courses › Beyond College

How computers find roots and areas step by step.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 54.

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Builds on: Using derivatives (not open yet) · Integrals (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Newton’s method par 30 s not tried yet

  worked example

  Apply one step of Newton’s method to f(x) = x³ − 58, starting from x₀ = 3. What is x₁? Give an exact fraction.

  Answer: 112/27

  1. x₁ = x₀ − f(x₀)/f′(x₀), with f′(x) = 3x².
  2. f(3) = 27 − 58 = −31 and f′(3) = 27.
  3. x₁ = 3 − (−31)/27 = 3 + 31/27 = 112/27 ≈ 4.148.

• Bisection par 30 s not tried yet

  worked example

  Start with [2, 3] for f(x) = x³ − 16. Test the midpoint and keep the half where f changes sign. Do this twice. What is the midpoint of the interval you end with? Give the answer as a decimal.

  Answer: 2.625

  1. 2.5³ = 15.625 < 16, so f(2.5) < 0: keep [2.5, 3].
  2. 2.75³ = 20.796875 > 16, so f(2.75) > 0: keep [2.5, 2.75].
  3. The midpoint of [2.5, 2.75] is 2.625.

• Trapezoid rule par 30 s not tried yet

  worked example

  Use the trapezoid rule with 4 strips of width 1 to estimate ∫₂⁶ (x² − 2x) dx. Give the estimate as a decimal.

  Answer: 38

  1. The points are x = 2, 3, 4, 5, 6, where f(x) = x² − 2x takes the values 0, 3, 8, 15, 24.
  2. (h/2)(f₀ + 2f₁ + 2f₂ + 2f₃ + f₄) = ½ × (0 + 2 × 3 + 2 × 8 + 2 × 15 + 24) = ½ × 76 = 38.
  3. The exact value is 112/3, so the estimate is 2/3 too high.

• Simpson’s rule par 30 s not tried yet

  worked example

  Use Simpson’s rule with 2 strips of width 1 to estimate ∫₀² (x³ − 3x² + x − 4) dx.

  Answer: -10

  1. The points are x = 0, 1, 2, where f(x) = x³ − 3x² + x − 4 takes the values −4, −5, −6.
  2. (h/3)(f₀ + 4f₁ + f₂) = ⅓ × (−4 + 4 × (−5) + (−6)) = ⅓ × (−30) = −10.
  3. Simpson’s rule is exact for polynomials up to degree 3, so −10 is also the exact value of the integral.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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