Section 3.4
        Exponential Growth and Decay

                   V63.0121.002.2010Su, Calculus I

                    ...
Announcements




           Review in second half of
           class today
           Office Hours after class
           ...
Objectives




V63.0121.002.2010Su, Calculus I (NYU)   Section 3.4 Exponential Growth and Decay   June 2, 2010   3 / 37
Outline


 Recall

 The differential equation y = ky

 Modeling simple population growth

 Modeling radioactive decay
   Ca...
Derivatives of exponential and logarithmic functions




                                                  y            y
...
Outline


 Recall

 The differential equation y = ky

 Modeling simple population growth

 Modeling radioactive decay
   Ca...
What is a differential equation?
 Definition
 A differential equation is an equation for an unknown function which
 includes ...
What is a differential equation?
 Definition
 A differential equation is an equation for an unknown function which
 includes ...
What is a differential equation?
 Definition
 A differential equation is an equation for an unknown function which
 includes ...
What is a differential equation?
 Definition
 A differential equation is an equation for an unknown function which
 includes ...
The Equation y = 2


 Example

         Find a solution to y (t) = 2.
         Find the most general solution to y (t) = 2...
The Equation y = 2


 Example

         Find a solution to y (t) = 2.
         Find the most general solution to y (t) = 2...
The Equation y = 2


 Example

         Find a solution to y (t) = 2.
         Find the most general solution to y (t) = 2...
The Equation y = 2


 Example

         Find a solution to y (t) = 2.
         Find the most general solution to y (t) = 2...
The Equation y = 2t



 Example

         Find a solution to y (t) = 2t.
         Find the most general solution to y (t) ...
The Equation y = 2t



 Example

         Find a solution to y (t) = 2t.
         Find the most general solution to y (t) ...
The Equation y = 2t



 Example

         Find a solution to y (t) = 2t.
         Find the most general solution to y (t) ...
The differential equation y = ky


 Example

         Find a solution to y (t) = y (t).
         Find the most general solu...
The differential equation y = ky


 Example

         Find a solution to y (t) = y (t).
         Find the most general solu...
The differential equation y = ky


 Example

         Find a solution to y (t) = y (t).
         Find the most general solu...
Kick it up a notch



 Example

         Find a solution to y = 2y .
         Find the general solution to y = 2y .




V6...
Kick it up a notch



 Example

         Find a solution to y = 2y .
         Find the general solution to y = 2y .


 Sol...
In general
 Example

         Find a solution to y = ky .
         Find the general solution to y = ky .




V63.0121.002....
In general
 Example

         Find a solution to y = ky .
         Find the general solution to y = ky .

 Solution

     ...
In general
 Example

         Find a solution to y = ky .
         Find the general solution to y = ky .

 Solution

     ...
Constant Relative Growth =⇒ Exponential Growth



 Theorem
 A function with constant relative growth rate k is an exponent...
Exponential Growth is everywhere


         Lots of situations have growth rates proportional to the current value
       ...
Outline


 Recall

 The differential equation y = ky

 Modeling simple population growth

 Modeling radioactive decay
   Ca...
Bacteria




          Since you need bacteria to
          make bacteria, the amount
          of new bacteria at any
   ...
Bacteria Example
 Example
 A colony of bacteria is grown under ideal conditions in a laboratory. At the
 end of 3 hours th...
Bacteria Example
 Example
 A colony of bacteria is grown under ideal conditions in a laboratory. At the
 end of 3 hours th...
Bacteria Example
 Example
 A colony of bacteria is grown under ideal conditions in a laboratory. At the
 end of 3 hours th...
Could you do that again please?

 We have

                                               10, 000 = y0 e k·3
             ...
Outline


 Recall

 The differential equation y = ky

 Modeling simple population growth

 Modeling radioactive decay
   Ca...
Modeling radioactive decay

 Radioactive decay occurs because many large atoms spontaneously give off
 particles.




V63.0...
Modeling radioactive decay

 Radioactive decay occurs because many large atoms spontaneously give off
 particles.


  This ...
Radioactive decay as a differential equation


 The relative rate of decay is constant:

                                  ...
Radioactive decay as a differential equation


 The relative rate of decay is constant:

                                  ...
Radioactive decay as a differential equation


 The relative rate of decay is constant:

                                  ...
Computing the amount remaining of a decaying
sample

 Example
 The half-life of polonium-210 is about 138 days. How much o...
Computing the amount remaining of a decaying
sample

 Example
 The half-life of polonium-210 is about 138 days. How much o...
Computing the amount remaining of a decaying
sample

 Example
 The half-life of polonium-210 is about 138 days. How much o...
Carbon-14 Dating

                                                                The ratio of carbon-14 to
              ...
Computing age with Carbon-14 content

 Example
 Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10%...
Computing age with Carbon-14 content

 Example
 Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10%...
Computing age with Carbon-14 content

 Example
 Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10%...
Computing age with Carbon-14 content

 Example
 Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10%...
Outline


 Recall

 The differential equation y = ky

 Modeling simple population growth

 Modeling radioactive decay
   Ca...
Newton’s Law of Cooling


          Newton’s Law of Cooling
          states that the rate of
          cooling of an obje...
Newton’s Law of Cooling


          Newton’s Law of Cooling
          states that the rate of
          cooling of an obje...
General Solution to NLC problems

 To solve this, change the variable y (t) = T (t) − Ts . Then y = T and
 k(T − Ts ) = ky...
General Solution to NLC problems

 To solve this, change the variable y (t) = T (t) − Ts . Then y = T and
 k(T − Ts ) = ky...
General Solution to NLC problems

 To solve this, change the variable y (t) = T (t) − Ts . Then y = T and
 k(T − Ts ) = ky...
General Solution to NLC problems

 To solve this, change the variable y (t) = T (t) − Ts . Then y = T and
 k(T − Ts ) = ky...
General Solution to NLC problems

 To solve this, change the variable y (t) = T (t) − Ts . Then y = T and
 k(T − Ts ) = ky...
General Solution to NLC problems

 To solve this, change the variable y (t) = T (t) − Ts . Then y = T and
 k(T − Ts ) = ky...
Computing cooling time with NLC

 Example
 A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. After 5
 minutes,...
Computing cooling time with NLC

 Example
 A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. After 5
 minutes,...
Finding k


                                            38 = T (5) = 80e 5k + 18
                                        2...
Finding k


                                             38 = T (5) = 80e 5k + 18
                                        ...
Finding t



                                                        t
                                        20 = 80e − ...
Computing time of death with NLC


  Example
  A murder victim is discovered at
  midnight and the temperature of
  the bo...
Solution

         Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29.
         We want to know the t for wh...
Solution

         Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29.
         We want to know the t for wh...
Solution

         Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29.
         We want to know the t for wh...
Outline


 Recall

 The differential equation y = ky

 Modeling simple population growth

 Modeling radioactive decay
   Ca...
Interest


         If an account has an compound interest rate of r per year
         compounded n times, then an initial...
Interest


         If an account has an compound interest rate of r per year
         compounded n times, then an initial...
Interest


         If an account has an compound interest rate of r per year
         compounded n times, then an initial...
Computing doubling time with exponential growth

 Example
 How long does it take an initial deposit of $100, compounded
 c...
Computing doubling time with exponential growth

 Example
 How long does it take an initial deposit of $100, compounded
 c...
I-banking interview tip of the day


                       ln 2
          The fraction      can also be
                 ...
Summary




         When something grows or decays at a constant relative rate, the
         growth or decay is exponenti...
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Lesson 15: Exponential Growth and Decay

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Lesson 15: Exponential Growth and Decay

  1. 1. Section 3.4 Exponential Growth and Decay V63.0121.002.2010Su, Calculus I New York University June 2, 2010 Announcements Review in second half of class today Office Hours after class today Midterm tomorrow
  2. 2. Announcements Review in second half of class today Office Hours after class today Midterm tomorrow V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 2 / 37
  3. 3. Objectives V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 3 / 37
  4. 4. Outline Recall The differential equation y = ky Modeling simple population growth Modeling radioactive decay Carbon-14 Dating Newton’s Law of Cooling Continuously Compounded Interest V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 4 / 37
  5. 5. Derivatives of exponential and logarithmic functions y y ex ex ax (ln a)ax 1 ln x x 1 1 loga x · ln a x V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 5 / 37
  6. 6. Outline Recall The differential equation y = ky Modeling simple population growth Modeling radioactive decay Carbon-14 Dating Newton’s Law of Cooling Continuously Compounded Interest V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 6 / 37
  7. 7. What is a differential equation? Definition A differential equation is an equation for an unknown function which includes the function and its derivatives. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 7 / 37
  8. 8. What is a differential equation? Definition A differential equation is an equation for an unknown function which includes the function and its derivatives. Example Newton’s Second Law F = ma is a differential equation, where a(t) = x (t). V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 7 / 37
  9. 9. What is a differential equation? Definition A differential equation is an equation for an unknown function which includes the function and its derivatives. Example Newton’s Second Law F = ma is a differential equation, where a(t) = x (t). In a spring, F (x) = −kx, where x is displacement from equilibrium and k is a constant. So k −kx(t) = mx (t) =⇒ x (t) + x(t) = 0. m V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 7 / 37
  10. 10. What is a differential equation? Definition A differential equation is an equation for an unknown function which includes the function and its derivatives. Example Newton’s Second Law F = ma is a differential equation, where a(t) = x (t). In a spring, F (x) = −kx, where x is displacement from equilibrium and k is a constant. So k −kx(t) = mx (t) =⇒ x (t) + x(t) = 0. m The most general solution is x(t) = A sin ωt + B cos ωt, where ω = k/m. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 7 / 37
  11. 11. The Equation y = 2 Example Find a solution to y (t) = 2. Find the most general solution to y (t) = 2. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 8 / 37
  12. 12. The Equation y = 2 Example Find a solution to y (t) = 2. Find the most general solution to y (t) = 2. Solution A solution is y (t) = 2t. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 8 / 37
  13. 13. The Equation y = 2 Example Find a solution to y (t) = 2. Find the most general solution to y (t) = 2. Solution A solution is y (t) = 2t. The general solution is y = 2t + C . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 8 / 37
  14. 14. The Equation y = 2 Example Find a solution to y (t) = 2. Find the most general solution to y (t) = 2. Solution A solution is y (t) = 2t. The general solution is y = 2t + C . Remark If a function has a constant rate of growth, it’s linear. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 8 / 37
  15. 15. The Equation y = 2t Example Find a solution to y (t) = 2t. Find the most general solution to y (t) = 2t. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 9 / 37
  16. 16. The Equation y = 2t Example Find a solution to y (t) = 2t. Find the most general solution to y (t) = 2t. Solution A solution is y (t) = t 2 . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 9 / 37
  17. 17. The Equation y = 2t Example Find a solution to y (t) = 2t. Find the most general solution to y (t) = 2t. Solution A solution is y (t) = t 2 . The general solution is y = t 2 + C . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 9 / 37
  18. 18. The differential equation y = ky Example Find a solution to y (t) = y (t). Find the most general solution to y (t) = y (t). V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 10 / 37
  19. 19. The differential equation y = ky Example Find a solution to y (t) = y (t). Find the most general solution to y (t) = y (t). Solution A solution is y (t) = e t . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 10 / 37
  20. 20. The differential equation y = ky Example Find a solution to y (t) = y (t). Find the most general solution to y (t) = y (t). Solution A solution is y (t) = e t . The general solution is y = Ce t , not y = e t + C . (check this) V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 10 / 37
  21. 21. Kick it up a notch Example Find a solution to y = 2y . Find the general solution to y = 2y . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 11 / 37
  22. 22. Kick it up a notch Example Find a solution to y = 2y . Find the general solution to y = 2y . Solution y = e 2t y = Ce 2t V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 11 / 37
  23. 23. In general Example Find a solution to y = ky . Find the general solution to y = ky . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 12 / 37
  24. 24. In general Example Find a solution to y = ky . Find the general solution to y = ky . Solution y = e kt y = Ce kt V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 12 / 37
  25. 25. In general Example Find a solution to y = ky . Find the general solution to y = ky . Solution y = e kt y = Ce kt Remark What is C ? Plug in t = 0: y (0) = Ce k·0 = C · 1 = C , so y (0) = y0 , the initial value of y . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 12 / 37
  26. 26. Constant Relative Growth =⇒ Exponential Growth Theorem A function with constant relative growth rate k is an exponential function with parameter k. Explicitly, the solution to the equation y (t) = ky (t) y (0) = y0 is y (t) = y0 e kt V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 13 / 37
  27. 27. Exponential Growth is everywhere Lots of situations have growth rates proportional to the current value This is the same as saying the relative growth rate is constant. Examples: Natural population growth, compounded interest, social networks V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 14 / 37
  28. 28. Outline Recall The differential equation y = ky Modeling simple population growth Modeling radioactive decay Carbon-14 Dating Newton’s Law of Cooling Continuously Compounded Interest V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 15 / 37
  29. 29. Bacteria Since you need bacteria to make bacteria, the amount of new bacteria at any moment is proportional to the total amount of bacteria. This means bacteria populations grow exponentially. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 16 / 37
  30. 30. Bacteria Example Example A colony of bacteria is grown under ideal conditions in a laboratory. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many bacteria were present initially? V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 17 / 37
  31. 31. Bacteria Example Example A colony of bacteria is grown under ideal conditions in a laboratory. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many bacteria were present initially? Solution Since y = ky for bacteria, we have y = y0 e kt . We have 10, 000 = y0 e k·3 40, 000 = y0 e k·5 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 17 / 37
  32. 32. Bacteria Example Example A colony of bacteria is grown under ideal conditions in a laboratory. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many bacteria were present initially? Solution Since y = ky for bacteria, we have y = y0 e kt . We have 10, 000 = y0 e k·3 40, 000 = y0 e k·5 Dividing the first into the second gives 4 = e 2k =⇒ 2k = ln 4 =⇒ k = ln 2. Now we have 10, 000 = y0 e ln 2·3 = y0 · 8 10, 000 So y0 = = 1250. 8 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 17 / 37
  33. 33. Could you do that again please? We have 10, 000 = y0 e k·3 40, 000 = y0 e k·5 Dividing the first into the second gives 40, 000 y0 e 5k = 10, 000 y0 e 3k =⇒ 4 = e 2k =⇒ ln 4 = ln(e 2k ) = 2k ln 4 ln 22 2 ln 2 =⇒ k = = = = ln 2 2 2 2 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 18 / 37
  34. 34. Outline Recall The differential equation y = ky Modeling simple population growth Modeling radioactive decay Carbon-14 Dating Newton’s Law of Cooling Continuously Compounded Interest V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 19 / 37
  35. 35. Modeling radioactive decay Radioactive decay occurs because many large atoms spontaneously give off particles. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 20 / 37
  36. 36. Modeling radioactive decay Radioactive decay occurs because many large atoms spontaneously give off particles. This means that in a sample of a bunch of atoms, we can assume a certain percentage of them will “go off” at any point. (For instance, if all atom of a certain radioactive element have a 20% chance of decaying at any point, then we can expect in a sample of 100 that 20 of them will be decaying.) V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 20 / 37
  37. 37. Radioactive decay as a differential equation The relative rate of decay is constant: y =k y where k is negative. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 21 / 37
  38. 38. Radioactive decay as a differential equation The relative rate of decay is constant: y =k y where k is negative. So y = ky =⇒ y = y0 e kt again! V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 21 / 37
  39. 39. Radioactive decay as a differential equation The relative rate of decay is constant: y =k y where k is negative. So y = ky =⇒ y = y0 e kt again! It’s customary to express the relative rate of decay in the units of half-life: the amount of time it takes a pure sample to decay to one which is only half pure. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 21 / 37
  40. 40. Computing the amount remaining of a decaying sample Example The half-life of polonium-210 is about 138 days. How much of a 100 g sample remains after t years? V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 22 / 37
  41. 41. Computing the amount remaining of a decaying sample Example The half-life of polonium-210 is about 138 days. How much of a 100 g sample remains after t years? Solution We have y = y0 e kt , where y0 = y (0) = 100 grams. Then 365 · ln 2 50 = 100e k·138/365 =⇒ k = − . 138 Therefore 365·ln 2 y (t) = 100e − 138 t = 100 · 2−365t/138 . V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 22 / 37
  42. 42. Computing the amount remaining of a decaying sample Example The half-life of polonium-210 is about 138 days. How much of a 100 g sample remains after t years? Solution We have y = y0 e kt , where y0 = y (0) = 100 grams. Then 365 · ln 2 50 = 100e k·138/365 =⇒ k = − . 138 Therefore 365·ln 2 y (t) = 100e − 138 t = 100 · 2−365t/138 . Notice y (t) = y0 · 2−t/t1/2 , where t1/2 is the half-life. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 22 / 37
  43. 43. Carbon-14 Dating The ratio of carbon-14 to carbon-12 in an organism decays exponentially: p(t) = p0 e −kt . The half-life of carbon-14 is about 5700 years. So the equation for p(t) is ln2 p(t) = p0 e − 5700 t Another way to write this would be p(t) = p0 2−t/5700 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 23 / 37
  44. 44. Computing age with Carbon-14 content Example Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10% of that in a living organism. How old is the fossil? V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 24 / 37
  45. 45. Computing age with Carbon-14 content Example Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10% of that in a living organism. How old is the fossil? Solution We are looking for the value of t for which p(t) = 0.1 p(0) V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 24 / 37
  46. 46. Computing age with Carbon-14 content Example Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10% of that in a living organism. How old is the fossil? Solution We are looking for the value of t for which p(t) = 0.1 p(0) From the equation we have t ln 0.1 2−t/5700 = 0.1 =⇒ − ln 2 = ln 0.1 =⇒ t = · 5700 ≈ 18, 940 5700 ln 2 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 24 / 37
  47. 47. Computing age with Carbon-14 content Example Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10% of that in a living organism. How old is the fossil? Solution We are looking for the value of t for which p(t) = 0.1 p(0) From the equation we have t ln 0.1 2−t/5700 = 0.1 =⇒ − ln 2 = ln 0.1 =⇒ t = · 5700 ≈ 18, 940 5700 ln 2 So the fossil is almost 19,000 years old. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 24 / 37
  48. 48. Outline Recall The differential equation y = ky Modeling simple population growth Modeling radioactive decay Carbon-14 Dating Newton’s Law of Cooling Continuously Compounded Interest V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 25 / 37
  49. 49. Newton’s Law of Cooling Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 26 / 37
  50. 50. Newton’s Law of Cooling Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. This gives us a differential equation of the form dT = k(T − Ts ) dt (where k < 0 again). V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 26 / 37
  51. 51. General Solution to NLC problems To solve this, change the variable y (t) = T (t) − Ts . Then y = T and k(T − Ts ) = ky . The equation now looks like dT dy = k(T − Ts ) ⇐⇒ = ky dt dt V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 27 / 37
  52. 52. General Solution to NLC problems To solve this, change the variable y (t) = T (t) − Ts . Then y = T and k(T − Ts ) = ky . The equation now looks like dT dy = k(T − Ts ) ⇐⇒ = ky dt dt Now we can solve! y = ky V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 27 / 37
  53. 53. General Solution to NLC problems To solve this, change the variable y (t) = T (t) − Ts . Then y = T and k(T − Ts ) = ky . The equation now looks like dT dy = k(T − Ts ) ⇐⇒ = ky dt dt Now we can solve! y = ky =⇒ y = Ce kt V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 27 / 37
  54. 54. General Solution to NLC problems To solve this, change the variable y (t) = T (t) − Ts . Then y = T and k(T − Ts ) = ky . The equation now looks like dT dy = k(T − Ts ) ⇐⇒ = ky dt dt Now we can solve! y = ky =⇒ y = Ce kt =⇒ T − Ts = Ce kt V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 27 / 37
  55. 55. General Solution to NLC problems To solve this, change the variable y (t) = T (t) − Ts . Then y = T and k(T − Ts ) = ky . The equation now looks like dT dy = k(T − Ts ) ⇐⇒ = ky dt dt Now we can solve! y = ky =⇒ y = Ce kt =⇒ T − Ts = Ce kt =⇒ T = Ce kt + Ts V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 27 / 37
  56. 56. General Solution to NLC problems To solve this, change the variable y (t) = T (t) − Ts . Then y = T and k(T − Ts ) = ky . The equation now looks like dT dy = k(T − Ts ) ⇐⇒ = ky dt dt Now we can solve! y = ky =⇒ y = Ce kt =⇒ T − Ts = Ce kt =⇒ T = Ce kt + Ts Plugging in t = 0, we see C = y0 = T0 − Ts . So Theorem The solution to the equation T (t) = k(T (t) − Ts ), T (0) = T0 is T (t) = (T0 − Ts )e kt + Ts V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 27 / 37
  57. 57. Computing cooling time with NLC Example A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. After 5 minutes, the egg’s temperature is 38 ◦ C. Assuming the water has not warmed appreciably, how much longer will it take the egg to reach 20 ◦ C? V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 28 / 37
  58. 58. Computing cooling time with NLC Example A hard-boiled egg at 98 ◦ C is put in a sink of 18 ◦ C water. After 5 minutes, the egg’s temperature is 38 ◦ C. Assuming the water has not warmed appreciably, how much longer will it take the egg to reach 20 ◦ C? Solution We know that the temperature function takes the form T (t) = (T0 − Ts )e kt + Ts = 80e kt + 18 To find k, plug in t = 5: 38 = T (5) = 80e 5k + 18 and solve for k. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 28 / 37
  59. 59. Finding k 38 = T (5) = 80e 5k + 18 20 = 80e 5k 1 = e 5k 4 1 ln = 5k 4 1 =⇒ k = − ln 4. 5 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 29 / 37
  60. 60. Finding k 38 = T (5) = 80e 5k + 18 20 = 80e 5k 1 = e 5k 4 1 ln = 5k 4 1 =⇒ k = − ln 4. 5 Now we need to solve t 20 = T (t) = 80e − 5 ln 4 + 18 for t. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 29 / 37
  61. 61. Finding t t 20 = 80e − 5 ln 4 + 18 t 2 = 80e − 5 ln 4 1 t = e − 5 ln 4 40 t − ln 40 = − ln 4 5 ln 40 5 ln 40 =⇒ t = 1 = ≈ 13 min 5 ln 4 ln 4 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 30 / 37
  62. 62. Computing time of death with NLC Example A murder victim is discovered at midnight and the temperature of the body is recorded as 31 ◦ C. One hour later, the temperature of the body is 29 ◦ C. Assume that the surrounding air temperature remains constant at 21 ◦ C. Calculate the victim’s time of death. (The “normal” temperature of a living human being is approximately 37 ◦ C.) V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 31 / 37
  63. 63. Solution Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29. We want to know the t for which T (t) = 37. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 32 / 37
  64. 64. Solution Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29. We want to know the t for which T (t) = 37. To find k: 29 = 10e k·1 + 21 =⇒ k = ln 0.8 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 32 / 37
  65. 65. Solution Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29. We want to know the t for which T (t) = 37. To find k: 29 = 10e k·1 + 21 =⇒ k = ln 0.8 To find t: 37 = 10e t·ln(0.8) + 21 1.6 = e t·ln(0.8) ln(1.6) t= ≈ −2.10 hr ln(0.8) So the time of death was just before 10:00 pm. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 32 / 37
  66. 66. Outline Recall The differential equation y = ky Modeling simple population growth Modeling radioactive decay Carbon-14 Dating Newton’s Law of Cooling Continuously Compounded Interest V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 33 / 37
  67. 67. Interest If an account has an compound interest rate of r per year compounded n times, then an initial deposit of A0 dollars becomes r nt A0 1 + n after t years. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 34 / 37
  68. 68. Interest If an account has an compound interest rate of r per year compounded n times, then an initial deposit of A0 dollars becomes r nt A0 1 + n after t years. For different amounts of compounding, this will change. As n → ∞, we get continously compounded interest r nt A(t) = lim A0 1 + = A0 e rt . n→∞ n V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 34 / 37
  69. 69. Interest If an account has an compound interest rate of r per year compounded n times, then an initial deposit of A0 dollars becomes r nt A0 1 + n after t years. For different amounts of compounding, this will change. As n → ∞, we get continously compounded interest r nt A(t) = lim A0 1 + = A0 e rt . n→∞ n Thus dollars are like bacteria. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 34 / 37
  70. 70. Computing doubling time with exponential growth Example How long does it take an initial deposit of $100, compounded continuously, to double? V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 35 / 37
  71. 71. Computing doubling time with exponential growth Example How long does it take an initial deposit of $100, compounded continuously, to double? Solution We need t such that A(t) = 200. In other words ln 2 200 = 100e rt =⇒ 2 = e rt =⇒ ln 2 = rt =⇒ t = . r For instance, if r = 6% = 0.06, we have ln 2 0.69 69 t= ≈ = = 11.5 years. 0.06 0.06 6 V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 35 / 37
  72. 72. I-banking interview tip of the day ln 2 The fraction can also be r approximated as either 70 or 72 divided by the percentage rate (as a number between 0 and 100, not a fraction between 0 and 1.) This is sometimes called the rule of 70 or rule of 72. 72 has lots of factors so it’s used more often. V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 36 / 37
  73. 73. Summary When something grows or decays at a constant relative rate, the growth or decay is exponential. Equations with unknowns in an exponent can be solved with logarithms. Your friend list is like culture of bacteria (no offense). V63.0121.002.2010Su, Calculus I (NYU) Section 3.4 Exponential Growth and Decay June 2, 2010 37 / 37
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