Newton’s Method

  1. For each of these quadratic polynomial functions, first calculate its roots using elementary techniques, and use a calculator to determine a decimal approximation of those roots if necessary. Then use Newton’s method to iteratively approximate a root of the quadratic, ensuring it converges to one of the previously computed roots. Can you tell which of each quadratic’s two roots Newton’s method will converge to base on the seed you start with?

    \( \displaystyle f(x) = x^2-8+15 \)
    \( \displaystyle g(x) = x^2-79 \)
    \( \displaystyle h(x) = x^2-2x-1 \)

  2. Use Newton’s method to compute a decimal approximation of each of the following numbers accurate to within \(\pm 0.00001\,.\)

    \( \displaystyle \sqrt{7} \)
    \( \displaystyle \sqrt[3]{181} \)
    \( \displaystyle \sqrt[5]{109} \)
  3. Use Newton’s method to approximate the coordinates of the point where the curves \(y = x\) and \(y = 2\sin(x)\) intersect. Use technology to plot the curves and visually verify your approximation.
  4. Use Newton’s method to approximate the \(x\)-coordinate of the point where the function \(f(x) = 2+x-x^4\) attains it maximum value. Use technology to plot the graph of the function and visually verify your approximation.
  5. Use Newton’s method to approximate the \(x\)-coordinates of any points in the interval \(\bigl(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\bigr)\) where the function \(f(x) = \ln\bigl(\cos(x)\bigr)+x^2 \) attains a local extreme value. Use technology to plot the graph of the function and visually verify your approximation.
  6. What is the shortest vertical distance between the graphs of \(\mathrm{e}^x\) and \(\ln(x)?\)
  7. Use Newton’s method to approximate the \(x\)-coordinate of the inflection point of the graph of \(f(x) = 2x^2-x^4-x^5.\) Use technology to plot the graph of the function and visually verify your approximation.