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 = 1.5

Apply Newton–Raphson to f(x) = ln x + x − 2 with the starting value x₀ = 1.5. Find x2.

f(x) = ln x + x − 2f′(x) = 1/x + 1x₀ = 1.5
x2 =
2.Easy

The bisection method starts on an interval of length 1. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.05?

b − a = 1ε = 0.05
n =
3.Medium
t (s)010203040v (m/s)68111110

A vehicle's speed is measured every 10 s (table). Estimate the distance s travelled from t = 0 to t = 40 s with the trapezoid rule.

h = 10 sn = 4
s = m
4.Medium
xy00.250.50.7511.251.5

Approximate ∫01.5 1/√(1 + x⁴) dx with the trapezoid rule using n = 6 strips. Give T.

f(x) = 1/√(1 + x⁴)[a, b] = [0, 1.5]n = 6
T =
5.Hard
00.10.20.30.411.21.4ty(0, 1)

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

y′ = y(2 − y)y(0) = 1h = 0.1
y4 =
6.Hard
00.20.40.600.5ty(0, 0)

Use Heun's method (improved Euler) with h = 0.2 for y′ = e−ty, y(0) = 0 to estimate y3 ≈ y(0.6).

y′ = e−tyy(0) = 0h = 0.2
y3 =

Answer key

  1. 1. x2 = 1.557
  2. 2. n = 5
  3. 3. s = 380 m
  4. 4. T = 1.197
  5. 5. y4 = 1.386
  6. 6. y3 = 0.5357

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