Problem 8 - Calculus Concepts

Question

a. indicate whether the function describes exponential growth or decay. b. give the constant percentage change. $$ A(t)=0.57\left(1.035^{t}\right) $$

Step-by-Step Solution

Answer

a. Exponential growth. b. 3.5% constant percentage increase.

Step 1: Identify the Base of the Exponent

The function given is \( A(t) = 0.57 \times 1.035^t \). To determine whether this function represents exponential growth or decay, we need to focus on the base of the exponent, which is 1.035. If this value is greater than 1, the function represents exponential growth. If it is less than 1, it represents exponential decay.

Step 2: Determine Growth or Decay

The base of the exponent in the function is 1.035, which is greater than 1. Therefore, the function represents exponential growth. If the base had been less than 1, it would have indicated exponential decay.

Step 3: Calculate the Constant Percentage Change

To find the constant percentage change for the growth, we use the formula \( (b - 1) \times 100\% \), where \( b \) is the base of the exponent. Here \( b = 1.035 \). So the calculation is: \[ (1.035 - 1) \times 100\% = 0.035 \times 100\% = 3.5\% \]This means the constant percentage change for the function is an increase of 3.5%.

Key Concepts

Exponential Decay

Exponential decay is a process where quantities decrease over time at a rate proportional to their current value. This type of decay happens in situations like radioactive decay, cooling of an object, or depreciation in value. In a mathematical function, exponential decay occurs when the base of the exponent is less than 1. For instance, consider a function of the form \( A(t) = a \times b^t \), where \( b < 1 \). When the base \( b \) is less than 1, each time period results in a decrease of the quantity by a certain percentage. This percentage decrease is known as the decay rate. To calculate this rate, you can use the formula:

• Decay Rate = \((1 - b) \times 100\%\)

This formula indicates by how much the quantity decreases as a percentage in each time period. Understanding exponential decay is crucial in fields like finance, chemistry, and physics. It helps predict how quickly a quantity will reduce. While the problem at hand shows exponential growth, understanding decay will build a foundation for recognizing this concept in various contexts.

Constant Percentage Change

Constant percentage change is a key feature of exponential functions, whether they describe growth or decay. For exponential growth, this change means the quantity increases by a fixed percent each period. Similarly, for exponential decay, as discussed earlier, the quantity decreases by a constant percent. To find the constant percentage change in exponential functions, we focus on the base \( b \) of the exponential component \( b^t \), where:

• For growth: The formula is \((b - 1) \times 100\% \)
• For decay: The formula is \((1 - b) \times 100\% \)

These formulae make it simple to compute the percentage change with just knowledge of \( b \), the base. In the given function, \( A(t) = 0.57 \times 1.035^t \), the base 1.035 gives a constant percentage change of 3.5%. This means the quantity grows by 3.5% every period. Understanding this helps in projections and modeling in fields like population studies, finance, and more, where predicting future values is essential.

Base of the Exponent

The base of the exponent in an exponential function is the determining factor for whether the function models growth or decay. This parameter, represented as \( b \), is a vital element in the equation \( A(t) = a \times b^t \). Here's how the base impacts the nature of the growth:

• If \( b > 1 \), the function describes exponential growth. Here, each increase in \( t \) leads to multiplication by \( b \), causing the quantity \( A(t) \) to grow.
• If \( b < 1 \), it indicates exponential decay, where each step reduces the value.

In the given function, \( A(t) = 0.57 \times 1.035^t \), the base \( b \) is 1.035. Since it is greater than 1, it confirms the function as representing exponential growth. Understanding the base is crucial. It simplifies recognizing the behavior of a function without computation. It is an essential concept in mathematics and various applications such as biology, economics, and engineering, where exponential patterns frequently appear.