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 = 0b = 1

Use the bisection method on f(x) = cos x − x with the starting interval [0, 1]. Find the midpoint c4 of step 4.

f(x) = cos x − x[a, b] = [0, 1]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
xy012

Approximate ∫02 x³ dx with the trapezoid rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = x³[a, b] = [0, 2]n = 2
|E| =
4.Medium
t (s)010203040v (m/s)04101517

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
5.Hard
00.20.40.50.60.7ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.25 for y′ = t² + y², y(0) = 0.5 to estimate y2 ≈ y(0.5).

y′ = t² + y²y(0) = 0.5h = 0.25
y2 =
6.Hard
00.51246ty(0, 2)

Use Euler's method with h = 0.25 for y′ = t + y, y(0) = 2 to estimate y4 ≈ y(1).

y′ = t + yy(0) = 2h = 0.25
y4 =

Answer key

  1. 1. c4 = 0.6875
  2. 2. n = 12
  3. 3. |E| = 1
  4. 4. s = 375 m
  5. 5. y2 = 0.717
  6. 6. y4 = 5.324

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