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) = cos x − x with the starting value x₀ = 0. Find x1.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 0
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
t (s)0102030405060v (m/s)281412141215

A vehicle's speed is measured every 10 s (table). Estimate the distance s travelled from t = 0 to t = 60 s with Simpson's rule.

h = 10 sn = 6
s = m
4.Medium

The trapezoid rule is used for ∫01 x³ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 1]. Find the smallest number of strips n that guarantees an error below 0.0005.

f(x) = x³[a, b] = [0, 1]ε = 0.0005
n =
5.Hard
00.20.40.611.52ty(0, 2)

Use Heun's method (improved Euler) with h = 0.25 for y′ = t − y, y(0) = 2 to estimate y3 ≈ y(0.75).

y′ = t − yy(0) = 2h = 0.25
y3 =
6.Hard
00.510.51ty(0, 1)

Use Euler's method with h = 0.2 for y′ = −2ty, y(0) = 1 to estimate y5 ≈ y(1).

y′ = −2tyy(0) = 1h = 0.2
y5 =

Answer key

  1. 1. x1 = 1
  2. 2. n = 9
  3. 3. s = 670 m
  4. 4. n = 32
  5. 5. y3 = 1.181
  6. 6. y5 = 0.3994

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