × 2 80 𝑑𝑎𝑦𝑠 = 1.77 ×105  L Exponential rise to maximum A negative exponential is subtracted from a constant (the maximum value) example: y = 𝑦0 × (1 − 𝑒 −𝑥 ), in this case 𝑦0 = 10, the maximum  L Exponential rise to maximum Physical example: charging a capacitor to a maximum charge of Q= 10 C the time constant  characterizes the speed of charging, the smaller  is, the faster the charging the actual value of the charge at a given time is q 𝑡 0 −𝜏 −𝜏 𝑞 = 𝑄(1 − 𝑒 ); when t = 0, 𝑞 0 = 𝑄 1 − 𝑒 =𝑄 1−1 =0 𝜏 −𝜏 when t = , 𝑞 = 𝑄 1 − 𝑒 = 𝑄 1 − 𝑒 −1 = 𝑄 1 − 0.368 = 0.632𝑄 one time constant after the start of the charging (t = ) the charge is q = 0.632Q. In this case q = 6.32 C The blue capacitor reaches it in 1 ms, the orange in 3 ms. If the attenuation coefficient of a metal is 7 mm-1 what fraction of the X-ray intensity will be absorbed by a 0.5 mm thick plate? How thick should the plate be to absorb 50 % of the entering intensity (I/ I0 = 0.5)?  If the attenuation coefficient of a metal is 7 mm-1 what fraction of the X-ray intensity will be absorbed by a 0.5 mm thick plate? 1 𝜇=7 𝐼 = 𝐼0 × 𝑒 −𝜇𝑥 𝑚𝑚 𝐼 −1 = 𝑒 −7 𝑚𝑚×0.5 𝑚𝑚 = 0.03 is transmitted, 97 % is absorbed 𝐼0 How thick should the plate be to absorb 50 % of the entering intensity (I/ I0 = 0.5)? 𝐼 = 𝐼0 × 𝑒 −𝜇𝑥 𝐼 𝐼 ln = 𝑒 −𝜇𝑥 𝐼0 ln 0.5 𝐼0 𝑥= = = 0.099 𝑚𝑚 −𝜇 −7 𝐼 ln = ln 𝑒 −𝜇𝑥 = −𝜇𝑥 𝐼0  The number of bacteria in a Petri dish at t=0 is 6 × 105. 6 hours later there are 2.4 × 106 What is their doubling time? How many bacteria will there be at t=24 hours?  The number of bacteria in a Petri dish at t=0 is 6 × 105. 6 hours later there are 2.4 × 106 What is their doubling time? How many bacteria will there be at t=24 hours? 6 2.4 × 106 =6× 105 × 2𝑇 6 4= 2𝑇 6 2= 𝑇 𝑇=3 24 5 N = 6 × 10 × 2𝑇 N = 6 × 105 × 28 = 1.536 × 108  S Exponential rise to maximum Physical example: charging a capacitor to a maximum charge of Q= 10 C 𝑡 −𝜏 𝑞 = 𝑄(1 − 𝑒 ) If the time constant is 5 ms, after how much time is q = 8 C?  S Exponential rise to maximum Physical example: charging a capacitor to a maximum charge of Q= 10 C 𝑡 −𝜏 𝑞 = 𝑄(1 − 𝑒 ) If the time constant is 5 ms, after how much time is q = 8 C? 𝑡 − 𝑞 = 𝑄(1 − 𝑒 𝜏 ) 𝑞 − 𝑡 𝑞 − 𝑡 =1−𝑒 𝜏 →1 − =𝑒 𝜏 𝑄 𝑄 𝑞 𝑡 𝑞 8 ln 1 − = − → 𝑡 = − ln 1 − 𝜏 = − ln 1 − × 5 = 8.05 𝑚𝑠 𝑄 𝜏 𝑄 10  Chapter 2 Motion in One Dimension How far is the trooper from his starting point? How far is the trooper from his starting point? x = 0.5  (3 m/s2)  16.92 = 428 m  Non-symmetrical Free Fall Need to divide the motion into segments Possibilities include Upward and downward portions The symmetrical portion back to the release point and then the non- symmetrical portion 36 37 38 39