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 − 2 with the starting value x₀ = 1. Find x1.

f(x) = x³ + 2x − 2f′(x) = 3x² + 2x₀ = 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 0.001?

b − a = 0.5ε = 0.001
n =
3.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 =
4.Medium
t (s)01020304050v (m/s)788101115

A vehicle's speed is measured every 10 s (table). Estimate the distance s travelled from t = 0 to t = 50 s with the trapezoid rule.

h = 10 sn = 5
s = m
5.Hard
00.511.52050100ty(0, 2)

Solve y′ = 2y, y(0) = 2 with Euler's method and step h = 0.5 up to t = 2. The exact solution is y = y₀e2t. Find the global error |E| = |y(2) − y4|.

y′ = 2yy(0) = 2h = 0.5
|E| =
6.Hard
00.51−4−2ty(0, −1)

Solve y′ = y − t², y(0) = −1 with Euler's method and step h = 0.25 up to t = 1.25. The exact solution is y = t² + 2t + 2 + (y₀ − 2)eᵗ. Find the global error |E| = |y(1.25) − y5|.

y′ = y − t²y(0) = −1h = 0.25
|E| =

Answer key

  1. 1. x1 = 0.8
  2. 2. n = 9
  3. 3. n = 222
  4. 4. s = 480 m
  5. 5. |E| = 77.2
  6. 6. |E| = 0.8028

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