Problem 28

Question

For the following exercises, refer to Table 7. $$\begin{array}{ccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} \\ {f(x)} & {1125} & {1495} & {2310} & {3294} & {4650} & {6361}\end{array}$$ Write the exponential function as an exponential equation with base $e .$

Step-by-Step Solution

Verified

Answer

The exponential equation is $ f(x) = ae^{bx} $ with specific $ a $ and $ b $ values calculated from the data.

1Step 1: Understand the Exponential Function

In an exponential function of the form $ f(x) = ae^{bx} $, $ a $ represents the initial amount when $ x = 0 $, and $ b $ controls the growth rate. Our task is to express the given set of values as such a function.

2Step 2: Identify Suitable Form for Equation

We need to write the equation in the form $ f(x) = ae^{bx} $. We will use two data points from the table to determine the values of $ a $ and $ b $.

3Step 3: Select Data Points

Choose the points $(1, 1125)$ and $(6, 6361)$ from the table to set up two equations. These points will allow us to solve for $ a $ and $ b $.

4Step 4: Set Up Equations for Data Points

Substitute the chosen points into the equation form: $ 1125 = ae^{b(1)} $ and $ 6361 = ae^{b(6)} $.

5Step 5: Solve for Parameter $ a $ and $ b $

From $ 1125 = ae^{b} $, we have $ a = \frac{1125}{e^b} $. Substitute this expression into the second equation: $ 6361 = \frac{1125}{e^b} \, e^{6b} $ to solve for $ b $.

6Step 6: Simplify and Solve for $ b $

Rearrange to get $ 6361 \cdot e^b = 1125 \cdot e^{6b} $, which simplifies to $ \frac{6361}{1125} = e^{5b} $. Taking natural log gives $ \ln\left( \frac{6361}{1125} \right) = 5b $.

7Step 7: Calculate $ b $

Calculate $ b $ as follows: $ b = \frac{\ln\left( \frac{6361}{1125} \right)}{5} $.

8Step 8: Solve for $ a $ Using $ b $

Plug the value of $ b $ back into the equation $ a = \frac{1125}{e^b} $ to find $ a $.

9Step 9: Write Final Exponential Equation

Insert the calculated values of $ a $ and $ b $ into the expression $ f(x) = ae^{bx} $ to obtain the exponential function.

Key Concepts

Base e in Exponential FunctionsUnderstanding Growth Rate in Exponential FunctionsSolving Exponential EquationsDetermining the Initial Amount: $ a $

Base e in Exponential Functions

The constant $ e $ is approximately equal to 2.71828. It is known as Euler's number, after the Swiss mathematician Leonhard Euler. This number is unique and fundamental in mathematics, particularly in the field of exponential functions. Exponential functions often have a general form, $ f(x) = ae^{bx} $, where $ e $ serves as the base of the exponent.
This base is important as it simplifies many mathematical operations. The derivative of $ e^x $ with respect to $ x $ is $ e^x $, making calculus operations straightforward and elegant.
Using $ e $ as the base also helps in modeling continuous growth, such as populations or compounded interest, because $ e $ best approximates real-world exponential processes.

Understanding Growth Rate in Exponential Functions

The growth rate in an exponential function is determined by the parameter $ b $ in the equation $ f(x) = ae^{bx} $. This parameter dictates whether the function is growing or decaying.

If $ b > 0 $, the function describes growth. The larger the $ b $, the faster the growth.
If $ b < 0 $, the function describes decay, which means it's decreasing over time.

Understanding growth rate is crucial for interpreting how quickly a quantity is increasing. This has applications in various fields like finance, biology, and physics. When solving exponential equations, determining $ b $ is key to understanding the nature of the growth process described by the function.

Solving Exponential Equations

Solving exponential equations involves finding the unknown values that satisfy the equation. Using the form $ f(x) = ae^{bx} $, we typically solve for constants $ a $ and $ b $ given specific data points.
The exercise provided involves selecting two data points to create a system of equations. Substituting these points into the form $ ae^{bx} $ leads to equations that we can solve.
To isolate $ b $, we might take the natural logarithm of both sides, using properties of logarithms to simplify. The logarithm allows the exponent to be brought down, turning the equation linear in terms of $ b $. This step is crucial for getting from complex exponential terms to a straightforward calculation.

Determining the Initial Amount: $ a $

In any exponential function described by $ f(x) = ae^{bx} $, the initial amount $ a $ represents the starting value of the function when $ x = 0 $. This constant provides the baseline from which growth or decay is measured.
To determine $ a $, one often needs the function value at a specific $ x $, typically at the smallest $ x $ in the data set, which might not be zero. After calculating $ b $, substituting it back will allow solving for $ a $. In practical applications, $ a $ is significant as it gives context to what is being measured or tracked over time, such as initial investments, starting populations, or other initial quantities.

Problem 28