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Exponential Functions and Percent Rate of Change (Grades 11-12)

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Exponential Functions and Percent Rate of Change

For the function [math]f(x) = 10 (1.05)^x[/math], what is the percent rate of change? Also, is this exponential growth or decay?
  1. 5%, growth
  2. 5%, decay
  3. 10%, growth
  4. 10%, decay
For the function [math]g(x) = 13*(0.92)^x[/math], what is the percent rate of change? Also, is this exponential growth or decay?
  1. 8%, growth
  2. 8%, decay
  3. 13%, growth
  4. 13%, decay
Simplify the function [math]f(x) = (1/2) 3^x (1/4)^x[/math], and then determine the percent rate of change. Also, is this exponential growth or decay?
  1. 225%, growth
  2. 75%, decay
  3. 200%, growth
  4. 25%, decay
Simplify the function [math]f(x) = 4^2 * 2^(-x) / (1/4)^x[/math]. Determine the percent rate of change, and whether it is exponential growth or decay.
  1. 87.5%, decay
  2. 50%, decay
  3. 12.5%, decay
  4. 100%, growth
An investment that is paid compound interest can be modeled by the function [math]A(t) = P*(1+r/n)^{n \ t}[/math], where [math]A[/math] is the current value of the investment after [math]t[/math] years, [math]P[/math] is the initial amount invested or principal, r is the interest rate, and [math]n[/math] is the number of compounding periods per year. $3,000 is invested in two accounts, Account I which pays 5% interest compounded quarterly, and Account II which pays 5% interest compounded yearly. Using the above function to model these two investments, which function has a greater percent rate of change?
  1. Account I, at 5%.
  2. Account II, at 20%.
  3. Account II, at about 22%.
  4. Account II, at about 5.1%.
A cup of coffee that has just been poured has a temperature of 210 °F. The temperature of the room is 75 °F. After 4 minutes, the coffee's temperature is 175 °F. The function [math] T(t) = T_a + (T_0 - T_a)e^(-k t) [/math], for [math]t>= 0[/math], can model the coffee's temperature as it cools, where [math]T_a[/math] is the ambient temperature, [math]T_0[/math] is the initial temperature, and [math]k[/math] is the constant of decay.
Using the information given, which of the following functions has the correct values for the constants [math]T_a, T_0, k ?[/math]
  1. [math]T(t) = 210 + 135 e^(-0.34 t)[/math]
  2. [math]T(t) = 75 + 135 e^(-0.075 t)[/math]
  3. [math]T(t) = 75 + 210 e^(-0.19 t)[/math]
  4. [math]T(t) = 75 + 35 e^(0.26 t)[/math]
Using laws of exponents, re-write the function [math]T(t)[/math] into the form [math]T(t) = a*b^t + c[/math], where [math]a,b,c in RR[/math].
  1. [math]T(t) = 1.445 * (1.078)^t + 75[/math]
  2. [math]T(t) = 0.692 * (0.928)^t + 75[/math]
  3. [math]T(t) = 135 * (1.078)^t + 75[/math]
  4. [math]T(t) = 135 * (0.928)^t + 75[/math]
Using the re-written function in the previous question, what is the percent rate of change of the function [math]T(t) ?[/math]
  1. 7.2%
  2. 7.8%
  3. 92.8%
  4. 69.2%
Find the percent rate of change for the function [math]T(t)[/math] between [math]t=1[/math] and [math]t=2[/math].
  1. 1.0%
  2. 9.0%
  3. 4.5%
  4. 95.5%
Why are the numbers for the percent rate of change calculated in the previous two questions not the same?
  1. Because the calculations didn't use exact numbers, and therefore there are rounding errors.
  2. Because the function has an added constant (it is not of the form [math]ab^x[/math]).
  3. Because the percent rate of change calculated from the different form of the function, [math]T(t) = a*b^t + c[/math], is a general rate of change, which is not the same as the specific rate of change from [math]t=1[/math] to [math]t=2[/math].
  4. Because an exponential function is not constant, therefore its percent rate of change shouldn't be constant.

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