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 = 1.5

Apply Newton–Raphson to f(x) = x³ + 2x − 10 with the starting value x₀ = 1.5. Find x2.

f(x) = x³ + 2x − 10f′(x) = 3x² + 2x₀ = 1.5
x2 =
2.Easy
xyx₀ = 0.5

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0.5. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x1 =
3.Medium

The trapezoid rule is used for ∫02 sin x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 2]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = sin x[a, b] = [0, 2]ε = 0.001
n =
4.Medium
xy00.51

Approximate ∫01 √x·e−x dx with the trapezoid rule using n = 2 strips. Give T.

f(x) = √x·e−x[a, b] = [0, 1]n = 2
T =
5.Hard
00.511.512ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.5 for y′ = 1 + sin(ty), y(0) = 0.5 to estimate y3 ≈ y(1.5).

y′ = 1 + sin(ty)y(0) = 0.5h = 0.5
y3 =
6.Hard
00.050.10.150.22.62.83ty(0, 3)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = cos t − y, y(0) = 3. Find y₁ ≈ y(0.2).

y′ = cos t − yy(0) = 3h = 0.2
y1 =

Answer key

  1. 1. x2 = 1.849
  2. 2. x1 = 0.5663
  3. 3. n = 26
  4. 4. T = 0.3064
  5. 5. y3 = 2.479
  6. 6. y1 = 2.636

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