Problem 3 - Essential Calculus Early Transcendentals

Question

Suppose the tangent line to the curve \(y=f(x)\) at the point \((2,5)\) has the equation \(y=9-2 x .\) If Newton's method is used to locate a root of the equation \(f(x)=0\) and the initial approximation is \(x_{1}=2,\) find the second approximation \(x_{2}\) .

Step-by-Step Solution

Answer

The second approximation, \( x_2 \), is 4.5.

Step 1: Understand Newton's Method

Newton’s method is an iterative root-finding technique. The formula for updating estimates in Newton's method is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] where \( f'(x_n) \) is the derivative of the function evaluated at \( x_n \).

Step 2: Identify Function Values from Tangent Equation

The tangent line equation \( y = 9 - 2x \) is given at the point \((2, 5)\), meaning that \( f(2) = 5 \). The slope \(-2\) is also the derivative of the function at \( x = 2 \), so \( f'(2) = -2 \).

Step 3: Set Initial Values from the Problem

The initial approximation provided is \( x_1 = 2 \). We will use this value as \( x_n \) in Newton’s method formula to find \( x_2 \).

Step 4: Apply Newton's Method Formula

Substitute the identified values into Newton's formula: \[ x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} = 2 - \frac{5}{-2} \]. Simplify the fraction as \( 2 + \frac{5}{2} \).

Step 5: Calculate Second Approximation

Compute \( x_2 \) by evaluating: \[ x_2 = 2 + \frac{5}{2} = 2 + 2.5 = 4.5 \]. So, \( x_2 = 4.5 \).

Key Concepts

Tangent Line

In mathematics, the tangent line to a curve at a particular point can often give us valuable information about the function itself. Imagine you have a curve on a graph. The tangent line to the curve at a point is simply the straight line that touches the curve at that point and has the same slope as the curve at that point. Think of it as a mini-flat section of the curve.

• If you draw a tangent to a curve at any point, it will only touch the curve at that exact point.
• The slope of this line equals the derivative of the function at that point.

For example, from the problem, we know the equation for the tangent line is given by \( y = 9 - 2x \) at the point \((2,5)\). This indicates that at \(x=2\), the slope of the tangent line or \(f'(2)\) is \(-2\). Hence, the tangent line tells us about the derivative at that point and helps in setting up Newton's method.

Root Finding

Root finding is a process in mathematical analysis used to determine the values (roots) for which a given function equals zero. Essentially, if you have a function \(f(x)\), finding its root means solving \(f(x) = 0\).

• Root finding is crucial because it allows for the determination of significant values in various calculations and applications.
• Newton's method is one of the efficient iterative techniques to find such roots, especially when the function is suitable for differentiation.

In this context, our goal is to use Newton's method to find a root of the equation \(f(x) = 0\). We start from an initial guess \(x_1 = 2\) and use the tangent information to move closer to the root for the next approximation \(x_2\), ultimately tightening in on the correct root through iteration.

Derivative

The derivative of a function at a certain point provides the slope of the tangent to the function at that point. It's like measuring how steep the function is at a particular moment.

• Derivatives can be visualized as the rate of change of a function with respect to one of its variables.
• They are foundational in calculus and essential in Newton's method.

Returning to our example, the slope of the given tangent line is \(-2\), indicating that the derivative of the function \(f\) at \(x = 2\) is \(f'(2) = -2\). We utilize this derivative in Newton's method formula to refine our approximation of the root. Using derivatives helps in adjusting our estimations and successfully navigating towards the root where the function is zero.