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 4. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.0001?

b − a = 4ε = 0.0001
n =
3.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 sin(x²) dx with Simpson's rule using n = 6 strips. Give S.

f(x) = sin(x²)[a, b] = [0, 1]n = 6
S =
4.Medium
xy00.33330.666711.3331.6672

Approximate ∫02 e−x² dx with Simpson's rule using n = 6 strips. Give S.

f(x) = e−x²[a, b] = [0, 2]n = 6
S =
5.Hard
00.10.20.31.71.81.92ty(0, 2)

Use Heun's method (improved Euler) with h = 0.1 for y′ = cos t − y, y(0) = 2 to estimate y3 ≈ y(0.3).

y′ = cos t − yy(0) = 2h = 0.1
y3 =
6.Hard
00.050.100.050.1ty(0, 0)

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

y′ = 1 + sin(ty)y(0) = 0h = 0.1
y1 =

Answer key

  1. 1. x1 = 1
  2. 2. n = 16
  3. 3. S = 0.3102
  4. 4. S = 0.882
  5. 5. y3 = 1.737
  6. 6. y1 = 0.1003

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