Lesson 12
The Number e
- Let’s learn about the number e.
Problem 1
Put the following expressions in order from least to greatest.
- e^3
- 2^2
- e^2
- 2e
- e^e
Problem 2
Here are graphs of three functions: f(x) = 2^x, g(x) = e^x, and h(x) = 3^x.
Which graph corresponds to each function? Explain how you know.
Problem 3
Which of the statements are true about the function f given by f(x) = 100 \boldcdot e^{\text- x}? Select all that apply.
The y-intercept of the graph of f is at (0,100).
The values of f increase when x increases.
The value of f when x = \text-1 is a little less than 40.
The value of f when x = 5 is less than 1.
The value of f is never 0.
Problem 4
Suppose you have $1 to put in an interest-bearing account for 1 year and are offered different options for interest rates and compounding frequencies (how often interest is calculated), as shown in the table. The highest interest rate is 100%, calculated once a year. The lower the interest rate, the more often it gets calculated.
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Complete the table with expressions that represent the amount you will have after one year, and then evaluate each expression to find its value in dollars (round to 5 decimal places).
interest rate frequency
per yearexpression value in dollars
after 1 year100% 1 1 \boldcdot (1+1)^1 10% 10 1 \boldcdot (1+0.1)^{10} 5% 20 1 \boldcdot (1+0.05)^{20} 1% 100 0.5% 200 0.1% 1,000 0.01% 10,000 0.001% 100,000 -
Predict whether the account value will be greater than $3 if there is an option for a 0.0001% interest rate calculated 1 million times a year. Check your prediction.
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What do you notice about the values of the account as the interest rate gets smaller and the frequency of compounding gets larger?
Problem 5
The function f is given by f(x) = (1+x)^{\frac{1}{x}}. How do the values of f behave for small positive and large positive values of x?
Problem 6
Since 1992, the value of homes in a neighborhood has doubled every 16 years. The value of one home in the neighborhood was $136,500 in 1992.
- What is the value of this home, in dollars, in the year 2000? Explain your reasoning.
- Write an equation that represents the growth in housing value as a function of time in t years since 1992.
- Write an equation that represents the growth in housing value as a function of time in d decades since 1992.
- Use one of your equations to find the value of the home, in dollars, 1.5 decades after 1992.
Problem 7
Write two equations—one in exponential form and one in logarithmic form—to represent each question. Use “?” for the unknown value.
- “To what exponent do we raise the number 5 to get 625?”
- “What is the log, base 3, of 27?”
Problem 8
Clare says that \log 0.1 = \text-1. Kiran says that \log (\text-10) = \text-1. Do you agree with either one of them? Explain your reasoning.