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Chapter 6: Problem 14

Repeat Prob. 6.13 except for \\[ \begin{array}{l} y=x^{2}+1 \\ y=2 \cos x \end{array} \\]

Short Answer

Expert verified
To find the points of intersection between the functions \(y = x^2 + 1\) and \(y = 2\cos x\), we first set the functions equal to each other and solve the resulting equation \(x^2 + 1 = 2\cos x\) for \(x\) using a numerical method, such as the Newton-Raphson method. After finding the values of \(x\) that satisfy the equation, we substitute those values back into either function to obtain the corresponding \(y\) values. The points of intersection are then represented as coordinates \((x_i, y_i)\).

Step by step solution

01

Set the two functions equal to each other

Since we want to find the points where the functions have the same \(y\) value, we can set the two functions equal to each other: \(x^2 + 1 = 2\cos x\)
02

Solve the equation for x

This step probably requires some numerical approximation methods since the equation involves both a quadratic and a trigonometric function. We can use a method such as the Newton-Raphson method, the bisection method, or any other suitable numerical method to solve for the values of \(x\). These methods typically require a starting point or an interval in which the solution lies, followed by iterating the method until the approximate solution converges. For this explanation, we will use the Newton-Raphson method. We first need to find the derivative of the function \(f(x) = x^2 + 1 - 2\cos x\), which is \(f'(x) = 2x + 2\sin x\). The Newton-Raphson formula is given by: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)
03

Apply the Newton-Raphson method

Choose a starting point, say \(x_0 = 0\), and calculate the successive values of \(x\) using the formula until the absolute difference between successive \(x\) values is smaller than a predetermined tolerance, say \(10^{-6}\). Keep iterating the formula to find other values of \(x\) that satisfy the equation. Note that there might be multiple points of intersection, depending on the graphs of the functions.
04

Calculate the corresponding y values

Once the values of \(x\) have been found, we can substitute them back into either of the original functions to find the corresponding \(y\) values. Remember that since both functions are equal at these points of intersection, it does not matter which function we use to calculate the \(y\) values. For example, if one of the found \(x\) values is \(x_1\), we can find the corresponding \(y\) value by substituting into either function: \(y_1 = x_1^2 + 1\) or \(y_1 = 2\cos x_1\)
05

Write down the points of intersection

Finally, we can list the points of intersection that we found. We represent the points as coordinates \((x_i, y_i)\), where \(x_i\) is each value of \(x\) found, and \(y_i\) is the corresponding value of \(y\). Now we have successfully solved the problem and found the points where the functions \(y = x^2 + 1\) and \(y = 2\cos x\) intersect.

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Key Concepts

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

Newton-Raphson method
The Newton-Raphson method is a powerful numerical technique used to find successively better approximations to the roots of a real-valued function. It's especially handy when dealing with complex equations that cannot be easily solved algebraically. The method exploits the idea of linear approximation.

Here's how it works:
  • Start with an initial guess, called \(x_0\), which should be as close as possible to the actual root.
  • Use the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) to compute successive approximations.
  • Continue iterating until the absolute value of the difference between successive approximations is less than a predetermined tolerance, such as \(10^{-6}\).
The precision of the Newton-Raphson method depends significantly on the choice of the initial guess. If good, convergence is very rapid. However, if the initial guess is far from the actual root or the function is not well-behaved, the method might fail to converge.
Bisection method
The bisection method is a very simple and robust method for finding roots of a function. Unlike the Newton-Raphson method, it does not require calculus, which makes it quite user-friendly for equations where the derivative is difficult to compute.

Here鈥檚 a simple breakdown of its operation:
  • Identify an interval \([a, b]\) where the function changes sign, i.e., where \(f(a)\cdot f(b) < 0\). This indicates there is likely a root within this interval.
  • Find the midpoint of the interval, \(c = \frac{a + b}{2}\).
  • Check the function value at the midpoint: if \(f(c) = 0\), you have found the root; if not, determine the half interval where the sign change occurs and repeat.
  • Continue this process until the interval is sufficiently small, which gives you an approximate root.
The beauty of the bisection method lies in its certainty for convergence, no matter how the function behaves within the interval, as long as the function is continuous and the initial interval is chosen correctly.
Trigonometric functions
Trigonometric functions, such as cosine, sine, and tangent, are fundamental in mathematics. They describe the relations between the angles and sides of triangles, and they are also periodic functions widely used in many applications in science and engineering.

For these functions:
  • Cosine (\(\cos x\)): Describes the x-coordinate of a point on the unit circle as a function of the angle from the positive x-axis. It's even and periodic with a period of \(2\pi\).
  • Sine (\(\sin x\)): Describes the y-coordinate of a point on the unit circle. It's odd and also has a period of \(2\pi\).
  • Tangent (\(\tan x\)): Defined as the ratio of sine to cosine. It has a period of \(\pi\) and is undefined where \(\cos x = 0\).
These functions allow for modeling wave-like behaviors, circular motion, and various oscillations. In the context of the given problem, the interest focuses mostly on how the cosine function interacts with algebraic functions.

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Most popular questions from this chapter

(a) Apply the Newton-Raphson method to the function \(f(x)=\tanh \left(x^{2}-9\right)\) to evaluate its known real root at \(x=3 .\) Use an initial guess of \(x_{0}=3.1\) and take a minimum of four iterations. (b) Did the method exhibit convergence onto ifs real root? Sketch the plot with the results for each iteration labeled.

Determine the lowest positive root of \(f(x)=8 \sin (x) e^{-x}-1\) (a) Graphically. (b) Using the Newton-Riphson method (three iterations, \(\left.x_{i}=0.3\right)\) (c) Using the secant method (three iterations, \(x_{i-1}=0.5\) and \(x_{i}=0.4\) ). (d) Using the modified secant method (five iterations, \(x_{i}=0.3\) \(\delta=0.01).\)

Employ the Newton-Raphson method to determine a real root for \(f(x)=-1+5.5 x-4 x^{2}+0.5 x^{3}\) using initial guesses of \((a) 4.52\) and (b) \(4.54 .\) Discuss and use graphical and analytical methods to explain any peculiarities in your results.

Determine the highest real root of \(f(x)=0.95 x^{3}-5.9 x^{2}+\) \(10.9 x-6:\) (a) Graphically. \(\begin{array}{lllll}\text { (b) Using } & \text { the } & \text { Newton-Raphson } & \text { method } & \text { (three } & \text { iterations. }\end{array}\) \(\left.x_{i}=3.5\right)\) (c) Using the secant method (three iterations, \(x_{i-1}=2.5\) and \(x_{i}=3.5 ?\) (d) Using the modified secant method (three iterations, \(x_{i}=3.5\) \(\delta=0.01).\)

Determine the real roots of \(f(x)=-1+5.5 x-4 x^{2}+0.5 x^{3}:\) (a) graphically and (b) using the Newton-Raphson method to within \(\varepsilon_{s}=0.01 \%.\)
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