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
xya = 1b = 2

Use the bisection method on f(x) = x³ − x − 1 with the starting interval [1, 2]. Find the midpoint c4 of step 4.

f(x) = x³ − x − 1[a, b] = [1, 2]n = 4
c4 =
2.Easy

The bisection method starts on an interval of length 3. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.001?

b − a = 3ε = 0.001
n =
3.Medium

The trapezoid rule is used for ∫02 x³ 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 0.0005.

f(x) = x³[a, b] = [0, 2]ε = 0.0005
n =
4.Medium
t (s)051015202530v (m/s)10121521242328

A vehicle's speed is measured every 5 s (table). Estimate the distance s travelled from t = 0 to t = 30 s with Simpson's rule.

h = 5 sn = 6
s = m
5.Hard
00.20.422.53ty(0, 3)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = cos t − y, y(0) = 3. Find y₁ ≈ y(0.5).

y′ = cos t − yy(0) = 3h = 0.5
y1 =
6.Hard
00.510.51ty(0, 0.2)

Solve y′ = y(2 − y), y(0) = 0.2 with Euler's method and step h = 0.25 up to t = 1. The exact solution is y = 2y₀/(y₀ + (2 − y₀)e−2t). Find the global error |E| = |y(1) − y4|.

y′ = y(2 − y)y(0) = 0.2h = 0.25
|E| =

Answer key

  1. 1. c4 = 1.313
  2. 2. n = 12
  3. 3. n = 127
  4. 4. s = 566.7 m
  5. 5. y1 = 2.195
  6. 6. |E| = 0.1181

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