Problem 78 - Calculus of a Single Variable

Question

Numerical, Graphical, and Analytic Analysis Consider the functions \(f(x)=x\) and \(g(x)=\sin x\) on the interval \((0, \pi) .\) (a) Complete the table and make a conjecture about which is the greater function on the interval \((0, \pi) .\) $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {0.5} & {1} & {1.5} & {2} & {2.5} & {3} \\\ \hline f(x) & {} & {} & {} \\ \hline g(x) & {} & {} & {} \\\ \hline\end{array} $$ (b) Use a graphing utility to graph the functions and use the graphs to make a conjecture about which is the greater function on the interval \((0, \pi) .\) (c) Prove that \(f(x)>g(x)\) on the interval \((0, \pi) .[\text { Hint: Show }\) that \(h^{\prime}(x)>0,\) where \(h=f-g .1\)

Step-by-Step Solution

Answer

Upon numerical analysis, we observe that \(f(x) > g(x)\) for all given x-values in the table. The graphs of the two functions confirm this observation. The function \(x\) lies above \(\sin(x)\) in the interval \((0, \pi)\). Analytically, deriving \(h(x)\) and evaluating the derivative shows that \(h'(x) > 0\) proving that \(f(x) > g(x)\) on the specified interval.

Step 1: Numerical Analysis - Completing the Table

Evaluate both functions \(f(x)\) and \(g(x)\) at each given \(x\)-value in the interval \((0, \pi)\), which in our case, is {\{0.5, 1, 1.5, 2, 2.5, 3\}}. Fill in the respective values into the table.

Step 2: Graphical Analysis - Graphing the Functions

Utilize a graphing software or tool to graph both functions \(f(x)\) and \(g(x)\) over the interval \((0, \pi)\). Compare the graphs to make a logical judgement about which function is greater on the given interval.

Step 3: Analytic Analysis - Using Derivatives

Define a new function \(h(x) = f(x) - g(x) = x - \sin(x)\). Compute the derivative \(h'(x)\) and analyze its sign. If \(h'(x) \) is positive on the interval \((0, \pi)\), it would mean that \(h(x)\) is increasing which implies that \(f(x) > g(x)\) on this interval.

Key Concepts

Numerical Analysis

In calculus, numerical analysis involves evaluating functions at specific points to gather insight into their behavior over an interval. For the functions \(f(x)=x\) and \(g(x)=\sin x\), numerical analysis allows us to compare these functions by filling in a table with values calculated at certain points in the interval \((0, \pi)\).

The process involves determining \(f(x)\) and \(g(x)\) for each given \(x\)-value, such as 0.5, 1, 1.5, 2, 2.5, and 3. This will help establish a trend and visualize which function yields larger values at these points. For example, for \(f(x)=0.5\), and calculating \(\sin(0.5)\) gives us approximately 0.479. Continuing this for all \(x\) values provides a concrete comparison between the two functions.

• For \(x = 0.5\), \(f(x) = 0.5\) and \(g(x) \approx 0.479\).
• For \(x = 1\), \(f(x) = 1\) and \(g(x) \approx 0.841\).
• Continue similarly for other points.

Filling out this table and observing the results allow us to make a conjecture about which function, \(f(x)\) or \(g(x)\), is greater over the interval.

Graphical Analysis

Graphical analysis provides a visual means to compare functions on an interval, helping us confirm numerical findings or find new insights. By plotting the functions \(f(x)=x\) and \(g(x)=\sin x\) on the interval \((0, \pi)\), you can visually assess the behavior of each function as \(x\) changes.

Using graphing utilities like graphing calculators or software, plot both functions. The graph of \(f(x)=x\) will appear as a straight line starting from the origin, increasing steadily. Meanwhile, \(g(x)=\sin x\) will form a curve starting at 0, reaching a peak near \(\pi/2\), and then returning towards 0 by \(\pi\).

• As you observe, \(g(x)\) initially is below \(f(x)\), surpasses it slightly, and then falls back below.
• The key takeaway is identifying regions where one function exceeds the other visually.

This visual comparison allows you to make informed conjectures about the relative size of \(f(x)\) and \(g(x)\) without relying solely on numerical calculations.

Analytic Analysis

Analytic analysis involves using calculus tools, such as derivatives, to rigorously prove assertions about functions. In this exercise, to prove that \(f(x) > g(x)\) on the interval \((0, \pi)\), we define a new function \(h(x) = f(x) - g(x) = x - \sin x\).

To show \(f(x) > g(x)\), we need \(h(x) > 0\). Checking if \(h(x)\) is increasing can help. Therefore, we compute the derivative \(h'(x)\):

• \(h'(x) = \frac{d}{dx}(x - \sin x) = 1 - \cos x\).

Since \(1 - \cos x > 0\) for \(x \in (0, \pi)\), \(h(x)\) is increasing over this interval. This increasing nature implies that \(f(x) = x\) exceeds \(g(x) = \sin x\) as \(x\) progresses through the interval.
Hence, analytic analysis using derivatives provides conclusive mathematical proof of the conjecture.

Functions

Functions are a cornerstone of calculus, representing relationships where each input, or \(x\)-value, corresponds to exactly one output, or \(y\)-value. In this exercise, we are dealing with two functions: \(f(x) = x\) and \(g(x) = \sin x\).

The function \(f(x) = x\) is a linear function, depicting a direct proportionality. As you increase \(x\), \(f(x)\) increases linearly without bounds. It’s straightforward, making it easy to predict and graph.

• Visualize it as a straight line through the origin.

Conversely, \(g(x) = \sin x\) is a trigonometric function exhibiting periodic behavior. Within its defined interval \((0, \pi)\), it starts at zero, peaks at one, and returns to zero.

• Sinusoidal curves like \(g(x)\) display a wave-like pattern.

Understanding these fundamental characteristics of functions helps explain the different paths they follow across an interval and aids in their comparison.

Derivatives

Derivatives in calculus measure the rate at which a function changes as its input changes. They are crucial for analyzing function behavior, such as finding where one function surpasses another.

In the current context, we use the derivative to explore the relationship between \(f(x)=x\) and \(g(x)=\sin x\). By examining the derivative of the difference \(h(x) = f(x) - g(x)\), denoted as \(h'(x)\), we assess function behavior.

• \(h'(x) = 1 - \cos x\) tells us how \(h(x)\) changes over \((0, \pi)\).

Positive \(h'(x)\) indicates that \(h(x)\) is increasing. Consequently, it implies that the gap between \(f(x)\) and \(g(x)\) widens as \(x\) increases in the specified interval.
Derivatives provide powerful insights into the progression and relationships within mathematical functions, offering clear evidence for reasoned conclusions.