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 = 0

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0
x1 =
2.Easy

The bisection method starts on an interval of length 4. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.005?

b − a = 4ε = 0.005
n =
3.Medium
t (s)0123456v (m/s)8101112121419

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

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

The trapezoid rule is used for ∫02 eˣ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 2]. Find the smallest number of strips n that guarantees an error below 0.0001.

f(x) = eˣ[a, b] = [0, 2]ε = 0.0001
n =
5.Hard
00.20.40.611.5ty(0, 1)

Solve y′ = y − t², y(0) = 1 with Euler's method and step h = 0.2 up to t = 0.6. The exact solution is y = t² + 2t + 2 + (y₀ − 2)eᵗ. Find the global error |E| = |y(0.6) − y3|.

y′ = y − t²y(0) = 1h = 0.2
|E| =
6.Hard
00.511.5123ty(0, 1)

Use Heun's method (improved Euler) with h = 0.5 for y′ = 1 + sin(ty), y(0) = 1 to estimate y3 ≈ y(1.5).

y′ = 1 + sin(ty)y(0) = 1h = 0.5
y3 =

Answer key

  1. 1. x1 = 0.5
  2. 2. n = 10
  3. 3. s = 72.5 m
  4. 4. n = 222
  5. 5. |E| = 0.05148
  6. 6. y3 = 2.904

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