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

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 1. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 1
x2 =
2.Easy

The bisection method starts on an interval of length 3. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.005?

b − a = 3ε = 0.005
n =
3.Medium

The trapezoid rule is used for ∫12 1/x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [1, 2]. Find the smallest number of strips n that guarantees an error below 1 × 10⁻⁵.

f(x) = 1/x[a, b] = [1, 2]ε = 1 × 10⁻⁵
n =
4.Medium

The trapezoid rule is used for ∫02 x³ 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) = x³[a, b] = [0, 2]ε = 0.001
n =
5.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 =
6.Hard
00.050.10.150.211.11.21.3ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = t² + y², y(0) = 1. Find y₁ ≈ y(0.2).

y′ = t² + y²y(0) = 1h = 0.2
y1 =

Answer key

  1. 1. x2 = 0.567
  2. 2. n = 10
  3. 3. n = 130
  4. 4. n = 90
  5. 5. y1 = 2.636
  6. 6. y1 = 1.253

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