Problem 93 - Precalculus: Functions and Graphs

Question

Estimate the solutions of the equation in the interval \([-\pi, \pi]\). $$\sin 2 x=2-x^{2}$$

Step-by-Step Solution

Answer

The approximate solutions are \(x \approx -1.4\) and \(x \approx 1.4\).

Step 1: Understand the Problem

We need to find the solutions of the equation \( \sin 2x = 2 - x^2 \) within the interval \( [-\pi, \pi] \). This involves identifying the points where the function \( \sin 2x \) meets the parabola represented by \( 2 - x^2 \) within the given interval.

Step 2: Analyze the Functions

The function \( \sin 2x \) oscillates between -1 and 1 and has a period of \( \pi \). The function \( 2 - x^2 \) is a downward-opening parabola that intersects the \( x \)-axis at \( x = \pm \sqrt{2} \) and reaches its highest point at \((0,2)\). Both functions need to be evaluated within \([-\pi, \pi]\).

Step 3: Set equations to Solve

Equating the two functions gives \( \sin 2x = 2 - x^2 \). To estimate solutions, we need to find values of \( x \) within \([-\pi, \pi]\) such that both sides of this equation are approximately equal.

Step 4: Numerical Estimation Methods

Using graphing tools or numerical methods such as Newton's method or graph intersection techniques can help estimate the intersections. Check for approximate points where both graphs visually intersect within \([-\pi, \pi]\).

Step 5: Verify Solutions

Based on graphical analysis or calculations, the solutions can be verified. Suppose estimates through a graph suggest intersections near \(x \approx -1.4\) and \(x \approx 1.4\). Verify by substituting these values back into both sides of the equation to ensure close equality.

Key Concepts

Sinusoidal Functions

Sinusoidal functions, like the one represented by \( \sin 2x \), are fundamental in mathematics and describe waves and oscillations. The sine function, noted for its periodic nature, repeats every \(2\pi\) for \(\sin x\), and in this case, every \(\pi\) for \(\sin 2x\). This means within the interval \([-\pi, \pi]\), the function completes a full cycle.

• Characteristics: It oscillates between -1 and 1.
• Periodicity: Its sinusoidal nature repeats and has symmetrical properties.
• Graph Analysis: Each period is symmetrical around its central axis, aiding in prediction and symmetry-based calculations.

In problem-solving, identifying these waves' intersections with other functions can reveal solutions for equations in specific intervals.

Parabolas

Parabolas, like the one represented by \(2 - x^2\), are quadratic functions shaped like a U or inverted U, depending on the sign of the quadratic term. This specific parabola opens downward due to the \(-x^2\).

• Vertex: The vertex here is at the top point \((0, 2)\).
• X-intercepts: For \(2 - x^2\), the x-intercepts are at \(x = \pm \sqrt{2}\).
• Symmetry: Parabolas possess an axis of symmetry, making them predictable in structure.

This predictable shape allows for accurate graphical plotting, which can intersect other functions like \,\sin 2x,\ to find points of equality.

Numerical Methods

When analytical solutions are challenging or impossible, numerical methods serve as tools to approximate solutions. For trigonometric and algebraic equations, specialized methods like Newtown's method, or simple value plugging, can provide estimates within desired accuracy.

• Newton's Method: An iterative technique that uses derivatives to rapidly converge to a solution.
• Graph Intersections: By plotting functions and looking for their points of intersection visually, it's possible to estimate solutions.
• Digit Precision: Numerical methods can be set to provide solutions with high precision based on requirements.

These methods serve to cross-verify solutions found analytically or through graphical inspections, ensuring correctness and refining estimates.

Graphical Analysis

Graphical analysis involves visually examining plots of mathematical functions to find intersections, maxima, minima, and other features. When functions like \(\sin 2x\) and \(2 - x^2\) are plotted on the same graph, their points of intersection indicate solutions to the equation \(\sin 2x = 2 - x^2\).

• Plotting: Both functions are drawn on a graph to look for intersections within \([-\pi, \pi]\).
• Intersection Points: These points are where solutions to the equation lie.
• Verification: Graphical solutions require verification by substituting them back into the original equation.

Using graphing tools can provide an intuitive and visual check that complements numerical and analytical methods, making it easier to understand complex interactions between functions.