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 = 2

Apply Newton–Raphson to f(x) = x³ − x − 3 with the starting value x₀ = 2. Find x2.

f(x) = x³ − x − 3f′(x) = 3x² − 1x₀ = 2
x2 =
2.Easy
xya = 0b = 1

Use the bisection method on f(x) = cos x − x with the starting interval [0, 1]. Find the midpoint c4 of step 4.

f(x) = cos x − x[a, b] = [0, 1]n = 4
c4 =
3.Medium
xy00.511.52

Approximate ∫02 √(1 + x³) dx with Simpson's rule using n = 4 strips. Give S.

f(x) = √(1 + x³)[a, b] = [0, 2]n = 4
S =
4.Medium
xy00.51

Approximate ∫01 e−x² dx with Simpson's rule using n = 2 strips. Give S.

f(x) = e−x²[a, b] = [0, 1]n = 2
S =
5.Hard
00.10.20.30.50.60.7ty(0, 0.5)

Solve y′ = t + y, y(0) = 0.5 with Euler's method and step h = 0.1 up to t = 0.3. The exact solution is y = −t − 1 + (y₀ + 1)eᵗ. Find the global error |E| = |y(0.3) − y3|.

y′ = t + yy(0) = 0.5h = 0.1
|E| =
6.Hard
00.050.10.150.211.11.2ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = 1 + sin(ty), y(0) = 1. Find y₁ ≈ y(0.2).

y′ = 1 + sin(ty)y(0) = 1h = 0.2
y1 =

Answer key

  1. 1. x2 = 1.674
  2. 2. c4 = 0.6875
  3. 3. S = 3.24
  4. 4. S = 0.7472
  5. 5. |E| = 0.02829
  6. 6. y1 = 1.223

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