Problem 38 - Contemporary Precalculus

Question

Write the rule of the function in the form \(\left.f(x)=P e^{k x} . \text { (See the discussion and box after Example } 11 .\right)\) $$g(x)=3.9\left(1.03^{x}\right)$$

Step-by-Step Solution

Answer

Question: Rewrite the function \(g(x) = 3.9(1.03^x)\) in the form of \(f(x) = P e^{kx}\), and identify the values of \(P\) and \(k\). Answer: The function \(g(x)=3.9(1.03^x)\) can be rewritten in the form of \(f(x) = P e^{kx}\) as \(f(x) = 3.9 e^{x \log(1.03)}\), with \(P = 3.9\) and \(k = \log(1.03)\).

Step 1: Identify the given function and its components.

The given function is \(g(x) = 3.9(1.03^x)\). The current form represents an exponential function where the initial value is \(3.9\) and the base of the exponent is \(1.03\).

Step 2: Identify the desired form and its components.

The desired form for the function is \(f(x) = P e^{kx}\). Here, \(P\) represents the initial value, and \(k\) represents the growth factor, which will be expressed within the exponent for \(e^x\) (Euler's number).

Step 3: Rewrite the given function using base e.

We need to rewrite the given function, \(g(x) = 3.9(1.03^x)\), using base \(e\). To achieve this, we can use the rule \(a^x = e^{x \log(a)}\). Apply this rule to rewrite the function: $$g(x) = 3.9 \left(e^{x \log(1.03)}\right)$$

Step 4: Identify the values for P and k.

Now that we have the function in the form of \(g(x) = 3.9 e^{x \log(1.03)}\), we can easily identify the initial value \(P\) and the growth factor \(k\). Here, \(P = 3.9\) and \(k = \log(1.03)\).

Step 5: Write the function in the desired form.

We now have all the components needed to write the function in the desired form. The function \(g(x)\) can be written in the form of \(f(x)\) as: $$f(x) = 3.9 e^{x \log(1.03)}$$ Therefore, the function in the requested form is \(f(x) = 3.9 e^{x \log(1.03)}\).

Key Concepts

Function Transformation

Function transformation is an essential concept in mathematics, particularly when dealing with functions like exponential ones. In essence, transforming a function involves changing its input, output, or both in a systematic manner. For our problem, the transformation focuses on expressing the given function involving the base of a number, such as 1.03, in terms of the natural exponential base, often denoted as \(e\).

This kind of transformation is beneficial because it allows for more straightforward application and understanding of calculus-related operations, such as differentiation and integration.

Here's how it works:

• Recognize the initial form of the function. In our case, it was \(g(x) = 3.9(1.03^x)\).
• Understand the target form, which was to express the function as \(f(x) = P e^{kx}\).
• Use known mathematical rules to transform the current function. Specifically, the rule \(a^x = e^{x \log(a)}\) is useful here to shift from base 1.03 to base \(e\).

By practicing these transformations, you'll develop a versatile mathematical toolkit for handling various types of functions.

Logarithmic Functions

Logarithmic functions are crucial in understanding and transforming exponential expressions. The log function, particularly a natural logarithm, helps transform regular exponential bases into the base \(e\), which is preferred in many mathematical and scientific computations.

When rewriting our given exponential function \(g(x) = 3.9(1.03^x)\) into a form involving \(e\), we applied a logarithmic transformation using the rule \(a^x = e^{x \log(a)}\). In this instance, \(\log(1.03)\) provides the necessary growth factor \(k\) when transforming the function into the desired form \(f(x) = P e^{kx}\).

Logarithms have several properties that make them especially useful for function manipulation:

• Converting multiplication into addition: \(\log(ab) = \log(a) + \log(b)\).
• Turning exponents into coefficients: \(\log(a^b) = b \log(a)\).
• Adjusting bases efficiently, such as converting base 1.03 into base \(e\).

Understanding these can simplify complex equations and make calculations more manageable.

Euler's Number (e)

Euler's Number, denoted as \(e\), is a fundamental mathematical constant approximately equal to 2.71828. It plays a pivotal role in exponential and logarithmic functions due to its unique properties, especially in continuous growth or decay processes.

In the context of our exercise, \(e\) provides an excellent base for rewriting exponential functions like \(g(x) = 3.9(1.03^x)\). By converting the base to \(e\), we create the expression \(f(x) = 3.9 e^{x \log(1.03)}\), which is advantageous for practical applications.

The significance of \(e\) can be summarized with its key features:

• It's the natural base for continuous compounding, making it prevalent in natural processes and financial mathematics.
• In calculus, the function \(e^x\) has a unique property where its derivative is itself, \(\frac{d}{dx}e^x = e^x\).
• It simplifies solving many real-world problems involving growth rates, like populations or investments.

Understanding \(e\) is crucial because it not only simplifies mathematical expressions but also provides a foundation for advanced mathematical concepts.