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The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential.

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Presentation on theme: "The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential."— Presentation transcript:

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3 The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time. Let P = population t years after the present. =.05P dP dt *label is people/yr Rate of change of the population

4 =.05P dP dt 4

5 The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time. P = 1000 e 0.05 t found equation *with this equation, however P = when t = 1. P should equal Must change up a little!!!! start over 4. To check accuracy of equation, find the population after one year given ….. The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year ; Use the equation we found to check if when t=1 the population comes out right! *when t = 1, P should equal (1000*.05= 50 plus original 1000)

6 The population of the little town of Scorpion Gulch is now 1000 people. The population is presently growing at about 5% per year. Write a differential equation that expresses this fact. Solve it to find an equation that expresses population as a function of time. *when t = 1, P should equal (1000*.05= 50 plus original 1000) *When t = 0, P = 1000 = kP dP dt We must find a new constant (k) since 0.05 didnt work!!! Constant of proportionality

7 *when t = 1, P should equal *When t = 0, P = 1000 = kP dP dt *Separate variables in = kP dP dt dP P = k dt ln | P | = k t + C solve

8 Now have: |P| = Ce k t Solve for C using: *You can solve for C as soon as you find antiderivative if there is not an absolute value involved……otherwise solve with C intact AND substitute j = ± C allowing a drop of absolute value!!! *When t = 0, P = 1000 Solve for k (what we wanted) using above equation and *when t = 1, P = P = j e k t

9 therefore: P = 1000e ln t In general: y = y o e k t where y o is original value at time, feet, or whatever =


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