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 = 1

Apply Newton–Raphson to f(x) = x³ − x − 3 with the starting value x₀ = 1. Find x2.

f(x) = x³ − x − 3f′(x) = 3x² − 1x₀ = 1
x2 =
2.Easy

The bisection method starts on an interval of length 1. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.001?

b − a = 1ε = 0.001
n =
3.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 sin(x²) dx with the trapezoid rule using n = 6 strips. Give T.

f(x) = sin(x²)[a, b] = [0, 1]n = 6
T =
4.Medium
xy00.66671.3332

Approximate ∫02 e−x² dx with the trapezoid rule using n = 3 strips. Give T.

f(x) = e−x²[a, b] = [0, 2]n = 3
T =
5.Hard
00.050.10.150.20.9991ty(0, 1)

Use Heun's method (improved Euler) with h = 0.1 for y′ = cos t − y, y(0) = 1 to estimate y2 ≈ y(0.2).

y′ = cos t − yy(0) = 1h = 0.1
y2 =
6.Hard
01212ty(0, 2)

Solve y′ = −0.5y, y(0) = 2 with Euler's method and step h = 0.5 up to t = 2.5. The exact solution is y = y₀e−0.5t. Find the global error |E| = |y(2.5) − y5|.

y′ = −0.5yy(0) = 2h = 0.5
|E| =

Answer key

  1. 1. x2 = 1.93
  2. 2. n = 10
  3. 3. T = 0.3128
  4. 4. T = 0.8796
  5. 5. y2 = 0.9986
  6. 6. |E| = 0.0984

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