Numerical methods worksheet with answers

Free printable Numerical methods worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
xyx₀ = 2

Apply Newton–Raphson to f(x) = x³ − x − 3 with the starting value x₀ = 2. Find x2.

f(x) = x³ − x − 3f′(x) = 3x² − 1x₀ = 2
x2 =
2.Easy
xyx₀ = 4

Apply Newton–Raphson to f(x) = ln x + x − 4 with the starting value x₀ = 4. Find x2.

f(x) = ln x + x − 4f′(x) = 1/x + 1x₀ = 4
x2 =
3.Medium

The trapezoid rule is used for ∫01 eˣ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 1]. Find the smallest number of strips n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣ[a, b] = [0, 1]ε = 1 × 10⁻⁵
n =
4.Medium
xy00.511.52

Approximate ∫02 eˣ dx with the trapezoid rule, n = 4, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = eˣ[a, b] = [0, 2]n = 4
|E| =
5.Hard
00.20.40.611.52ty(0, 1)

Use Heun's method (improved Euler) with h = 0.25 for y′ = 1 + sin(ty), y(0) = 1 to estimate y3 ≈ y(0.75).

y′ = 1 + sin(ty)y(0) = 1h = 0.25
y3 =
6.Hard
00.10.20.30.411.5ty(0, 1)

Use Euler's method with h = 0.1 for y′ = t + y, y(0) = 1 to estimate y4 ≈ y(0.4).

y′ = t + yy(0) = 1h = 0.1
y4 =

Answer key

  1. 1. x2 = 1.674
  2. 2. x2 = 2.926
  3. 3. n = 151
  4. 4. |E| = 0.1326
  5. 5. y3 = 2.136
  6. 6. y4 = 1.528

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