Exponential Functions and Applications in Business Calculus

Exponential Functions

Definition and Properties

An exponential function with base a is defined as:

$f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Domain: All real numbers ($(-\infty, \infty)$)

• Range: All positive real numbers ($(0, \infty)$)

• One-to-one property: For each $y$ in the range, there is exactly one $x$ in the domain such that $a^x = y$.

• Horizontal asymptote: $y = 0$

• Behavior: If $a > 1$, $f(x)$ increases as $x$ increases. If $0 < a < 1$, $f(x)$ decreases as $x$ increases.

Graphing Exponential Functions

Graphs of $f(x) = 2^x$ and $f(x) = (\frac{1}{2})^x = 2^{-x}$ illustrate the difference between growth ($a > 1$) and decay ($0 < a < 1$).

x	y = 2^x	y = 2^{-x}
-3	1/8	8
-2	1/4	4
-1	1/2	2
0	1	1
1	2	1/2
2	4	1/4
3	8	1/8

Transformations:

• Shifting $f(x) = 2^x$ right by 2 units: $g(x) = 2^{x-2}$

• Shifting $f(x) = 2^x$ up by 2 units: $h(x) = 2^x + 2$

• Reflecting $f(x) = 2^x$ across the y-axis: $k(x) = 2^{-x}$

Domain and Range Examples:

• $g(x) = 2^{x-2}$: Domain $(-\infty, \infty)$, Range $(0, \infty)$

• $h(x) = 2^x + 2$: Domain $(-\infty, \infty)$, Range $(2, \infty)$

• $k(x) = 2^{-x}$: Domain $(-\infty, \infty)$, Range $(0, \infty)$

Solving Exponential Equations

To solve equations involving exponential functions, use the one-to-one property:

• If $a^u = a^v$, then $u = v$.

Example:

• $(\frac{1}{2})^{x-1} = (\frac{1}{4})^{x+1}$ $\left(\frac{1}{2}\right)^{x-1} = \left(\frac{1}{2}\right)^{2(x+1)}$ $x-1 = 2(x+1)$ $x-1 = 2x+2$ $-x = 3$ $x = -3$

• $9^{x} = 27^{3-x}$ $3^{2x} = 3^{3(3-x)}$ $2x = 9 - 3x$ $5x = 9$ $x = \frac{9}{5}$

The Number e and Continuous Growth

Definition of e

The mathematical constant e is defined as:

$e = \lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m \approx 2.71828$

m	$(1 + \frac{1}{m})^m$
1	2
10	2.593742
100	2.704813
10,000	2.718145
1,000,000	2.718280
1,000,000,000	2.718281

Exponential Functions with Base e

The function $f(x) = e^x$ is a special exponential function used in continuous growth and decay models.

x	$f(x) = e^x$
-1	0.4
0	1
1	2.7
2	7.4

Transformations: $g(x) = -e^{x+1} + 3$ shifts and reflects the graph.

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, 3)$

Applications: Interest and Investment

Simple Interest

Simple interest is calculated only on the principal:

$I = Prt$

• P: Principal (amount invested or borrowed)

• r: Annual interest rate (as a decimal)

• t: Time in years

Compound Interest

Compound interest is calculated on both the principal and the accumulated interest.

After	Interest earned	Total amount on deposit
1 year	$P \cdot r$	$P + Pr = P(1 + r)$
2 years	$P(1 + r) \cdot r$	$P(1 + r)^2$
3 years	$P(1 + r)^2 \cdot r$	$P(1 + r)^3$
n years	$P(1 + r)^{n-1} \cdot r$	$P(1 + r)^n$

If interest is compounded m times per year:

$A = P \left(1 + \frac{r}{m}\right)^{mt}$

Continuous Compounding

As the number of compounding periods per year increases without bound ($m \to \infty$), the formula becomes:

$A = Pe^{rt}$

Example: Investment Calculations

• Quarterly compounding: $P = 5000$, $r = 0.06$, $t = 10$, $m = 4$ $A = 5000 \left(1 + \frac{0.06}{4}\right)^{40} = 5000(1.015)^{40} \approx 8,107.08$

• Continuous compounding: $A = 5000 e^{0.06 \times 10} = 5000 e^{0.6} \approx 8,107.08$

Business Application: Exponential Growth of GDP

Modeling GDP with Exponential Functions

The U.S. Gross Domestic Product (GDP) can be modeled using an exponential function:

Year (1940+x)	GDP (billions)
1940	103
1950	300
1960	542
1970	1073
1980	2857
1990	5963
2000	10,252
2010	14,992
2019	21,429

Finding the Exponential Model:

• Let $x = 0$ correspond to 1940.

• Find $f(x) = b a^x$ that fits the data at 1940 and 2019.

• Using $x_1 = 0$ and $x_2 = 79$ (2019): $f(x) = 103 \cdot (1.0644)^x$

Estimating Doubling Time:

• Set $f(x) = 2 \times 21,429$ and solve for $x$ to estimate when GDP will double from its 2019 value.

Graphical Representation: The exponential model fits the GDP data, showing rapid growth over time.

Summary Table: Exponential Function Properties

Property	Description
Domain	$(-\infty, \infty)$
Range	$(0, \infty)$
Horizontal Asymptote	$y = 0$
Growth/Decay	Growth if $a > 1$, Decay if $0 < a < 1$
One-to-one	Each $y$ in range corresponds to one $x$ in domain

Additional info: These notes cover foundational concepts in exponential functions, their properties, graphing techniques, and applications in finance and economics, which are central topics in Business Calculus.