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 = 2.5

Apply Newton–Raphson to f(x) = ln x + x − 4 with the starting value x₀ = 2.5. Find x1.

f(x) = ln x + x − 4f′(x) = 1/x + 1x₀ = 2.5
x1 =
2.Easy
xyx₀ = 2.5

Apply Newton–Raphson to f(x) = ln x + x − 4 with the starting value x₀ = 2.5. Find x2.

f(x) = ln x + x − 4f′(x) = 1/x + 1x₀ = 2.5
x2 =
3.Medium
xy11.752.53.254

Approximate ∫14 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, 4]n = 4
|E| =
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 0.0005.

f(x) = x³[a, b] = [0, 2]ε = 0.0005
n =
5.Hard
00.20.411.52ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = t² + y², y(0) = 1. Find y₁ ≈ y(0.5).

y′ = t² + y²y(0) = 1h = 0.5
y1 =
6.Hard
00.050.10.150.20.9991ty(0, 1)

Use Heun's method (improved Euler) with h = 0.1 for y′ = cos t − y, y(0) = 1 to estimate y2 ≈ y(0.2).

y′ = cos t − yy(0) = 1h = 0.1
y2 =

Answer key

  1. 1. x1 = 2.917
  2. 2. x2 = 2.926
  3. 3. |E| = 0.0418
  4. 4. n = 127
  5. 5. y1 = 2.055
  6. 6. y2 = 0.9986

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