Exponential growth and decay #12 - Questions and Answers

Question 1 A car's value decreases at a rate of 5% annually. The car was worth $32,000 in 2010. Find the car's 2013 market value. Step 1: Decide whether it is growing or decaying Step 2: Solve for the rate: Step 3: solve Answer Given: worth of car is 32,000 Rate of depreciation is 5% year. since the car value is depreciating. It is an exponential decay function. Rate = 5% = 0.05 Exponential decay formula is A = P(1 - r)^t where P is the initial value r is the rate t is the time period p=32000, r=0.05, t=2013-2010=3 years A = P(1 - r)^t where P is the initial value A=32000(1-0.05)^3 A=32000(0.857375) A=27436 The value of car in 2013 is $27,436 Question 2 Exponential Growth /Decay Model Given exponential growth or decay the amount P after time t is given by the following formula: P = P₀e^(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 y-describes 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 be the rate of change of the variable y. Then, as per the question dy/dx proportional to y. that is, dy/dx = Ky where, K is the constant of proportionality. Hence the differential equation is dy/dx = Ky, where K is the constant of proportionality. Question 3 Polynomial function graphs have similarities depending on their degree. Explain how you can determine the best regression model by using finite differences. Then determine the end behavior of the graph based on the degree of a function and information gathered from a data table. Answer Step 1 Determining the best regression model by using finite differences. The method of finite differences is used for curve fitting with polynomial models. It is best introduced by an example as shown in table given below. For the sequence of x value inputs at the regularly spaced points -3, -2, -1, ..., 3, the y value outputs are shown in the table. Table 1 is given below. From the table, we can understand that this is third difference finite function of third degree. This explains f(x) = ax³ + bx² + cx + d. Step 2 Here we need to find the value of a, b, c and d a = -6 / 3! a = -6 / 6 a = -1 Step 3 Taking x = 0, -1, 1 for calculating the value of rest of the co-efficients. f(x) = -x³ + bx² + cx + d f(-1) = 1 + b - c + d x = -1 y = 0 0 = 1 + b - c + d -7 = b - c (Table showing the finite difference calculation) Also f(1) = -1 + b + c + d 8 = b + c + 5 b + c = 3 Moreover, f(0) = 0 + 0 + 0 + d x = 0 y = 6 6 = d Thus, finding b and c by subtracting and adding equation (1) and (2) 2b = -4 b = -2 2c = 10 c = 5 At last the finite function becomes: f(x) = -x³ - 2x² + 5x + 6 Step 4 Sketching a graph from the function found above Thus, end behaviour of f(x) = -x³ - 2x² + 5x + 6 is described from the table as below As x → ∞, f(x) → -∞ and As x → -∞, f(x) → ∞ (Image of graph showing a cubic function with the described end behavior) Question 4 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) = Ce⁰ = 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