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Exponential Function

Exponential Function. y = a x. If we look at the graphs of y = a x for different values of a, we can see that the gradient of the exponential graph increases as the value of a increases a  gradient . y = a x. a increasing. Gradient of y = 2 x at (0,1) is 0.693

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Exponential Function

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  1. Exponential Function

  2. y = ax If we look at the graphs of y = ax for different values of a, we can see that the gradient of the exponential graph increases as the value of a increases a  gradient 

  3. y = ax a increasing

  4. Gradient of y = 2x at (0,1) is 0.693 • Gradient of y = 3x at (0,1) is 1.099 • So there must be a value (between 2 and 3) where the gradient at (0,1) is exactly 1 • This number is 2.71828… and is called e

  5. Natural Logarithms

  6. By drawing the graph of y = ex we can see that there is an inverse function

  7. If y = ex Take logs of both sides logey = logeex logey = xlogee but logee = 1 So x = logey Interchanging x and y gives y = logex (written as y = lnx) This is the inverse function

  8. Questions 1. Make x the subject of lnx – lnA = kt 2. Make t the subject of s = e-kt 3. Make x the subject of y-5 = (A - 5)ex

  9. Questions • A colony of human settler on a previously uninhabited planet. After t years, their population, P, is given by P = 100e0.056t a. Sketch the graph of P against t b. How many settlers were there planet initially? c. How long does it take for the population to reach 1 million?

  10. Answers • lnx – lnA = ktln(x/A) = kt x/A = ekt x = Aekt • s = e-ktlns =-kt t =-lns/k 3. y-5 = (A - 5)ex ex =(y-5)/(A - 5) x =ln((y-5)/(A - 5))

  11. Answers ctd • P = 100e0.056t • When t= 0P = 100e0 = 100 c. P = 100e0.056t = 1,000,000 e0.056t = 10,000 0.056t = ln 10,000= 9.2103 t = 9.2103/0.056 = 164.47 .47 years =12x.47 months = 5.64months It takes 164.5 years to 4 s.f. or 164 years 6 months to the nearest month

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