Problem 16 - Calculus Early Transcendentals

Question

Use a graphing utility to determine the number of times the curves intersect and then apply Newton’s Method, where needed, to approximate the \(x\)-coordinates of all intersections. \(y=\sin x\) and \(y=x^{3}-2 x^{2}+1\)

Step-by-Step Solution

Answer

The curves intersect three times. Use Newton’s method to find the approximate intersection points: where f(x) crosses zero.

Step 1: Plot the Curves

First, use a graphing utility to plot the curves of the functions. Plot the first curve, \( y = \sin x \), and the second curve, \( y = x^3 - 2x^2 + 1 \). Take a look at the graph to visually identify the points where the curves intersect.

Step 2: Identifying Intersections

Upon examining the graph, observe how many times the curves intersect. You will typically see the intersections where the curves cross each other.

Step 3: Approximating Intersection Points with Newton's Method

Now, apply Newton's Method to find more accurate \(x\)-coordinates of the intersections. The general formula for Newton’s Method is \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), where \( f(x) = \sin x - (x^3 - 2x^2 + 1) \). Start with an initial guess, \(x_0\), near each intersection point identified from the graph.

Step 4: Derivative Calculation

Calculate the derivative \( f'(x) \). For this problem, \(f(x) = \sin x - (x^3 - 2x^2 + 1) \), so \( f'(x) = \cos x - (3x^2 - 4x) \).

Step 5: Iteration Process

Using your starting guess from the graph, iteratively apply Newton's formula, \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), to each point until the values stabilize. Continue this process for each intersection to find the precise \(x\)-coordinates of the intersection points.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool in visualizing mathematical concepts and solving equations graphically. By using a graphing utility, like Desmos or a graphing calculator, we can plot mathematical functions to observe their behavior.

In our exercise, we plotted the curves of "\( y = \sin x \)" and "\( y = x^3 - 2x^2 + 1 \)". This step is crucial as it gives a visual representation, helping us identify where the two curves intersect.

The function \( y = \sin x \) is periodic and oscillates between -1 and 1. On the other hand, \( y = x^3 - 2x^2 + 1 \) is a polynomial that tends to dominate with its cubic term.

With the graphing utility, intersections are the points where both functions share the same \( y \)-value. By visually inspecting the graph, we can count these intersections and use them as a starting point for further analysis with Newton's Method.

Newton's Method

Newton's Method is an iterative numerical technique used to approximate roots of a real-valued function. This method is particularly useful for finding more precise intersection points after we've identified them with a graphing utility.

To apply Newton's Method, we need a function \( f(x) \) representing the difference between the two original functions, so \( f(x) = \sin x - (x^3 - 2x^2 + 1) \). Our goal is to find \( x \)-values where this function equals zero, which signifies the curves are intersecting.

The formula for Newton's Method is straightforward:

• Start with an initial guess, \( x_0 \), close to where the graph indicates an intersection.
• Use the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \).
• Repeat this process, updating \( x_n \) each time until the result stabilizes to a consistent value.

This iterative approach refines our guess, leveraging the derivative to direct us closer to the actual root.

Derivative Calculation

In Newton's Method, calculating the derivative of our function \( f(x) = \sin x - (x^3 - 2x^2 + 1) \) is a critical step. The derivative, denoted as \( f'(x) \), tells us how the function changes at a particular point, guiding our iterative process.

First, break down the function into simpler parts:

• The derivative of \( \sin x \) is \( \cos x \).
• For the polynomial \( x^3 - 2x^2 + 1 \), use the power rule:
• The derivative of \( x^3 \) is \( 3x^2 \).
• For \( -2x^2 \), it is \( -4x \).
• The constant \( 1 \) has a derivative of zero.

Thus, our combined derivative is \( f'(x) = \cos x - (3x^2 - 4x) \). This is plugged into Newton's formula to iteratively find the precise values of \( x \) where our two curves intersect.

The derivative not only helps in moving towards the solution with Newton's Method but also provides insight into how the function behaves, ensuring our iterative approach converges effectively.