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) = e−x − x with the starting value x₀ = 0. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0
x2 =
2.Easy
xyx₀ = 0.5

Apply Newton–Raphson to f(x) = cos x − x with the starting value x₀ = 0.5. Find x1.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 0.5
x1 =
3.Medium

The trapezoid rule is used for ∫01 eˣ 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) = eˣ[a, b] = [0, 1]ε = 0.0005
n =
4.Medium
t (s)0102030405060v (m/s)510887710

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
5.Hard
00.10.20.300.2ty(0, 0)

Use Heun's method (improved Euler) with h = 0.1 for y′ = 1 + sin(ty), y(0) = 0 to estimate y3 ≈ y(0.3).

y′ = 1 + sin(ty)y(0) = 0h = 0.1
y3 =
6.Hard
00.10.20.333.54ty(0, 3)

Use Heun's method (improved Euler) with h = 0.1 for y′ = t + y, y(0) = 3 to estimate y3 ≈ y(0.3).

y′ = t + yy(0) = 3h = 0.1
y3 =

Answer key

  1. 1. x2 = 0.5663
  2. 2. x1 = 0.7552
  3. 3. n = 22
  4. 4. s = 483.3 m
  5. 5. y3 = 0.3096
  6. 6. y3 = 4.097

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