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 − 2 with the starting value x₀ = 2. Find x2.

f(x) = x³ − x − 2f′(x) = 3x² − 1x₀ = 2
x2 =
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 0.001?

b − a = 4ε = 0.001
n =
3.Medium
xy11.251.51.752

Approximate ∫12 1/x dx with the trapezoid rule, n = 4, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = 1/x[a, b] = [1, 2]n = 4
|E| =
4.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 √(1 + x³) dx with Simpson's rule using n = 6 strips. Give S.

f(x) = √(1 + x³)[a, b] = [0, 1]n = 6
S =
5.Hard
00.20.40.600.5ty(0, 0)

Use Heun's method (improved Euler) with h = 0.2 for y′ = e−ty, y(0) = 0 to estimate y3 ≈ y(0.6).

y′ = e−tyy(0) = 0h = 0.2
y3 =
6.Hard
00.20.400.5ty(0, 0)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = 1 + sin(ty), y(0) = 0. Find y₁ ≈ y(0.5).

y′ = 1 + sin(ty)y(0) = 0h = 0.5
y1 =

Answer key

  1. 1. x2 = 1.53
  2. 2. n = 12
  3. 3. |E| = 0.003877
  4. 4. S = 1.111
  5. 5. y3 = 0.5357
  6. 6. y1 = 0.5434

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