Problem 35 - Intermediate Algebra

Question

For Problems \(33-38\), graph each of the exponential functions. $$ f(x)=2 e^{x} $$

Step-by-Step Solution

Answer

Graph is an exponential curve with a y-intercept at 2, increasing as x increases.

Step 1: Understand the Function Form

The given function is an exponential function of the form \( f(x) = 2e^x \). Here, \( e \) is the base of the natural logarithm, approximately equal to 2.718. The function suggests an exponential growth because the exponent, \( x \), increases the function’s value rapidly as \( x \) increases.

Step 2: Identify Key Characteristics

For \( f(x) = 2e^x \), there are several key features:- The y-intercept occurs at \( x = 0 \), which gives \( f(0) = 2e^0 = 2 \).- The function tends towards zero as \( x \) tends to negative infinity (\( x \to -\infty \)).- The function increases (grows rapidly) as \( x \) becomes positive.

Step 3: Plot Points

Calculate specific values:- \( f(-1) = 2e^{-1} \approx 2 \times 0.368 = 0.736 \).- \( f(0) = 2e^0 = 2 \).- \( f(1) = 2e^1 \approx 2 \times 2.718 = 5.436 \). - \( f(2) = 2e^2 \approx 2 \times 7.389 = 14.778 \).

Step 4: Sketch the Graph

Using the points calculated from Step 3, plot these on the Cartesian plane:1. Point (-1, 0.736).2. Point (0, 2) - the y-intercept.3. Point (1, 5.436).4. Point (2, 14.778).Connect these points smoothly and extend the curve to indicate sharp upward growth as \( x \) increases, and a flat trend towards zero as \( x \) decreases past negative values.

Step 5: Analyze the Graph

The graph represents an exponential function, which should have a smooth curve upwards to the right and approaching the x-axis (zero) on the left. This confirms the behavior described in Step 2.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions like \( f(x) = 2e^x \) allows us to visualize how these functions behave over different values of \( x \). A key characteristic of exponential functions is their consistent growth or decay rate.

To graph \( f(x) = 2e^x \), we follow these steps:

• Identify the y-intercept. For \( f(x) \), the y-intercept is \( f(0) = 2(e^0) = 2 \).
• Plot several points using calculated values, such as \( f(-1) \), \( f(0) \), \( f(1) \), and \( f(2) \). These show us where the graph starts and how it grows.
• The graph should extend smoothly, curving upwards as \( x \) increases, indicating exponential growth.

Exponential graphs have an important feature: they show rapid increases, making them perfect for modeling growth over time, like population increases or investments. The graph declines smoothly as \( x \) moves towards negative infinity, showing a slow approach to zero.

Exponential Growth

Exponential growth occurs when a function’s value increases at a constant rate relative to its current value. In the function \( f(x) = 2e^x \), the base \( e \) (approximately 2.718) ensures this growth.

Characteristics of exponential growth:

• A small increase in \( x \) results in a large change in \( f(x) \), illustrating rapid growth.
• The factor by which \( f(x) \) grows is controlled by \( e \); hence, \( e^x \) grows uniquely because of this constant base.
• The y-intercept provides a starting point; here it's 2, which scales the exponential function linearly.

Exponentials are evident in natural phenomena where quantities double or triple over specific intervals, like money compounding in a bank or cells dividing in biology. These real-life applications make understanding exponential growth critical to various fields.

Natural Logarithm

The natural logarithm (ln) connects deeply to exponential functions due to its unique base, \( e \). It is essentially the inverse of raising "e" to a power.

Important points:

• \( \ln(e^x) = x \), meaning the natural logarithm measures how many times you multiply \( e \) to get a value.
• The natural logarithm is useful for "undoing" the exponential function, returning to the original exponent before growth was applied.
• In calculus, the derivative of \( e^x \) is \( e^x \), while the derivative of \( \ln(x) \) is \( \frac{1}{x} \).

Understanding natural logs helps in solving exponential equations and analyzing exponential growth. They simplify complex calculations involving exponential functions, especially in continuous growth models or when deciphering growth rates.