Problem 22 - Thomas Calculus

Question

Intersection of curves At what value(s) of \(x\) does \(\cos x=-x ?\)

Step-by-Step Solution

Answer

The curves intersect at approximately \( x = -0.739 \).

Step 1: Set the Equations Equal

To find the intersection, equate the equations given: \( \cos x = -x \). This represents where the two curves intersect.

Step 2: Graphical Analysis

Plot both curves, \( y = \cos x \) and \( y = -x \), on a coordinate plane. Observe where the two graphs intersect to estimate the values of \( x \) visually.

Step 3: Solve Numerically

Use a numerical method or calculator to find the intersections. Start with an approximate guess from the graph and refine it using numerical computation tools like the Newton-Raphson method. Calculators or computer software can be used to find \( x \approx -0.739 \) as an intersection point.

Step 4: Check the Intersection

Substitute \( x = -0.739 \) back into the equations to verify the intersection. Check that the values are equal (\( \cos(-0.739) = -(-0.739) \)).

Key Concepts

Graphical Analysis

Graphical analysis plays a crucial role when determining where two curves intersect. It's an intuitive way to visualize solutions. In this exercise, you're examining the intersection of the curves represented by the equations:

• \( y = \cos x \)
• \( y = -x \)

To start, graph both functions on the same coordinate plane. You'll notice that \( y = \cos x \) is a wave-like function that oscillates between -1 and 1, while \( y = -x \) is a straight line with a negative slope that extends infinitely in both directions.
Look closely at where these two might intersect. This visual approach allows you to estimate that the point of intersection is somewhere around \( x = -0.739 \). Graphical analysis provides a first step that can guide more precise methods later. It's an essential skill for understanding broader concepts in math and physics.

Numerical Methods

Once you've estimated the intersection point through graphical analysis, the next step often involves numerical methods. These methods are an excellent tool for getting a more precise solution. Numerical methods are used extensively when exact solutions are difficult to calculate analytically, which often happens with transcendental equations involving trigonometric functions like cosine.
Several numerical methods can be applied here, such as the bisection method or the Newton-Raphson method, and often involve the use of computer programs or calculators. In this problem, starting from our graphical estimate of \( x \), the value gets refined through iterative approximations. Different numerical techniques might provide slightly different approximations, but they generally converge to a more accurate solution, which, in this case, is approximately \( x = -0.739 \). This value satisfies the condition \( \cos x = -x \). Numerical methods not only increase accuracy but also provide insights into how equations behave.

Newton-Raphson Method

The Newton-Raphson method is a powerful numerical technique used for finding successively better approximations to the roots (or zeroes) of a real-valued function. This method uses the concept of linearization, leveraging the function's derivative to hone in on the root.
For our equation, \( \cos x = -x \) is equivalent to finding the root of \( f(x) = \cos x + x \). The Newton-Raphson formula is given by:

• \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)

In this case:

• \( f(x) = \cos x + x \)
• \( f'(x) = -\sin x + 1 \)

Starting with an initial guess close to the graphical estimate, this method involves computing successive iterations:

• Substitute the initial guess into the formula
• Calculate the next approximation

Repeat these steps until the value converges to a stable solution, in this problem approximately \( x = -0.739\). This iterative process is fundamental in finding precise intersection points between curves.