Problem 62 - Essential Calculus

Question

Graphical Analysis In Exercises 61 and 62, graph \(f(x) / g(x)\) and \(f^{\prime}(x) / g^{\prime}(x)\) near \(x=0 .\) What do you notice about these ratios as \(x \rightarrow 0\) ? How does this illustrate L'Hôpital's Rule? \(f(x)=e^{3 x}-1, \quad g(x)=x\)

Step-by-Step Solution

Answer

The step by step analysis revealed that as \(x\) approaches 0, both \(f(x)/g(x)\) and \(f'(x)/g'(x)\) approach a similar value. This behavior illustrates L'Hopital's rule, as the ratio of the functions and the ratio of their derivatives have the same limit as \(x\) approaches 0.

Step 1: Calculating the derivate of each function

The first derivatives of \(f(x)\) and \(g(x)\) are required in order to evaluate \(f'(x) / g'(x)\). Here, we apply derivative rules to compute them : \[f'(x) = 3e^{3x}\] and \[g'(x) = 1\]

Step 2: Graphing the functions

Using the original function and the derivatives, plot the following ratios near \(x=0\): \[f(x)/g(x)\] and \[f'(x)/g'(x)\]. You can use a graphing calculator or online tool to do this

Step 3: Observations about the behavior as \(x \rightarrow 0\)

From your graph, note the behavior of the ratios as \(x \rightarrow 0\). Your graph should show that both ratios approach the same value as \(x \rightarrow 0\).

Step 4: Application of L'Hôpital's Rule

L'Hôpital's rule states that the limit as \(x \rightarrow c\) of \((f(x)-f(c))/(g(x)-g(c))\) is equal to \((f'(x)-f'(c))/(g'(x)-g'(c))\) if the limit exists. In this case, as \(x \rightarrow 0\), the ratios \(f(x)/g(x)\) and \(f'(x)/g'(x)\) approach the same value, demonstrating L'Hôpital's rule.

Key Concepts

Graphical Analysis

Graphical analysis can be a great tool to visualize complex mathematical concepts such as ratios, derivatives, and limits. To better understand how the ratios of functions behave, it's useful to plot them graphically. When you graph the functions \(f(x) = e^{3x} - 1\) and \(g(x) = x\), you'll be interested in both the function itself and the behavior of its derivative.

By graphing \(f(x)/g(x)\) and \(f'(x)/g'(x)\) near \(x=0\), you can observe how these functions behave as \(x\) approaches zero. Typically, when graphing such functions, you'll find that they could initially seem to have undefined values but may reveal patterns upon closer inspection. As you examine the graph, note if both ratios head towards a common point. This act of observation lays the groundwork for applying L'Hôpital's Rule, which is commonly used to resolve indeterminate forms in limits.

Visual tools can efficiently convey how values change, precisely why graphical analysis is a powerful technique in understanding these concepts.

Derivatives

The derivative of a function represents the rate at which a function is changing at any point and is fundamental in understanding the behavior of ratios near specific points. For the functions \(f(x) = e^{3x} - 1\) and \(g(x) = x\), calculating their derivatives allows us to examine the rate at which each function changes as \(x\) nears zero. The derivative of \(f(x)\) is \(f'(x) = 3e^{3x}\), and for \(g(x)\), it is \(g'(x) = 1\).

Understanding the derivatives helps in graphing \(f'(x)/g'(x)\) alongside \(f(x)/g(x)\). This comparison gives insight into how closely these functions track with one another, especially as \(x\) gets closer to zero. By analyzing the derivatives, you can predict and confirm the behavior of the original functions' ratio as \(x\) approaches a limit. This is especially useful when dealing with indeterminate forms, such as when both the numerator and denominator approach zero. Derivatives provide us the mathematical machinery to apply L'Hôpital's Rule effectively.

Limits

Limits are a fundamental concept in calculus, helping us understand the behavior of functions as they approach specific points. A 'limit' describes the value a function approaches as the input approaches some value. In the context of our exercise, we are interested in the behavior of \(f(x)/g(x)\) as \(x\rightarrow0\).

Limits are particularly useful when dealing with indeterminate forms, such as \(0/0\), which is the case with \(f(x)=e^{3x}-1\) and \(g(x)=x\) as \(x\) approaches zero. L'Hôpital's Rule is a powerful tool that enables us to find these elusive limits. By taking the derivatives of both the numerator and the denominator, L'Hôpital's Rule allows the limit to be evaluated using derivatives:

• If \(\lim_{{x \to c}} f(x)/g(x)\) is indeterminate, then \(\lim_{{x \to c}} f'(x)/g'(x)\) provides a possible solution.

Utilizing limits in this way helps in simplifying the deduction of the behavior of a function around points where it might otherwise be unclear.