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Chapter 4: Problem 16

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. $$y=\sin x \text { and } y=x^{3}-2 x^{2}+1$$

Short Answer

Expert verified
The curves intersect three times; Newton's Method confirms these intersections.

Step by step solution

01

Graph the Functions

Use a graphing utility like a graphing calculator or software to plot the functions \( y = \sin x \) and \( y = x^3 - 2x^2 + 1 \). Look at the graph to identify apparent intersection points.
02

Identify Intersection Points

From the graph, visually inspect and count how many times the two curves intersect. Notice that without zooming deeply, intersections can occur multiple times; therefore, seeing the graph at different zoom levels might be helpful.
03

Choose Initial Guesses for Newton's Method

Based on the graph from Step 1, estimate the \(x\)-coordinates of each intersection. These estimates will be your initial guesses for Newton's Method. Try to choose points close to the actual intersections.
04

Apply Newton's Method

For each guessed intersection \(x_0\), apply Newton's Method. Use the formula:\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]Where \( f(x) = x^3 - 2x^2 + 1 - \sin x \) and \( f'(x) = 3x^2 - 4x - \cos x \). Calculate few iterations for each initial guess until you reach desired accuracy.
05

Verify Intersection Points

After calculating the approximate \(x\)-coordinates, plug these values back into both functions to ensure that the \(y\)-coordinates are indeed equal, confirming the intersection points.

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Key Concepts

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

Intersection of Curves
When dealing with two functions, finding where they intersect is like figuring out where they both agree at the same point on a graph. For the functions \( y = \sin x \) and \( y = x^3 - 2x^2 + 1 \), an intersection point means they have the same \( y \)-value for a particular \( x \)-coordinate. This happens when the equation \( \sin x = x^3 - 2x^2 + 1 \) holds true. The solution to this equation represents the intersection points on the graph.
  • To find intersections, we need to solve \( f(x) = 0 \) where \( f(x) = x^3 - 2x^2 + 1 - \sin x \).
  • These intersections give us a graphical and numerical solution showing where both functions meet.
Understanding intersections helps in studying how two curves relate and touch each other, playing a crucial role in calculus and graph theory.
Graphing Techniques
Graphing is a powerful technique to visually interpret mathematical functions. When plotting the curves \( y = \sin x \) and \( y = x^3 - 2x^2 + 1 \), it's essential to use appropriate scaling and zooming to capture all intersection points. Using tools like graphing calculators or graphing software makes this easier.
  • Scaling: Properly adjusting the axes ensures all critical points are visible without overcrowding the graph.
  • Zooming: Allows closer inspection of areas where intersections may not be clearly visible at first glance.
  • Overlay and Compare: By plotting both functions on the same graph, it becomes straightforward to compare and detect intersections quickly.
Effective graphing techniques provide a visual intuition for understanding where and how curves intersect, making subsequent calculations like finding precise intersection points more accurate.
Newton's Method
Newton's Method is a handy iterative technique for finding more precise solutions to equations, especially useful when other methods fall short. When using it to find the intersections of \( y = \sin x \) and \( y = x^3 - 2x^2 + 1 \), it involves refining guesses for those \( x \)-coordinates where the curves meet.
  • Start with an initial guess based on graphical data.
  • Use the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), where \( f(x) = x^3 - 2x^2 + 1 - \sin x \) and \( f'(x) = 3x^2 - 4x - \cos x \).
  • Iterate the process, updating \( x \) until it converges to a stable solution closely matching the intersection point.
Newton's Method helps bridge the gap between graphical guessing and numerical precision, ensuring results aligns accurately with real intersection points.

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Most popular questions from this chapter

Explain the difference between a relative maximum and an absolute maximum. Sketch a graph that illustrates a function with a relative maximum that is not an absolute maximum, and sketch another graph illustrating an absolute maximum that is not a relative maximum. Explain how these graphs satisfy the given conditions.

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=x \ln x$$

(a) Use the Constant Difference Theorem (4.8.3) to show that if \(f^{\prime}(x)=g^{\prime}(x)\) for all \(x\) in \((-\infty,+\infty),\) and if \(f\left(x_{0}\right)-g\left(x_{0}\right)=c\) at some point \(x_{0},\) then $$ f(x)-g(x)=c $$ for all \(x\) in \((-\infty,+\infty)\) (b) Use the result in part (a) to show that the function $$ h(x)=(x-1)^{3}-\left(x^{2}+3\right)(x-3) $$ is constant for all \(x\) in \((-\infty,+\infty),\) and find the constant. (c) Check the result in part (b) by multiplying out and simplifying the formula for \(h(x)\)

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$f(x)=x^{3}+x-4 ;[-1,2]$$

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval; then use those graphs to estimate the \(x\) -coordinates of the inflection points of \(f\), the intervals on which \(f\) is concave up or down, and the intervals on which \(f\) is increasing or decreasing. Check your estimates by graphing \(f\). $$f(x)=\frac{1}{1+x^{2}}, \quad-5 \leq x \leq 5$$
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