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 = 2

Apply Newton–Raphson to f(x) = x³ + 2x − 10 with the starting value x₀ = 2. Find x1.

f(x) = x³ + 2x − 10f′(x) = 3x² + 2x₀ = 2
x1 =
3.Medium
xy00.3750.751.1251.5

Approximate ∫01.5 1/(1 + x³) dx with Simpson's rule using n = 4 strips. Give S.

f(x) = 1/(1 + x³)[a, b] = [0, 1.5]n = 4
S =
4.Medium
xy01.53

Approximate ∫03 sin x dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.

f(x) = sin x[a, b] = [0, 3]n = 2
|E| =
5.Hard
00.10.20.30.400.20.4ty(0, 0)

Use Heun's method (improved Euler) with h = 0.2 for y′ = 1 + sin(ty), y(0) = 0 to estimate y2 ≈ y(0.4).

y′ = 1 + sin(ty)y(0) = 0h = 0.2
y2 =
6.Hard
00.511.520.511.52ty(0, 2)

Solve y′ = t − y, y(0) = 2 with Euler's method and step h = 0.5 up to t = 2. The exact solution is y = t − 1 + (y₀ + 1)e−t. Find the global error |E| = |y(2) − y4|.

y′ = t − yy(0) = 2h = 0.5
|E| =

Answer key

  1. 1. x2 = 0.567
  2. 2. x1 = 1.857
  3. 3. S = 1.011
  4. 4. |E| = 0.07556
  5. 5. y2 = 0.4245
  6. 6. |E| = 0.2185

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