Section 6.2 Calculus AP/Dual, Revised ©2018 viet.dang@humbleisd.net Growth and Decay Section 6.2 Calculus AP/Dual, Revised ©2018 viet.dang@humbleisd.net 5/8/2019 12:56 PM §6.2: Growth and Decay

Review Solve the differential equation, 𝒅𝒚 𝒅𝒙 =𝟒−𝒙 5/8/2019 12:56 PM §6.2: Growth and Decay

Algebra Review Directly Proportional: 𝒚=𝒌𝒙 or 𝒚 𝒙 =𝒌 Inversely Proportional: 𝒚𝒙=𝒌 or 𝒚= 𝒌 𝒙 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 1 The rate of change of 𝑷 with respect to 𝒕 is proportional to 𝟏𝟎−𝒕. Write the equation of the function. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 2 The rate of change of 𝑵 with respect to 𝒔 is proportional to 𝟐𝟓𝟎−𝑵. Write the equation of the function. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 2 The rate of change of 𝑵 with respect to 𝒔 is proportional to 𝟐𝟓𝟎−𝑵. Write the equation of the function. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 3 The rate of change of 𝒚 is inversely proportional to 𝒚 5/8/2019 12:56 PM §6.2: Growth and Decay

Your Turn The rate of change of 𝒚 is proportional to the square root of 𝒚. Find the function. 5/8/2019 12:56 PM §6.2: Growth and Decay

Growth and Decay 5/8/2019 12:56 PM §6.2: Growth and Decay

Growth and Decay 5/8/2019 12:56 PM §6.2: Growth and Decay

Growth and Decay In many applications, the rate of change of a variable is proportional to the value of 𝒚 If 𝒅𝒚 𝒅𝒕 =𝒌𝒚: The equation, 𝒚=𝑪 𝒆 𝒌𝒕 𝑪 is the initial value of 𝒚 𝑲 is the proportional constant Exponential Growth occurs when 𝒌>𝟎 Exponential Decay occurs when 𝒌<𝟎 5/8/2019 12:56 PM §6.2: Growth and Decay

Equation 𝑪 is the initial value of 𝒚 𝒆 is the natural base 𝑲 is the proportional constant Exponential Growth occurs when 𝒌 > 𝟎 Exponential Decay occurs when 𝒌 < 𝟎 5/8/2019 12:56 PM §6.2: Growth and Decay

…is the same as… Growth and Decay 5/8/2019 12:56 PM

Steps Separate the variables Integrate SOLVE for the initial amount (𝑪) Then, solve for other missing parts (𝒌 typically) Solve for 𝒄 or 𝒚 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 4 The weight of a population of yeast is given by a differentiable function 𝒚, where 𝒚 𝒕 is measured in grams and 𝒕 is measured in minutes. The weight of the yeast population increases according to the equation 𝒅𝒚 𝒅𝒕 =𝒌𝒚, where 𝒌 is a constant. At time 𝒕=𝟎, the weight of the yeast population is 144 grams and is increasing at the rate of 24 grams per minute. Write an expression for 𝒚 𝒕 . 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 4 The weight of a population of yeast is given by a differentiable function 𝒚, where 𝒚 𝒕 is measured in grams and 𝒕 is measured in minutes. The weight of the yeast population increases according to the equation 𝒅𝒚 𝒅𝒕 =𝒌𝒚, where 𝒌 is a constant. At time 𝒕=𝟎, the weight of the yeast population is 144 grams and is increasing at the rate of 24 grams per minute. Write an expression for 𝒚 𝒕 . 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 4 The weight of a population of yeast is given by a differentiable function 𝒚, where 𝒚 𝒕 is measured in grams and 𝒕 is measured in minutes. The weight of the yeast population increases according to the equation 𝒅𝒚 𝒅𝒕 =𝒌𝒚, where 𝒌 is a constant. At time 𝒕=𝟎, the weight of the yeast population is 144 grams and is increasing at the rate of 24 grams per minute. Write an expression for 𝒚 𝒕 . 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 4 The weight of a population of yeast is given by a differentiable function 𝒚, where 𝒚 𝒕 is measured in grams and 𝒕 is measured in minutes. The weight of the yeast population increases according to the equation 𝒅𝒚 𝒅𝒕 =𝒌𝒚, where 𝒌 is a constant. At time 𝒕=𝟎, the weight of the yeast population is 144 grams and is increasing at the rate of 24 grams per minute. Write an expression for 𝒚 𝒕 . 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 5 During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, then write the equation where there are 12 days after the epidemic is first discovered. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 5 During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, then write the equation where there are 12 days after the epidemic is first discovered. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 5 During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, then write the equation where there are 12 days after the epidemic is first discovered. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 5 During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, then write the equation where there are 12 days after the epidemic is first discovered. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 5 During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. If 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later, then write the equation where there are 12 days after the epidemic is first discovered. 5/8/2019 12:56 PM §6.2: Growth and Decay

Calculator Instruction [2nd] [+] [7] [1] [2] to ensure clarity ALWAYS in radian mode, NEVER in degree mode Round decimal answers to 4 DECIMAL places. DO NOT round the third number of a decimal answer. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6 Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. Write a differential equation that states this fact. Evaluate the proportionality constant if, at the time of zero, the pressure is 35 psi and decreasing at a rate of .28 psi/min. Solve the differential equation subject to the initial condition in 𝒂. Sketch the graph of the function without a calculator. Show its behavior a long time after the tire is punctured. What would the pressure be 10 minutes after the tire was punctured? The car is safe to drive as long as the tire pressure is 12 psi or greater. For how long after the puncture will the car be safe to drive? 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6a Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. Write a differential equation that states this fact. Evaluate the proportionality constant if, at the time of zero, the pressure is 35 psi and decreasing at a rate of .28 psi/min. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6b Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. B. Solve the differential equation subject to the initial condition in 𝒂. Initial pressure is 35 psi. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6b Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. B. Solve the differential equation subject to the initial condition in 𝒂. Initial pressure is 35 psi. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6c Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. C. Sketch the graph of the function without a calculator. Show its behavior a long time after the tire is punctured. 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6d Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. D. What would the pressure be 10 minutes after the tire was punctured? 5/8/2019 12:56 PM §6.2: Growth and Decay

Example 6e Your tires have just run over a nail. As air leaks out of the car’s tire, the rate of change of the air pressure inside the tire is directly proportional to the pressure. E. The car is safe to drive as long as the tire pressure is 12 psi or greater. For how long after the puncture will the car be safe to drive? 5/8/2019 12:56 PM §6.2: Growth and Decay

AP Multiple Choice Practice Question 1 (non-calculator) The rate of change of atmospheric pressure 𝑷 with respect to the altitude 𝒉 is proportional to 𝑷, provided that the temperature is constant. Which of the following differential equations represents this problem? (A) 𝒅𝑷 𝒅𝒉 =𝑷−𝟎.𝟏𝟓 (B) 𝒅𝑷 𝒅𝒉 =−𝟎.𝟏𝟓𝑷 (C) 𝒅𝑷 𝒅𝒉 = 𝒆 −𝟎.𝟏𝟓𝒉 (D) 𝒅𝑷 𝒅𝒉 = 𝒆 −𝟎.𝟏𝟓𝑷 5/8/2019 12:56 PM §6.2: Growth and Decay

AP Multiple Choice Practice Question 1 (non-calculator) The rate of change of atmospheric pressure 𝑷 with respect to the altitude 𝒉 is proportional to 𝑷, provided that the temperature is constant. Which of the following differential equations represents this problem? Vocabulary Connections and Process Answer and Justifications 5/8/2019 12:56 PM §6.2: Growth and Decay

AP Multiple Choice Practice Question 2 (non-calculator) The number of employees in an office who have heard a rumor at time 𝒕 hours is modeled by the function 𝑷, the solution to a differential equation. At noon, 50 of the office’s 500 employees have heard the rumor. Also at noon, 𝑷 is increasing at a rate of 18 employees per hour. Which of the following could be a differential equation? (A) 𝒅𝑷 𝒅𝒕 = 𝟏 𝟑𝟔𝟎 𝑷 𝟓𝟎𝟎−𝑷 (B) 𝒅𝑷 𝒅𝒕 = 𝟏 𝟏𝟐𝟓𝟎 𝑷 𝟓𝟎𝟎−𝑷 (C) 𝒅𝑷 𝒅𝒕 = 𝟏 𝟐𝟒𝟓𝟎 𝑷 𝟓𝟎𝟎−𝑷 (D) 𝒅𝑷 𝒅𝒕 = 𝟓 𝟒𝟕 𝑷 𝟓𝟎𝟎−𝑷 5/8/2019 12:56 PM §6.2: Growth and Decay

AP Multiple Choice Practice Question 2 (non-calculator) The number of employees in an office who have heard a rumor at time 𝒕 hours is modeled by the function 𝑷, the solution to a differential equation. At noon, 50 of the office’s 500 employees have heard the rumor. Also at noon, 𝑷 is increasing at a rate of 18 employees per hour. Which of the following could be a differential equation? Vocabulary Connections and Process Answer and Justifications 5/8/2019 12:56 PM §6.2: Growth and Decay

Assignment Worksheet 5/8/2019 12:56 PM §6.2: Growth and Decay