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 = 0

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0
x1 =
2.Easy
xyx₀ = 0

Apply Newton–Raphson to f(x) = cos x − x with the starting value x₀ = 0. Find x1.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 0
x1 =
3.Medium
xy00.51

Approximate ∫01 eˣ dx with the trapezoid rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = eˣ[a, b] = [0, 1]n = 2
|E| =
4.Medium
xy00.33330.666711.3331.6672

Approximate ∫02 1/(1 + x³) dx with the trapezoid rule using n = 6 strips. Give T.

f(x) = 1/(1 + x³)[a, b] = [0, 2]n = 6
T =
5.Hard
00.050.10.50.510.520.53ty(0, 0.5)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.1 for y′ = t² + y², y(0) = 0.5. Find y₁ ≈ y(0.1).

y′ = t² + y²y(0) = 0.5h = 0.1
y1 =
6.Hard
00.10.20.30.4−1.6−1.4−1.2−1ty(0, −1)

Solve y′ = y − t², y(0) = −1 with Euler's method and step h = 0.1 up to t = 0.4. The exact solution is y = t² + 2t + 2 + (y₀ − 2)eᵗ. Find the global error |E| = |y(0.4) − y4|.

y′ = y − t²y(0) = −1h = 0.1
|E| =

Answer key

  1. 1. x1 = 0.5
  2. 2. x1 = 1
  3. 3. |E| = 0.03565
  4. 4. T = 1.089
  5. 5. y1 = 0.5267
  6. 6. |E| = 0.03676

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