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) = ln x + x − 2 with the starting value x₀ = 2. Find x2.

f(x) = ln x + x − 2f′(x) = 1/x + 1x₀ = 2
x2 =
2.Easy
xyx₀ = 0.5

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

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

The trapezoid rule is used for ∫13 1/x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [1, 3]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = 1/x[a, b] = [1, 3]ε = 0.001
n =
4.Medium
xy00.51

Approximate ∫01 sin x dx with the trapezoid rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = sin x[a, b] = [0, 1]n = 2
|E| =
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
00.050.10.50.520.54ty(0, 0.5)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.1 for y′ = cos t − y, y(0) = 0.5. Find y₁ ≈ y(0.1).

y′ = cos t − yy(0) = 0.5h = 0.1
y1 =

Answer key

  1. 1. x2 = 1.557
  2. 2. x1 = 0.7552
  3. 3. n = 37
  4. 4. |E| = 0.009617
  5. 5. y1 = 2.055
  6. 6. y1 = 0.5474

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