Problem 111 - Calculus and its applications

Question

Graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) $$ f(x)=2 e^{0.3 x} $$

Step-by-Step Solution

Answer

Graph the exponential functions: \( f(x) = 2e^{0.3x} \), \( f'(x) = 0.6e^{0.3x} \), \( f''(x) = 0.18e^{0.3x} \).

Step 1: Original Function

The original function given is \( f(x) = 2e^{0.3x} \). This is an exponential function with base \( e \) and a growth rate of \( 0.3 \). The entire function is scaled by a factor of 2.

Step 2: Calculate the First Derivative

To find the first derivative \( f'(x) \), apply the derivative rule for exponential functions. The derivative of \( e^{ax} \) with respect to \( x \) is \( ae^{ax} \). Therefore, \( f'(x) = 0.3 \cdot 2e^{0.3x} = 0.6e^{0.3x} \).

Step 3: Calculate the Second Derivative

To find the second derivative \( f''(x) \), differentiate \( f'(x) = 0.6e^{0.3x} \) again. Applying the same rule, \( f''(x) = 0.3 \cdot 0.6e^{0.3x} = 0.18e^{0.3x} \).

Step 4: Sketch the Graphs

To graph these functions, note that for all three — \( f(x) = 2e^{0.3x} \), \( f'(x) = 0.6e^{0.3x} \), and \( f''(x) = 0.18e^{0.3x} \) — they are all increasing exponential functions. They have the same base with different coefficients, which affects their steepness. \( f(x) \) will be the steepest, followed by \( f'(x) \) and \( f''(x) \). Plot these on a coordinate plane with \( x \) on the horizontal axis and the functions on the vertical axis.

Step 5: Interpretation of Graphs

The graphs will show that \( f(x) \) is the fastest-growing, \( f'(x) \) indicates the rate of growth of \( f(x) \), and \( f''(x) \) indicates the acceleration of \( f(x) \). As all are exponential functions and have the same exponential base, their shapes will be similar, just scaled by their coefficients.

Key Concepts

Derivative of Exponential Function

The derivative of an exponential function like \( f(x) = 2e^{0.3x} \) involves applying rules specific to exponential forms. When you determine the derivative of \( e^{ax} \) with respect to \( x \), it results in \( ae^{ax} \). This rule is crucial because it keeps the structure of the exponential function while adding a new coefficient. For our function, this leads to the first derivative, \( f'(x) = 0.6e^{0.3x} \). The coefficient 0.6 here comes from multiplying the original coefficient 2 by the growth rate 0.3.

Finding a second derivative involves differentiating once more, using the same rule, resulting in \( f''(x) = 0.18e^{0.3x} \). This reveals exponential functions have derivatives that are also exponential, maintaining the essential form of \( e^{0.3x} \). Each step down in differentiation introduces a smaller scaling coefficient, delineated by repeated multiplication of 0.3, illustrating how derivatives scale with the same factor.

Graphing Derivatives

Plotting derivatives involves observing how each successive derivative changes the graph's steepness. When graphing \( f(x) = 2e^{0.3x} \), the function grows significantly, reflecting its rapid increase due to the coefficients and base \( e \). The first derivative \( f'(x) = 0.6e^{0.3x} \) indicates the rate at which our original function grows, appearing less steep than \( f(x) \) since the scaling coefficient is smaller.

The second derivative \( f''(x) = 0.18e^{0.3x} \), with its further reduced coefficient, manifests a gentler slope, representing the acceleration of the original function’s growth, rather than growth itself. When graphed together:

• \( f(x) \) shows as the steepest, illustrating direct exponential growth.
• \( f'(x) \) appears a middle gradient, displaying the rate of change of \( f(x) \).
• \( f''(x) \) is the least steep, detailing how the rate of change itself is accelerating.

This visual comparison provides a clear representation of how derivatives of exponential functions relate to the change and rate of change in the initial function.

Understanding Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, leading to rapid escalation over time. In this exercise, \( f(x) = 2e^{0.3x} \), we witness a classic form of exponential growth.

• The base \( e \), approximately 2.718, is a constant that naturally manifests in continuously growing processes.
• The growth rate 0.3, a positive constant, ensures increasing values for our function as \( x \) progresses.

This setup leads to the exponential functions' characteristic sharp upward curve. Such behavior is noticeable in real-world scenarios like population growth, where the greater the population size, the faster it grows.

By examining this function and its derivatives, it's clear that exponential functions grow subtly at first, but swiftly pick up pace, illustrating why understanding their derivatives is crucial. It aids in predicting where and how fast such growth will occur, and in applications ranging from finance to biology, this comprehension becomes invaluable, allowing informed decision-making.