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) = x³ + 2x − 5 with the starting value x₀ = 1. Find x2.

f(x) = x³ + 2x − 5f′(x) = 3x² + 2x₀ = 1
x2 =
2.Easy
xyx₀ = 0.5

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0.5. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x2 =
3.Medium
t (s)0123456v (m/s)891211161615

A vehicle's speed is measured every 1 s (table). Estimate the distance s travelled from t = 0 to t = 6 s with Simpson's rule.

h = 1 sn = 6
s = m
4.Medium
xy00.66671.3332

Approximate ∫02 1/√(1 + x⁴) dx with the trapezoid rule using n = 3 strips. Give T.

f(x) = 1/√(1 + x⁴)[a, b] = [0, 2]n = 3
T =
5.Hard
00.511.5200.51ty(0, 1)

Solve y′ = −2ty, y(0) = 1 with Euler's method and step h = 0.5 up to t = 2. The exact solution is y = y₀e−t². Find the global error |E| = |y(2) − y4|.

y′ = −2tyy(0) = 1h = 0.5
|E| =
6.Hard
00.10.20.30.40.60.8ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.2 for y′ = t + y, y(0) = 0.5 to estimate y2 ≈ y(0.4).

y′ = t + yy(0) = 0.5h = 0.2
y2 =

Answer key

  1. 1. x2 = 1.331
  2. 2. x2 = 0.5671
  3. 3. s = 74.33 m
  4. 4. T = 1.35
  5. 5. |E| = 0.01832
  6. 6. y2 = 0.8326

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