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 = 1

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 1. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 1
x2 =
2.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 =
3.Medium
xy00.511.52

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

f(x) = sin x[a, b] = [0, 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
00.510.30.40.50.6ty(0, 0.5)

Solve y′ = t − y, y(0) = 0.5 with Euler's method and step h = 0.25 up to t = 1. The exact solution is y = t − 1 + (y₀ + 1)e−t. Find the global error |E| = |y(1) − y4|.

y′ = t − yy(0) = 0.5h = 0.25
|E| =
6.Hard
00.20.400.20.4ty(0, 0)

Use Heun's method (improved Euler) with h = 0.25 for y′ = e−ty, y(0) = 0 to estimate y2 ≈ y(0.5).

y′ = e−tyy(0) = 0h = 0.25
y2 =

Answer key

  1. 1. x2 = 0.567
  2. 2. x1 = 0.7552
  3. 3. |E| = 0.02963
  4. 4. |E| = 0.5238
  5. 5. |E| = 0.07721
  6. 6. y2 = 0.4585

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