Problem 85 - Essential Calculus

Question

Numerical, Graphical, and Analytic Analysis Consider the functions \(f(x)=x\) and \(g(x)=\tan x\) on the interval \((0, \pi / 2)\) (a) Complete the table and make a conjecture about which is the greater function on the interval \((0, \pi / 2)\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 0.25 & 0.5 & 0.75 & 1 & 1.25 & 1.5 \\ \hline f(x) & & & & & & \\ \hline g(x) & & & & & & \\ \hline \end{array} $$ (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 \((0, \pi / 2)\). (c) Prove that \(f(x)0,\) where \(h=g-f .\)

Step-by-Step Solution

Answer

For the interval \((0, \pi / 2)\), after numeric analysis, graphic analysis and analytic proves, conclude that \(f(x)

Step 1: Numeric Analysis

Calculate \(f(x)\) and \(g(x)\) for the given values of \(x\). For example, \(f(0.25)=0.25\) and \(g(0.25)=\tan(0.25)\). Fill all the values of \(f(x)\) and \(g(x)\) in the table.

Step 2: Graphic Analysis

Plot \(f(x)=x\) and \(g(x)=\tan x\) on a graphing utility for the interval \((0, \pi / 2)\). Notice how the graph of \(g(x)\) is above the graph of \(f(x)\) indicating \(f(x)

Step 3: Analytic Analysis

Prove \(f(x) f(x)\) in the interval \((0, \pi / 2)\). Calculate \(h^{\prime}(x)\) and show that it is positive to complete your proof.

Key Concepts

Graphical Analysis

When conducting a graphical analysis, one of the effective ways to understand how two functions behave in relation to each other is by plotting them on a graph. In this case, we are comparing the functions \(f(x) = x\) and \(g(x) = \tan x\) within the interval \( \(0, \pi / 2\)\). By graphing these two functions, we can visually examine which function is greater.The function \(f(x) = x\) is a straight line through the origin with a consistent increase as \(x\) increases. Meanwhile, \(g(x) = \tan x\) is a curve that starts at the origin but rapidly increases as \(x\) approaches \(\pi/2\).

• \(f(x) = x\) is a linear function.
• \(g(x) = \tan x\) increases exponentially as it nears \(\pi/2\).

This difference in growth rates means that on a graph, \(g(x)\) will sit above \(f(x)\), visually demonstrating that \(g(x) > f(x)\) on the interval \((0, \pi / 2)\). This powerful visual tool helps us confirm our numerical observations and make confident conjectures about the behavior of these functions.

Analytic Proof

An analytic proof uses mathematical reasoning to demonstrate the inequalities between functions. In our exercise, we need to show that \(f(x) = x\) is less than \(g(x) = \tan x\) for the interval \((0, \pi/2)\).To establish this inequality analytically, we define a new function \(h(x) = g(x) - f(x) = \tan x - x\). The task is to determine if \(h(x)\) is greater than zero for all \(x\) in the given interval. To do this, calculate the derivative of \(h(x)\): \[ h'(x) = \frac{d}{dx}(\tan x) - \frac{d}{dx}(x) = \sec^2 x - 1.\]Since \(\sec^2 x\) is always greater than 1 for \(xeq 0\) in the interval \((0, \pi / 2)\), we can see that \(h'(x) > 0\). A positive derivative indicates that \(h(x)\) is increasing, and hence \(\tan x > x\) throughout the interval. By showing this analytically, we demonstrate that \(f(x)\) is indeed less than \(g(x)\) over the specified interval, confirming our graphical analysis findings.

Functions Comparison

When comparing functions, it’s important to consider various criteria to fully understand how they interact and relate. For the functions \(f(x) = x\) and \(g(x) = \tan x\), we explore numerical calculation, graphical representation, and analytic proof.Numerically, evaluate these functions at different points within the interval \((0, \pi/2)\):

• \(f(0.25) = 0.25, g(0.25) = \tan(0.25)\), compare values.
• Continue for \(x = 0.5, 0.75, 1, 1.25, 1.5\).

Observed values will typically show that \(g(x)\) is greater than \(f(x)\).Graphically, the curve for \(\tan x\) rises steeply compared to the linear increase of \(x\), reinforcing that \(g(x)\) surpasses \(f(x)\).Analytically, we proved \(\tan x > x\) through the positive derivative \(h'(x) = \sec^2 x - 1\). This method confirms supremity across the interval.Through these methods, understanding becomes clearer. Each approach corroborates the others, forming a comprehensive picture of how these functions behave compared to one another.

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Key Concepts

Graphical Analysis

When conducting a graphical analysis, one of the effective ways to understand how two functions behave in relation to each other is by plotting them on a graph. In this case, we are comparing the functions \(f(x) = x\) and \(g(x) = \tan x\) within the interval \( \(0, \pi / 2\)\). By graphing these two functions, we can visually examine which function is greater.The function \(f(x) = x\) is a straight line through the origin with a consistent increase as \(x\) increases. Meanwhile, \(g(x) = \tan x\) is a curve that starts at the origin but rapidly increases as \(x\) approaches \(\pi/2\).

• \(f(x) = x\) is a linear function.
• \(g(x) = \tan x\) increases exponentially as it nears \(\pi/2\).

This difference in growth rates means that on a graph, \(g(x)\) will sit above \(f(x)\), visually demonstrating that \(g(x) > f(x)\) on the interval \((0, \pi / 2)\). This powerful visual tool helps us confirm our numerical observations and make confident conjectures about the behavior of these functions.

Analytic Proof

An analytic proof uses mathematical reasoning to demonstrate the inequalities between functions. In our exercise, we need to show that \(f(x) = x\) is less than \(g(x) = \tan x\) for the interval \((0, \pi/2)\).To establish this inequality analytically, we define a new function \(h(x) = g(x) - f(x) = \tan x - x\). The task is to determine if \(h(x)\) is greater than zero for all \(x\) in the given interval. To do this, calculate the derivative of \(h(x)\): \[ h'(x) = \frac{d}{dx}(\tan x) - \frac{d}{dx}(x) = \sec^2 x - 1.\]Since \(\sec^2 x\) is always greater than 1 for \(xeq 0\) in the interval \((0, \pi / 2)\), we can see that \(h'(x) > 0\). A positive derivative indicates that \(h(x)\) is increasing, and hence \(\tan x > x\) throughout the interval. By showing this analytically, we demonstrate that \(f(x)\) is indeed less than \(g(x)\) over the specified interval, confirming our graphical analysis findings.

Functions Comparison

When comparing functions, it’s important to consider various criteria to fully understand how they interact and relate. For the functions \(f(x) = x\) and \(g(x) = \tan x\), we explore numerical calculation, graphical representation, and analytic proof.Numerically, evaluate these functions at different points within the interval \((0, \pi/2)\):

• \(f(0.25) = 0.25, g(0.25) = \tan(0.25)\), compare values.
• Continue for \(x = 0.5, 0.75, 1, 1.25, 1.5\).

Observed values will typically show that \(g(x)\) is greater than \(f(x)\).Graphically, the curve for \(\tan x\) rises steeply compared to the linear increase of \(x\), reinforcing that \(g(x)\) surpasses \(f(x)\).Analytically, we proved \(\tan x > x\) through the positive derivative \(h'(x) = \sec^2 x - 1\). This method confirms supremity across the interval.Through these methods, understanding becomes clearer. Each approach corroborates the others, forming a comprehensive picture of how these functions behave compared to one another.