3.3 The special number

Introduction

The natural base

Why any exponential function can be written in terms of

Summary

Open a new Desmos worksheet and define the following functions: $f(t) = 2^t$, $g(t) = 3^t$, $h(t) = (\frac{1}{3})^t$, and $p(t) = f(kt)$. After you define $p$, accept the slider for $k$, and set the range of the slider to be $-2 \le k \le 2$.

By experimenting with the value of $k$, find a value of $k$ so that the graph of $p(t) = f(kt) = 2^{kt}$ appears to align with the graph of $g(t) = 3^t$. What is the value of $k$?

What is the meaning of $A(0.5)$ in terms of the function $f$ and its graph?

Compute the value of $A(h)$ for at least $6$ different small values of $h$, both positive and negative. For instance, one value to try might be $h = 0.0001$. Record a table of your results.

What do you notice about the values you found in (b)? How do they compare to an important number?

Explain why the following sentence makes sense: “The function $e^t$ is increasing at an average rate that is about the same as its value on small intervals near $t = 1$.”

Adjust your definition of $A$ in Desmos by changing $1$ to $2$ so that

. How does the value of $A(h)$ compare to $f(2)$ for small values of $h$?

$e^t = 2$

$e^{3t} = 5$

$2e^t - 4 = 7$

$3e^{0.25t} + 2 = 6$

$4 - 2e^{-0.7t} = 3$

$2e^{1.2t} = 1.5e^{1.6t}$

When a single investment of principal, ＄$P$, is invested in an account that returns interest at an annual rate of $r$ (a decimal that corresponds to the percentage rate, such as $r = 0.05$ corresponding to $5$%) that is compounded $n$ times per year, the amount of money in the account after $t$ years is given by $A(t) = P(1 + \frac{r}{n})^{nt}$.

Suppose we invest ＄$100$ in an account that earns $8$% annual interest. We investigate the effects of different rates of compounding.

• Compute $A(1)$ if interest is compounded quarterly ($n = 4$). - Compute $A(1)$ if interest is compounded monthly. - Compute $A(1)$ if interest is compounded weekly. - Compute $A(1)$ if interest is compounded daily. - If we let the number of times that interest is compounded increase without bound, we say that the interest is “compounded continuously”. When interest is compounded continuously, it turns out that the amount of money an account with initial investment ＄$P$ after $t$ years at an annual interest rate of $r$ is $A(t) = Pe^{rt}$, where $e$ is the natural base. Compute $A(1)$ in the same context as the preceding questions but where interest is compounded continuously. - How much of a difference does continuously compounded interest make over interest compounded quarterly in one year's time? How does your answer change over $25$ years' time?

In Desmos, define the function $g(t) = e^{kt}$ and accept the slider for $k$. Set the range of the slider to $-2 \le k \le 2$, and assume that $k \ne 0$. Experiment with a wide range of values of $k$ to see the effects of changing $k$.

• For what values of $k$ is $g$ always increasing? For what values of $k$ is $g$ always decreasing? - For which value of $k$ is the average rate of change of $g$ on $[0,1]$ greater: when $k = -0.1$ or when $k = -0.05$? - What is the long-term behavior of $g$ when $k \lt 0$? Why does this occur? - Experiment with the slider to find a value of $k$ for which $g(2) = \frac{1}{2}$. Test your estimate by computing $e^{2k}$. How accurate is your estimate?

A can of soda is removed from a refrigerator at time $t = 0$ (in minutes) and its temperature, $F(t)$, in degrees Fahrenheit, is computed at regular intervals. Based on the data, a model is formulated for the object's temperature, given by

.

• What is the long-term behavior of the function $g(t) = e^{-0.05t}$? Why? - What is the long-term behavior of the function $F(t) = 74.4 - 38.8e^{-0.05t}$? What is the meaning of this value in the physical context of the problem? - What is the temperature of the refrigerator? Why? - Compute the average rate of change of $F$ on the intervals $[10,20]$, $[20,30]$, and $[30,40]$. Write a careful sentence, with units, to explain the meaning of each, and write an additional sentence to describe any overall trends in how the average rate of change of $F$ is changing.