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) = cos x − x with the starting value x₀ = 1. Find x2.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 1
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
xy012

Approximate ∫02 e−x² dx with Simpson's rule using n = 2 strips. Give S.

f(x) = e−x²[a, b] = [0, 2]n = 2
S =
4.Medium
xy012

Approximate ∫02 1/√(1 + x⁴) dx with the trapezoid rule using n = 2 strips. Give T.

f(x) = 1/√(1 + x⁴)[a, b] = [0, 2]n = 2
T =
5.Hard
00.10.20.30.434ty(0, 3)

Solve y′ = t + y, y(0) = 3 with Euler's method and step h = 0.1 up to t = 0.4. The exact solution is y = −t − 1 + (y₀ + 1)eᵗ. Find the global error |E| = |y(0.4) − y4|.

y′ = t + yy(0) = 3h = 0.1
|E| =
6.Hard
00.10.20.30.400.20.4ty(0, 0)

Use Heun's method (improved Euler) with h = 0.2 for y′ = e−ty, y(0) = 0 to estimate y2 ≈ y(0.4).

y′ = e−tyy(0) = 0h = 0.2
y2 =

Answer key

  1. 1. x2 = 0.7391
  2. 2. x2 = 2.926
  3. 3. S = 0.8299
  4. 4. T = 1.328
  5. 5. |E| = 0.1109
  6. 6. y2 = 0.3778

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