Problem 57 - Precalculus

Question

Refer to the following: In calculus, we find the derivative, \(f^{\prime}(x),\) of a function \(f(x)\) by allowing \(h\) to approach 0 in the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of functions involving exponential functions. Find the difference quotient of the exponential growth model \(f(x)=P e^{k x},\) where \(P\) and \(k\) are positive constants.

Step-by-Step Solution

Answer

The difference quotient is \( \frac{P e^{k x} (e^{k h} - 1)}{h} \).

Step 1: Write the Definition of the Difference Quotient

The difference quotient is given by:\[\frac{f(x+h)-f(x)}{h}\]For the function \( f(x) = P e^{k x} \), we substitute to get:\[\frac{P e^{k(x + h)} - P e^{k x}}{h}\]

Step 2: Simplify the Exponential Expression

Use the properties of exponents to expand the expression:\[P e^{k(x+h)} = P e^{k x} e^{k h}\]Substitute back into the difference quotient:\[\frac{P e^{k x} e^{k h} - P e^{k x}}{h}\]

Step 3: Factor Out Common Terms

Factor out the common term \( P e^{k x} \) from the numerator:\[P e^{k x} \left( e^{k h} - 1 \right)\]Now, the difference quotient becomes:\[\frac{P e^{k x} \left( e^{k h} - 1 \right)}{h}\]

Step 4: Final Form of the Difference Quotient

The expression \( \frac{P e^{k x} \left( e^{k h} - 1 \right)}{h} \) is the difference quotient for the function \( f(x) = P e^{k x} \), not simplified for \( h \rightarrow 0 \) yet, as the main request was the difference quotient itself.

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in the form of \( f(x) = a \, b^{x} \), where \(a\) is a constant, \(b\) is the base of the exponential, and \(x\) is the exponent. Exponential functions involve variables in the exponent and are a key component in modeling many natural phenomena.These functions exhibit rapid growth or decay, making them perfect for describing processes in which change occurs multiplicatively. Due to their distinct curve shape, which starts slow and drastically increases or decreases, they differ from linear or polynomial functions. A specific type of exponential function, the natural exponential function, uses Euler's number \(e\) as the base.Euler's number, approximately equal to 2.718, emerges in calculus and real-world growth models, like population dynamics or radioactive decay. The general equation \( f(x) = Pe^{kx} \) is used extensively to model exponential growth, where \( P \) is the initial amount and \( k \) determines the rate of growth.

Derivative

In calculus, a derivative represents the rate of change of a function concerning its variable. It can be thought of as the slope of a function or curve at a particular point and is essential in understanding how functions behave.The fundamental expression for finding the derivative is the difference quotient:\[ \frac{f(x+h) - f(x)}{h} \]This expression calculates the average rate of change over an interval \( h \), which then evaluates the instantaneous rate of change as \( h \) approaches zero. It's the core method used to find derivatives in functions.For exponential functions, like \( f(x) = Pe^{kx} \), derivatives play a significant role in identifying growth rates. By utilizing the properties of the exponential function and differentiation rules, it's possible to compute complex behaviors in various contexts regularly.Understanding derivatives encompasses recognizing their rules, such as the power rule, product rule, and chain rule, all of which assist in simplifying complex derivative calculations.

Exponential Growth Model

The exponential growth model is a mathematical framework used to describe processes that double or increase at a constant rate over time. It is depicted by the formula \( P(t) = P_0 e^{kt} \), where \( P_0 \) is the starting quantity, \( e \) is Euler's number, \( k \) is the growth rate constant, and \( t \) is time.In this model:

• When \( k > 0 \), the function displays growth, indicating that the quantity is increasing.
• When \( k < 0 \), it showcases decay, which is how diminishing processes are modeled.

This model applies to a wide range of scenarios such as population growth, compound interest, and even biological processes like bacteria proliferation.
The beauty of the exponential growth model is its ability to illustrate consistent, proportional increases over time, offering clear insights through its mathematical simplicity. When visually graphed, exponential growth models create a curve that rapidly ascends, highlighting the dynamic nature of exponential growth compared to linear or other growth forms.