Problem 73 - Algebra and Trigonometry Real Mathematics, Real People

Question

Consider the functions given by $$f(x)=\tan \frac{\pi x}{2} \quad \text { and } \quad g(x)=\frac{1}{2} \sec \frac{\pi x}{2}$$ on the interval (-1,1), (a) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (b) Approximate the interval in which \(f(x)

Step-by-Step Solution

Answer

An approximation of the functions \(f(x)\) and \(g(x)\) leads to comparison of the functions, after which \(2f(x)\) and \( 2g(x)\) are compared. If the intervals in both comparisons match, it shows that multiplication of a function by a scalar doesn't affect the inequality. The intervals obtained in both comparisons are the solution to the exercise provided.

Step 1: Approximation of f(x) and g(x)

Given the domain (-1,1), \(f(x)= tan(\frac{\pi x}{2})\) and \(g(x)= \frac{1}{2} sec(\frac{\pi x}{2})\). firstly, calculate the value of these functions at different points within this domain.

Step 2: Comparison of the Functions

Compute \(f(x)

Step 3: Comparison of 2f(x) and 2g(x)

Compute \(2f(x)<2g(x)\) for x in the domain (-1,1). This will give the sets of x for which this inequality holds true. The results form the intervals where \(2f(x) < 2g(x)\).

Step 4: Comparison of Results from Steps 2 and 3

Compare the intervals obtained in the last two steps. If multiplication by 2 on both sides of the function does not affect the inequality, then it means that the scaling factor does not affect the validity of original inequality \(f(x)

Key Concepts

Tangent and Secant Functions

The tangent and secant functions are among the basic trigonometric functions used in various mathematical analyses. The function \( f(x) = \tan (\frac{\pi x}{2}) \) represents the tangent function, which is periodic and has vertical asymptotes where the function tends towards infinity. Specifically for \( f(x) \), these asymptotes occur as \( x \) approaches \( \pm 1 \).
On the other hand, \( g(x) = \frac{1}{2}\sec (\frac{\pi x}{2}) \) relates to the secant function which is the reciprocal of the cosine function. It also has vertical asymptotes where the cosine function equals zero. Within the interval \( (-1, 1) \), the secant function also approaches infinity at the endpoints.
Both functions exhibit similar behavior with respect to asymptotes in this interval because they amplify around angles where traditional trigonometric functions (cosine and sine) reach values of zero.

Inequalities

Mathematical inequalities describe how two expressions compare to each other. In this exercise, the inequality to evaluate is \( f(x) < g(x) \). Breaking it down, this inequality signifies that the values of the tangent function are smaller than the corresponding secant function values at specific points within the given domain (-1, 1).
To explore this, you compare the outputs of \( f(x) \) and \( g(x) \) across different approximation points. This process of checking each point helps determine where one function consistently lags behind the other across the interval, thus forming an interval range where the inequality holds true. Essentially, these comparison points help identify the crucial domains of differences between \( f(x) \) and \( g(x) \).

Interval Approximation

Interval approximation involves estimating where certain conditions are satisfied consistently across a range of values. For our task with \( f(x) < g(x) \) and later with \( 2f(x) < 2g(x) \), it requires evaluating these inequalities across the interval (-1, 1).
By defining points within this interval and calculating the respective function values, we identify sections where the inequalities hold true or false. This forms the basis for estimating the actual intervals, which can be more nuanced around points where the behavior of trigonometric functions dramatically changes, such as approaching vertical asymptotes.
It's important to recognize that despite scaling, such as multiplying by 2, the intervals where inequalities hold can remain unaffected, acting as a testament to the multiplicative properties of inequalities.

Graphing Utilities

Graphing utilities are essential tools that aid in visualizing functions and their behaviors over specific domains. In this exercise, graphing utilities help plot \( f(x) \) and \( g(x) \) together on a viewing window which offers immediate insight into how these functions compare.
By visualizing, we can see where \( f(x) \) lies below \( g(x) \), thus identifying intervals where the inequality \( f(x) < g(x) \) holds. Using a tool, those points of crossing or divergence where comparisons are clearer help demystify the function interactions.
Moreover, graphing utilities allow for dynamic checking with sliders or inputs to vary \( x \) values. This interaction can be extremely helpful in diagnostic approaches to understanding where inequalities are persistent, making interval approximations more precise and approachable.

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Key Concepts

Tangent and Secant Functions

The tangent and secant functions are among the basic trigonometric functions used in various mathematical analyses. The function \( f(x) = \tan (\frac{\pi x}{2}) \) represents the tangent function, which is periodic and has vertical asymptotes where the function tends towards infinity. Specifically for \( f(x) \), these asymptotes occur as \( x \) approaches \( \pm 1 \).
On the other hand, \( g(x) = \frac{1}{2}\sec (\frac{\pi x}{2}) \) relates to the secant function which is the reciprocal of the cosine function. It also has vertical asymptotes where the cosine function equals zero. Within the interval \( (-1, 1) \), the secant function also approaches infinity at the endpoints.
Both functions exhibit similar behavior with respect to asymptotes in this interval because they amplify around angles where traditional trigonometric functions (cosine and sine) reach values of zero.

Inequalities

Mathematical inequalities describe how two expressions compare to each other. In this exercise, the inequality to evaluate is \( f(x) < g(x) \). Breaking it down, this inequality signifies that the values of the tangent function are smaller than the corresponding secant function values at specific points within the given domain (-1, 1).
To explore this, you compare the outputs of \( f(x) \) and \( g(x) \) across different approximation points. This process of checking each point helps determine where one function consistently lags behind the other across the interval, thus forming an interval range where the inequality holds true. Essentially, these comparison points help identify the crucial domains of differences between \( f(x) \) and \( g(x) \).

Interval Approximation

Interval approximation involves estimating where certain conditions are satisfied consistently across a range of values. For our task with \( f(x) < g(x) \) and later with \( 2f(x) < 2g(x) \), it requires evaluating these inequalities across the interval (-1, 1).
By defining points within this interval and calculating the respective function values, we identify sections where the inequalities hold true or false. This forms the basis for estimating the actual intervals, which can be more nuanced around points where the behavior of trigonometric functions dramatically changes, such as approaching vertical asymptotes.
It's important to recognize that despite scaling, such as multiplying by 2, the intervals where inequalities hold can remain unaffected, acting as a testament to the multiplicative properties of inequalities.

Graphing Utilities

Graphing utilities are essential tools that aid in visualizing functions and their behaviors over specific domains. In this exercise, graphing utilities help plot \( f(x) \) and \( g(x) \) together on a viewing window which offers immediate insight into how these functions compare.
By visualizing, we can see where \( f(x) \) lies below \( g(x) \), thus identifying intervals where the inequality \( f(x) < g(x) \) holds. Using a tool, those points of crossing or divergence where comparisons are clearer help demystify the function interactions.
Moreover, graphing utilities allow for dynamic checking with sliders or inputs to vary \( x \) values. This interaction can be extremely helpful in diagnostic approaches to understanding where inequalities are persistent, making interval approximations more precise and approachable.