University:
Stanford University
Course:
MATH 20 | Calculus
Academic year:

2023
Views:

30

Pages:

4
Author:

MathPaladin

Questions and Answers Sheet 2 Exponential Growth and Decay Question #1 Determine whether each function represents exponential growth or decay. Write the base in terms of the rate of growth or decay, identify r, and interpret the rate of growth or decay 7 𝑓(𝑥) = 12000 ( ) 𝑥 10 Answer: For an exponential function in the form 𝑦 = 𝑎 × 𝑏 𝑥 If b>1, the function is increasing and is an exponential growth function. If 0 1, t is a growth factor. Rewrite the function in exponential growth form. 𝑦 = 2(1 + 0.7)𝑡 Therefore, the function represents exponential growth. The function represents exponential growth. Question #3 Tell whether the function represents exponential growth or exponential decay. Then graph the function. 𝑦 = (1.8) 𝑥 Answer: The base of the function 𝑦 = 1.8𝑥 𝑖𝑠 1.8 > 1, so the function represents exponential growth. Look at the base to determine whether it is growth or decay. If the base is greater than 1, the function represents exponential growth. If the base is less than 1, the function represents exponential decay. To sketch the graph, find a few points (x,y) by choosing some value x and calculating its pair 𝑦 = 1.8𝑥 , then draw smooth curve through those points. exponential growth Question #4 Explain when a function in the form 𝑦 = 𝑎 × 𝑏𝑥 models exponential growth and when it models exponential decay. Answer: Recall the property: 𝑎−𝑛 = 1 𝑎𝑛 Possible answers will be: 𝑥 −5 = 1 𝑥5 𝑥 −1 𝑦 −1 = 𝑎 −2 𝑏 −3 = 𝑏3 𝑎2 1 𝑥𝑦 Question #5 Give an example of an exponential function in the form 𝑦 = 𝑎 × 𝑏 𝑥 that is neither an exponential growth function nor an exponential decay function. Explain your reasoning. Answer: An exponential function of the form 𝑦 = 𝑎 × 𝑏 𝑥 is an exponential growth or decay function if a > 0. Therefore any function of the form 𝑦 = 𝑎 × 𝑏 𝑥 where a < 0 would not be an exponential growth or decay function. A possible function could then be 𝑦 = −2 × 3𝑥 𝑦 = −2.3𝑥 is a possible function since 𝑦 = 𝑎 × 𝑏 𝑥 must have a >0 to be an exponential growth or decay function. Question #6 The equation = 𝑦8000(1.09) 𝑥 describes the amount of an investment (in dollars) based on the number of years it's been allowed to grow. (a) Identify what the equation = 𝑦8000(1.09) 𝑥 illustrates. Quadratic growth, linear decay. linear growth, quadratic decay, expotential growth, or expotential decay Answer: We have the given equation is 𝑦 = 8000(1.9)𝑥 After seeing the equation it is clear that it is neither quadratic growth nor quadratic decay and it is also same for linear. We also know that the exponential growth and decay is given by the formula 𝑦 = 𝑎𝑏 𝑥 where 𝑎 ≠ 0 , the base 𝑏 ≠ 1 and x is any real number. Now, if b>1 then the function represents exponential growth and if 01 Therefore, we can say that the given equation is an exponential growth. Question #7 Describe what the values of C and k represent in the exponential growth and decay model 𝑦 = 𝐶𝑒 𝑘𝑡 . Answer: For t=0, 𝑦 = 𝐶𝑒 𝑘(0) = 𝐶𝑒 0 = 𝐶(1) = 𝐶 So the initial value is C If k is a positive number then it is the growth rate, if k is negative then it is the decay rate. Answer: C=initial amount k=growth rate/decay rate Question #8 Exponential Growth /Decay Model Given exponential growth or decay the amoun tP after time tis given by the following formula: 𝑃 = 𝑃0 𝑒 𝑘𝑡 Here 𝑃0 is the initial amount and k is the exponential growth/decay rate. Now you are ready to complete the following: a) As stated previously the statement - The rate of change of variable y is proportional to the value of ydescribes a differential equation. Write this differential equation. Answer: a) The rate of change of variable y is proportional to the value of y. Let 𝑑𝑦 𝑑𝑥 𝑑𝑦𝑑𝑥 be the rate of change of the variable y. Then, as per the question that is, 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥 proportional to y dydx proportional to y. 𝑑𝑦𝑑𝑥 = 𝐾𝑦 where, K is the constant of proportionality. Hence the differential equation is 𝑑𝑦 𝑑𝑥 𝑑𝑦𝑑𝑥 = 𝐾𝑦 , where K is the constant of proportionality.