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 = 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 =
3.Medium

The trapezoid rule is used for ∫01 eˣ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 1]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = eˣ[a, b] = [0, 1]ε = 0.001
n =
4.Medium
xy00.66671.3332

Approximate ∫02 sin(x²) dx with the trapezoid rule using n = 3 strips. Give T.

f(x) = sin(x²)[a, b] = [0, 2]n = 3
T =
5.Hard
00.050.10.150.20.50.55ty(0, 0.5)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = t² + y², y(0) = 0.5. Find y₁ ≈ y(0.2).

y′ = t² + y²y(0) = 0.5h = 0.2
y1 =
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 = 0.7504
  2. 2. x2 = 1.674
  3. 3. n = 16
  4. 4. T = 0.6868
  5. 5. y1 = 0.5584
  6. 6. y1 = 0.5434

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