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 = 0.5

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0.5. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x2 =
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
xy00.751.52.253

Approximate ∫03 sin x dx with the trapezoid rule, n = 4, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = sin x[a, b] = [0, 3]n = 4
|E| =
4.Medium
t (s)0510152025v (m/s)776868

A vehicle's speed is measured every 5 s (table). Estimate the distance s travelled from t = 0 to t = 25 s with the trapezoid rule.

h = 5 sn = 5
s = m
5.Hard
00.511234ty(0, 1)

Use Euler's method with h = 0.2 for y′ = t + y, y(0) = 1 to estimate y5 ≈ y(1).

y′ = t + yy(0) = 1h = 0.2
y5 =
6.Hard
01212ty(0, 0.5)

Solve y′ = y(2 − y), y(0) = 0.5 with Euler's method and step h = 0.5 up to t = 2.5. The exact solution is y = 2y₀/(y₀ + (2 − y₀)e−2t). Find the global error |E| = |y(2.5) − y5|.

y′ = y(2 − y)y(0) = 0.5h = 0.5
|E| =

Answer key

  1. 1. x2 = 0.5671
  2. 2. n = 9
  3. 3. |E| = 0.09417
  4. 4. s = 172.5 m
  5. 5. y5 = 2.977
  6. 6. |E| = 0.03943

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