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

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## Exponential Functions and Percent Rate of Change

1.

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?

- 5%, growth
- 5%, decay
- 10%, growth
- 10%, decay

2.

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?

- 8%, growth
- 8%, decay
- 13%, growth
- 13%, decay

3.

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?

- 225%, growth
- 75%, decay
- 200%, growth
- 25%, decay

4.

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.

- 87.5%, decay
- 50%, decay
- 12.5%, decay
- 100%, growth

5.

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?

- Account I, at 5%.
- Account II, at 20%.
- Account II, at about 22%.
- Account II, at about 5.1%.

6.

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.

A.

Using the information given, which of the following functions has the correct values for the constants [math]T_a, T_0, k ?[/math]

- [math]T(t) = 210 + 135 e^(-0.34 t)[/math]
- [math]T(t) = 75 + 135 e^(-0.075 t)[/math]
- [math]T(t) = 75 + 210 e^(-0.19 t)[/math]
- [math]T(t) = 75 + 35 e^(0.26 t)[/math]

B.

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].

- [math]T(t) = 1.445 * (1.078)^t + 75[/math]
- [math]T(t) = 0.692 * (0.928)^t + 75[/math]
- [math]T(t) = 135 * (1.078)^t + 75[/math]
- [math]T(t) = 135 * (0.928)^t + 75[/math]

C.

Using the re-written function in the previous question, what is the percent rate of change of the function [math]T(t) ?[/math]

- 7.2%
- 7.8%
- 92.8%
- 69.2%

D.

Find the percent rate of change for the function [math]T(t)[/math] between [math]t=1[/math] and [math]t=2[/math].

- 1.0%
- 9.0%
- 4.5%
- 95.5%

E.

Why are the numbers for the percent rate of change calculated in the previous two questions not the same?

- Because the calculations didn't use exact numbers, and therefore there are rounding errors.
- Because the function has an added constant (it is not of the form [math]ab^x[/math]).
- 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].
- Because an exponential function is not constant, therefore its percent rate of change shouldn't be constant.

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