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 = 0.5

Apply Newton–Raphson to f(x) = cos x − x with the starting value x₀ = 0.5. Find x1.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 0.5
x1 =
2.Easy
xyx₀ = 0

Apply Newton–Raphson to f(x) = cos x − x with the starting value x₀ = 0. Find x2.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 0
x2 =
3.Medium
xy11.251.51.752

Approximate ∫12 1/x dx with the trapezoid rule, n = 4, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = 1/x[a, b] = [1, 2]n = 4
|E| =
4.Medium
xy012

Approximate ∫02 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, 2]n = 2
|E| =
5.Hard
00.20.40.60.823ty(0, 3)

Use Euler's method with h = 0.2 for y′ = t − y, y(0) = 3 to estimate y4 ≈ y(0.8).

y′ = t − yy(0) = 3h = 0.2
y4 =
6.Hard
00.050.10.150.211.11.2ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = 1 + sin(ty), y(0) = 1. Find y₁ ≈ y(0.2).

y′ = 1 + sin(ty)y(0) = 1h = 0.2
y1 =

Answer key

  1. 1. x1 = 0.7552
  2. 2. x2 = 0.7504
  3. 3. |E| = 0.003877
  4. 4. |E| = 0.5238
  5. 5. y4 = 1.438
  6. 6. y1 = 1.223

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