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 x2.

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

Apply Newton–Raphson to f(x) = x³ − x − 3 with the starting value x₀ = 1. Find x1.

f(x) = x³ − x − 3f′(x) = 3x² − 1x₀ = 1
x1 =
3.Medium
xy012

Approximate ∫02 e−x² dx with the trapezoid rule using n = 2 strips. Give T.

f(x) = e−x²[a, b] = [0, 2]n = 2
T =
4.Medium
t (s)010203040v (m/s)4481114

A vehicle's speed is measured every 10 s (table). Estimate the distance s travelled from t = 0 to t = 40 s with Simpson's rule.

h = 10 sn = 4
s = m
5.Hard
00.510.40.60.81ty(0, 0.5)

Use Euler's method with h = 0.2 for y′ = y − t², y(0) = 0.5 to estimate y5 ≈ y(1).

y′ = y − t²y(0) = 0.5h = 0.2
y5 =
6.Hard
00.050.111.051.1ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.1 for y′ = 1 + sin(ty), y(0) = 1. Find y₁ ≈ y(0.1).

y′ = 1 + sin(ty)y(0) = 1h = 0.1
y1 =

Answer key

  1. 1. x2 = 0.7504
  2. 2. x1 = 2.5
  3. 3. T = 0.877
  4. 4. s = 313.3 m
  5. 5. y5 = 0.9699
  6. 6. y1 = 1.105

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