Unit 4, Lesson 12
The Number $e$
Integrated Math 3
Not all roles available for this page.
Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
Put these expressions in order from least to greatest.
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.
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.
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 rate is adjusted so that the rate divided by the frequency per year remains 100%.
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 year |
expression | value in dollars after 1 year |
|---|---|---|---|
| 100% | 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.
What do you notice about the value of the account as the frequency of compounding gets larger?
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 \(x\)?
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.
Write two equations—one in exponential form and one in logarithmic form—to represent each question. Use “?” for the unknown value.
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.