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Chapter 4: Problem 17

When solving. You may need to use appropriate technology (such as a calculator or a computer). Intersecting curves The curve \(y=\tan x\) crosses the line \(y=2 x\) between \(x=0\) and \(x=\pi / 2 .\) Use Newton's method to find where.

Short Answer

Expert verified
The intersection occurs at approximately \( x = 0.76 \).

Step by step solution

01

Set up the Equation

To find the intersection of the curves, equate the two equations, \ \( y = \tan x = 2x \). \ Re-arrange this to form \ \( f(x) = \tan x - 2x \). \ We want to find the root of the equation \( f(x) = 0 \).
02

Compute the Derivative

For applying Newton's Method, we need the derivative of \( f(x) \). \ \( f'(x) = \frac{d}{dx}[\tan x] - \frac{d}{dx}[2x] = \sec^2 x - 2 \).
03

Choose an Initial Guess

Select an initial guess \( x_0 \) between 0 and \( \frac{\pi}{2} \). \A reasonable initial guess could be \( x_0 = 0.5 \) as it is halfway through the interval.
04

Apply Newton's Method

Newton鈥檚 method formula is \ \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). \ Calculate \( x_1 \) using \ \( x_1 = 0.5 - \frac{\tan(0.5) - 2 \times 0.5}{\sec^2(0.5) - 2} \). \ Continue iterating using this formula until the change in \( x \) values is negligible.
05

Iterate Until Convergence

Continue with Newton's iterations to refine the estimate of \( x \) until the values stabilize. \Monitor each iteration's result to ensure it falls below a small tolerance level, say \( 10^{-5} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Method
Newton's Method is a powerful technique in numerical methods used to find successively better approximations to the roots (or zeroes) of a real-valued function. It is particularly useful when the function is complex and other algebraic methods are cumbersome or impossible. To understand Newton's Method:
  • Start with an initial guess for the root based on prior knowledge or analysis of the graph.
  • Utilize the formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). This means you take the current guess \( x_n \) and subtract the function value difference divided by its derivative at that point.
  • Iterate this process. Calculate the new \( x_{n+1} \) and use it as the next \( x_n \), repeating until the change between \( x_n \) and \( x_{n+1} \) is negligible.
Newton's Method requires the function \( f(x) \), its derivative \( f'(x) \), and an initial guess to start with. The method converges quickly in many cases, making it very effective if the initial guess is reasonable and the function behaves well around the root.
Differentiation
Differentiation is a fundamental concept of calculus that involves finding the derivative of a function. The derivative represents how a function changes as its input changes. In the context of Newton's Method:
  • The derivative \( f'(x) \) is crucial because it provides the slope of the tangent to the curve at point \( x_n \).
  • To differentiate a function like \( \tan x - 2x \), compute the derivative for each term separately.
  • For \( \tan x \), the derivative is \( \sec^2 x \), and for \( 2x \), it is simply \( 2 \).
  • Thus, for our example, the derivative \( f'(x) = \sec^2 x - 2 \).
Differentiation allows us to determine the behavior of the function, which is critical for Newton's Method because it aids in finding the slope at which we approach the root. This slope informs how quickly or slowly we approach the root iteration by iteration.
Tangent Function
The tangent function, \( \tan x \), is one of the trigonometric functions, and it is periodic with a period \( \pi \). It is defined as \( \tan x = \frac{\sin x}{\cos x} \). When the tangent of an angle within a right triangle is considered, it represents the ratio of the opposite side to the adjacent side.
  • The graph of \( \tan x \) has vertical asymptotes where \( x = \frac{\pi}{2} + k\pi \), for integers \( k \), because \( \cos x = 0 \) at these points.
  • Understanding the behavior of \( \tan x \) and its asymptotes is essential, especially since intersections involving \( \tan x \) can be influenced by these discontinuities.
  • The derivative of \( \tan x \), which is \( \sec^2 x \), gives the slope of the tangent line to the curve at any point \( x \).
In problems like finding the intersection with other functions, considering the characteristic properties of \( \tan x \) helps in setting the correct expectations and choosing a valid initial guess for numerical methods such as Newton's.
Intersection of Curves
Finding the intersection of curves is a common problem in mathematics, involving solving equations where two functions equate. For the functions \( y = \tan x \) and \( y = 2x \):
  • Set the equations equal: \( \tan x = 2x \).
  • This equality signifies that the points where the two curves meet fulfill both functional forms.
  • Transform this into an equation \( f(x) = \tan x - 2x \), where we aim to find \( f(x) = 0 \).
  • Using Newton's Method, iteratively solve for \( x \) where \( f(x) \) approaches zero, ensuring convergence to a point where the tangent and line intersect.
The intersection analysis helps in visualizing solutions graphically and confirms that analysis with numerical methods like Newton's can yield precise results when traditional algebraic solutions are difficult to apply. This method provides practical value across fields requiring precise intersection data, such as physics and engineering.

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Most popular questions from this chapter

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(2 e^{x}-3 e^{-2 x}\right) d x$$

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{\sec ^{3} \theta}{3}+C\) b. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \tan ^{2} \theta+C\) c. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \sec ^{2} \theta+C\)

Verify the formulas by differentiation. $$\int \frac{1}{(x+1)^{2}} d x=-\frac{1}{x+1}+C$$

a. Find a curve \(y=f(x)\) with the following properties: i) \(\frac{d^{2} y}{d x^{2}}=6 x\) ii) Its graph passes through the point (0,1) and has a horizontal tangent line there. b. How many curves like this are there? How do you know?

Solve the initial value problems. $$\frac{d y}{d x}=\frac{1}{x^{2}}+x, \quad x>0 ; \quad y(2)=1$$
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