3.7.1: Resources and Key Concepts

Math 400: Calculus I - Differential Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Resources

    Videos

    Key Concepts

    Definitions

    • Newton's Method Formula: An iterative process to approximate a root of \(f(x)=0\), given by \(x_{n+1} = x_n - \frac{f(x_n)}{f^{\prime}(x_n)}\), where \(x_0\) is an initial guess.
    • Iterative Process: A process that creates a list of numbers \(x_0, x_1, x_2, \dots, x_n, \dots\) where each term \(x_n\) (for \(n \ge 1\)) is defined in terms of the previous term \(x_{n-1}\) by repeating the same function or formula (e.g., \(x_n = F(x_{n-1})\)).
    • Fixed Point (of an iterative process): If an iterative process \(x_n = F(x_{n-1})\) generates a list of numbers \(x_1, x_2, \dots\) that approaches a finite number \(x^*\), then \(x^*\) satisfies \(x^* = F(x^*)\), and \(x^*\) is called a fixed point of \(F\).
    • 2-cycle (in an iterative process): An iterative process where the sequence of values bounces back and forth between two distinct values.
    • Sensitive Dependence on Initial Conditions: A property of chaotic iterative processes where small changes in initial conditions can generate drastically different behavior in the iterative process.
    • Mandelbrot Set: A well-known example of a set of points generated by the iterative chaotic behavior of a relatively simple function.

    Common Mistakes

    • Choosing a Poor Initial Guess (\(x_0\)):
      • If \(x_0\) is too far from the desired root, Newton's method might converge to a different root or not converge at all.
      • If \(f^{\prime}(x_n) = 0\) (or is very close to 0) for some approximation \(x_n\), the method fails because the tangent line is horizontal (or nearly so) and doesn't intersect the x-axis (or intersects it very far away).
    • Function Behavior Leading to Non-Convergence:
      • The approximations may fail to approach a root entirely, possibly alternating between values.
    • Using Degree Mode for Trigonometric Functions: If \(f(x)\) involves trigonometric functions, calculators must be in radian mode.

    3.7.1: Resources and Key Concepts is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.