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) = ln x + x − 4 with the starting value x₀ = 1. Find x2.

f(x) = ln x + x − 4f′(x) = 1/x + 1x₀ = 1
x2 =
2.Easy
xya = 1b = 4

Use the bisection method on f(x) = ln x + x − 4 with the starting interval [1, 4]. Find the midpoint c3 of step 3.

f(x) = ln x + x − 4[a, b] = [1, 4]n = 3
c3 =
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 0.0001.

f(x) = sin x[a, b] = [0, 2]ε = 0.0001
n =
4.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 =
5.Hard
00.20.40.60.51ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.25 for y′ = e−ty, y(0) = 0.5 to estimate y3 ≈ y(0.75).

y′ = e−tyy(0) = 0.5h = 0.25
y3 =
6.Hard
00.511.5212ty(0, 0.5)

Solve y′ = y(2 − y), y(0) = 0.5 with Euler's method and step h = 0.5 up to t = 2. The exact solution is y = 2y₀/(y₀ + (2 − y₀)e−2t). Find the global error |E| = |y(2) − y4|.

y′ = y(2 − y)y(0) = 0.5h = 0.5
|E| =

Answer key

  1. 1. x2 = 2.917
  2. 2. c3 = 2.875
  3. 3. n = 82
  4. 4. n = 130
  5. 5. y3 = 1.046
  6. 6. |E| = 0.08412

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