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) = x² − 3 with the starting value x₀ = 1. Find x1.

f(x) = x² − 3f′(x) = 2xx₀ = 1
x1 =
2.Easy

The bisection method starts on an interval of length 0.5. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.0001?

b − a = 0.5ε = 0.0001
n =
3.Medium
t (s)02468v (m/s)1010121115

A vehicle's speed is measured every 2 s (table). Estimate the distance s travelled from t = 0 to t = 8 s with the trapezoid rule.

h = 2 sn = 4
s = m
4.Medium
t (s)02468v (m/s)288912

A vehicle's speed is measured every 2 s (table). Estimate the distance s travelled from t = 0 to t = 8 s with Simpson's rule.

h = 2 sn = 4
s = m
5.Hard
00.10.20.30.40.91ty(0, 1)

Use Euler's method with h = 0.1 for y′ = −2ty, y(0) = 1 to estimate y4 ≈ y(0.4).

y′ = −2tyy(0) = 1h = 0.1
y4 =
6.Hard
00.050.10.150.200.10.2ty(0, 0)

Use Heun's method (improved Euler) with h = 0.1 for y′ = 1 + sin(ty), y(0) = 0 to estimate y2 ≈ y(0.2).

y′ = 1 + sin(ty)y(0) = 0h = 0.1
y2 =

Answer key

  1. 1. x1 = 2
  2. 2. n = 13
  3. 3. s = 91 m
  4. 4. s = 65.33 m
  5. 5. y4 = 0.8844
  6. 6. y2 = 0.203

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