The Newton-Raphson Method · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Sliding down the tangent to the root

Mathematics · Calculus & Analysis · ages 17-18
Name ______________________   Date ____________

  1. Use one Newton step from x0 = 1 to estimate a root of f(x) = x squared minus 2. What is x1?

    Answer: ______________

  2. Use one Newton step from x0 = 1 to estimate a root of f(x) = x squared minus 2. What is x1?

    Answer: ______________

  3. For f(x) = x squared minus 9 with x0 = 1, what is x1 after one Newton step?

    • 5
    • 3
    • 4

  4. In the Newton formula, what point is x1 on the graph?

    • Where the curve crosses the y-axis
    • Where the tangent at x0 crosses the x-axis
    • The midpoint between x0 and the true root

  5. For f(x) = x squared minus 9 with x0 = 1, what is x1 after one Newton step?

    • 5
    • 4
    • 3
    • 6

  6. Starting at a point with zero slope still gives a usable next Newton guess.

    Circle one:   True   False

  7. For f(x) = x squared minus 5 with x0 = 2, one Newton step gives which x1?

    Answer: ______________

  8. For f(x) = x squared minus 5 with x0 = 2, one Newton step gives which x1?

    Answer: ______________

  9. For f(x) = x squared minus 9 with x0 = 5, one Newton step gives which x1?

    Answer: ______________

  10. For f(x) = x squared minus 9 with x0 = 5, one Newton step gives which x1?

    Answer: ______________

LightMySky · lightmysky.comW1-mt_jkjJre5ETd-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Sliding down the tangent to the root W1-mt_jkjJre5ETd-s1

  1. 1.5 · f(1) is minus 1 and the slope there is 2, so the tangent crosses the axis at 1.5.
  2. 1.5 · f(1) is minus 1 and the slope there is 2, so the tangent crosses the axis at 1.5.
  3. 5 · f(1) is minus 8 and the slope is 2, so x1 = 1 plus 4.
  4. Where the tangent at x0 crosses the x-axis · The formula follows the tangent line down to the axis, and that crossing becomes the next guess.
  5. 5 · f(1) is minus 8 and the slope is 2, so x1 = 1 plus 4, which is 5.
  6. False · A flat tangent never crosses the axis, so the method has nowhere to go.
  7. 2.25 · f(2) is minus 1 with slope 4, so x1 = 2 plus 0.25.
  8. 2.25 · f(2) is minus 1 with slope 4, so x1 = 2 plus 0.25, which is 2.25.
  9. 3.4 · f(5) is 16 with slope 10, so x1 = 5 minus 1.6, already much closer to 3.
  10. 3.4 · f(5) is 16 with slope 10, so x1 = 5 minus 1.6, which is 3.4, already much closer to 3.
Worksheet · LightMySky