Problem 70 - Precalculus: Functions and Graphs

Question

Use a graph to solve the Inequality on the interval \([-\pi, \pi]\) \(\frac{1}{4} \tan \left(\frac{1}{3} x^{2}\right)<\frac{1}{2} \cos 2 x+\frac{1}{5} x^{2}\)

Step-by-Step Solution

Answer

Graph f(x) and g(x), find where f(x) < g(x) on [-π, π].

Step 1: Understand the Problem

We are tasked with solving the inequality \( \frac{1}{4} \tan \left( \frac{1}{3} x^{2}\right) < \frac{1}{2} \cos 2x + \frac{1}{5} x^{2} \) over the interval \([-\pi, \pi]\). This means we need to find all the values of \(x\) within this interval where the left-hand side is less than the right-hand side.

Step 2: Define Functions

Let \( f(x) = \frac{1}{4} \tan \left( \frac{1}{3} x^{2} \right) \) and \( g(x) = \frac{1}{2} \cos 2x + \frac{1}{5} x^{2} \). Our task is to examine where \( f(x) < g(x) \) on the graph.

Step 3: Draw the Graphs

Start by drawing the graph of \( f(x) \) and \( g(x) \) on the same coordinate plane within the interval \([-\pi, \pi]\). For \( f(x) \), plot the points considering the behavior of the \( \tan \) function, which has asymptotes and can take very large values. For \( g(x) \), notice it's a combination of a cosine function and a quadratic term, which generally gives a wavy pattern.

Step 4: Identify Intersection Points

Look for points where the graphs of \( f(x) \) and \( g(x) \) intersect, as these points could be potential boundaries for the inequality solution. Use methods like estimation or software tools to find intersections.

Step 5: Test Intervals

Once you have potential boundary intersection points, test intervals between these points to determine where \( f(x) < g(x) \). For each interval, choose a test point, substitute it into both functions, and check the inequality.

Step 6: Conclude the Interval

After testing, identify and confirm the complete range of intervals within \([-\pi, \pi]\) where \( f(x) < g(x) \). Conclude if there are single or multiple intervals.

Key Concepts

Trigonometric Inequalities

Trigonometric inequalities involve expressions that include trigonometric functions like sine, cosine, and tangent, compared to other expressions or constants. In this context, the inequality is given as \( \frac{1}{4} \tan \left( \frac{1}{3} x^{2} \right) < \frac{1}{2} \cos 2x + \frac{1}{5} x^{2} \).
To solve such inequalities, we typically compare the values of the two functions on either side of the inequality within a specified interval. This involves identifying where one function is less than or greater than the other graphically.
By using a graphical approach, we gain a visual understanding of how these functions behave over the interval and where they intersect. These intersections can provide potential solution boundary points, which then need to be tested further.
It's important to be aware of the behavior typical of the involved trigonometric function. For example, the tangent function has asymptotes, where it could reach very high or low values and convert smoothly between them. This can affect the intervals where the inequality holds true.

Function Graphing

Graphing functions is an essential step in solving inequalities graphically. In this exercise, we focus on two distinct functions: \( f(x) = \frac{1}{4} \tan \left( \frac{1}{3} x^{2} \right) \) and \( g(x) = \frac{1}{2} \cos 2x + \frac{1}{5} x^{2} \). By plotting these on the same coordinate plane, we can visually inspect their interaction over the interval \([-\pi, \pi]\).

• The function \( f(x) \) involves a tangent component, meaning it will have vertical asymptotes where the tangent function is undefined. Careful attention is required to avoid these asymptotes when evaluating graph sections.
• The function \( g(x) \) is more stable due to the combination of the cosine function and a quadratic term. It typically results in a wavy, yet smooth, curve that fluctuates in a predictable manner.

By drawing these graphs carefully, we can identify key points such as intersections, which are clues to where inequalities change direction. Thus, understanding how to graph these functions appropriately is critical in reaching an accurate solution.

Interval Notation

Interval notation is a concise way of describing sets of numbers, particularly those that represent the solution to inequalities. When we identify intervals on a graph where one function is less than another, we use interval notation to express these ranges compactly.
When documenting these solutions, brackets are used to represent intervals, with:

• Round brackets \((a, b)\) indicating that the endpoints \(a\) and \(b\) are not included in the interval.
• Square brackets \([a, b]\) indicating that the endpoints are included in the interval.

For the exercise, we examine intervals on the graph of \( f(x) \) and \( g(x) \) to identify where \( f(x) < g(x) \). Any interval where this holds true is part of our solution, using the proper notation to describe it clearly. This way, we ensure the solution is both precise and easy to understand, helpful for further interpretation or analysis.