Locating Roots and Iterative Methods · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Trapping a root, then walking in

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

  1. For f(x) = x cubed minus 2x minus 5, what is f(2)?

    Answer: ______________

  2. Opposite signs at f(1) and f(2) guarantee exactly one root between 1 and 2. True or false?

    Circle one:   True   False

  3. For f(x) = x cubed - 2x - 5, what is f(2)?

    Answer: ______________

  4. A continuous f has f(1) = -2 and f(2) = 3. What follows?

    • x = 1 is a root
    • x = 2 is a root
    • a root lies between 1 and 2

  5. Iterate x_{n+1} = (x_n + 3)/2 from x_0 = 1. What is x_2?

    Answer: ______________

  6. Continue x0 = 1, x next = (x + 3)/2, with x1 = 2 and x2 = 2.5. What is x3?

    • 2.75
    • 2.5
    • 2.625

  7. Which is a valid rearrangement of x cubed + x = 7 into x = g(x) form?

    • x = the cube root of (7 - x)
    • x = 7 + x cubed
    • x = (7 - x) cubed
    • x = 7/x - x

  8. Iterate x next = (x + 3)/2 from x0 = 1. What is x2?

    Answer: ______________

  9. An iteration heads to 2.5 with values 2.4, 2.48, 2.496. Which statement is correct?

    • 2.48 is correct to 2 decimal places
    • the error is growing
    • 2.496 is correct to 1 decimal place

  10. An iteration heads to alpha = 2.5 with values 2.4, 2.48, 2.496. Which statement is correct?

    • 2.48 is correct to 2 decimal places
    • 2.496 is correct to 1 decimal place
    • 2.4 is correct to 1 decimal place
    • the error is growing
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Answer key

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

Trapping a root, then walking in W1-mt_2COO6TWerM-s1

  1. -1 · Substituting gives 8 - 4 - 5 = -1.
  2. False · False. A sign change guarantees at least one root, but the graph may cross several times.
  3. -1 · -1. Substituting gives 8 - 4 - 5 = -1.
  4. a root lies between 1 and 2 · Opposite signs force the graph to cross the axis between 1 and 2.
  5. 2.5 · 2.5. One step gives 2, and the next gives (2 + 3)/2 = 2.5.
  6. 2.75 · Substituting gives (2.5 + 3)/2 = 2.75, climbing toward 3.
  7. x = the cube root of (7 - x) · x = the cube root of (7 - x). Isolating x cubed first and taking roots solves for x.
  8. 2.5 · One step gives 2, and the next gives (2 + 3)/2 = 2.5.
  9. 2.496 is correct to 1 decimal place · 2.496 sits within 0.004 of 2.5, inside the 0.05 tolerance for 1 decimal place.
  10. 2.496 is correct to 1 decimal place · 2.496 is correct to 1 decimal place. It sits within 0.004 of 2.5, inside the 0.05 tolerance, while 2.48 misses 2 decimal place accuracy.
Worksheet · LightMySky