Does the equation y=45(1.82)x represent exponential decay or growth and percentage increase

Identifying Exponential Decay or Growth

The given equation is:

This type of equation represents an exponential function. To determine whether it signifies exponential growth or exponential decay, we need to examine the base of the exponent, which in this case is $1.82$.

Exponential Growth

If the base of the exponent $b$ (in this case, $1.82$) is greater than $1$, the function represents exponential growth. Here, since $1.82 > 1$, the given equation clearly represents exponential growth.

Percentage Increase

To find the percentage increase, we use the formula:

$\text{Percentage Increase} = (b - 1) \times 100\%$

Substituting $b = 1.82$:

Conclusion

The equation $y = 45(1.82)^x$ represents exponential growth with an 82% percentage increase.

Understanding Exponential Function Analysis

Exponential function analysis is a field within mathematics that deals with functions of the form:

where $e$ is the base of natural logarithms, approximately equal to 2.71828, and $a$ and $b$ are constants.

Key Properties

1. Growth Rate:

   • Exponential Growth occurs when $b > 0$. The function increases rapidly as $x$ increases.
   • Exponential Decay happens when $b < 0$. The function decreases rapidly as $x$ increases.

2. Derivatives and Integrals:

   • The derivative of an exponential function remains proportional to the original function:
   $$\frac{d}{dx} \left( a e^{bx} \right) = ab e^{bx}$$
   • The integral of an exponential function is given by:
   $$\int a e^{bx}\, dx = \frac{a}{b} e^{bx} + C$$

   where $C$ is the constant of integration.

Applications

• Modeling Population Growth: Exponential functions effectively model populations in biology where the growth rate is proportional to the current size.
• Compound Interest: In finance, exponential functions are used to calculate compound interest.
• Radioactive Decay: Physics uses exponential decay to describe the process by which unstable atoms lose energy.

Example

Suppose we want to model the growth of a bacteria colony. If the initial population $P_0$ doubles every hour, we can represent this with the function:

where $k$ is the growth constant.

Conclusion

Exponential function analysis serves as a powerful tool to describe various natural processes and phenomena, invariably linking mathematics with real-world applications through its properties and behaviors.

Understanding Base Comparison for Growth or Decay Determination

The concept of "base comparison for growth or decay determination" is essential in understanding the behavior of exponential functions. Exponential functions generally take the form:

Here, $a$ is a constant, $b$ is the base, and $x$ is the exponent. The base $b$ plays a crucial role in determining whether the function is exhibiting growth or decay.

Exponential Growth

• The function exhibits growth when the base $b$ is greater than 1.
• This can be mathematically expressed as:

• For instance, consider the function $f(x) = 2 \cdot 3^x$. Since $3 > 1$, this function represents exponential growth.

Exponential Decay

• The function exhibits decay when the base $b$ is between 0 and 1.
• This can be expressed as:

• For example, $f(x) = 5 \cdot (0.5)^x$ is a decaying function since $0.5$ lies in the interval $0 < 0.5 < 1$.

Summary

In essence, comparing the base to 1 is the key step in determining whether an exponential function is growing or decaying:

• Growth when $b > 1$
• Decay when $0 < b < 1$

This simple comparison allows us to quickly understand the nature of the exponential function without complex calculations.