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

The bisection method starts on an interval of length 4. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.001?

b − a = 4ε = 0.001
n =
2.Easy
xyx₀ = 2

Apply Newton–Raphson to f(x) = x² − 3 with the starting value x₀ = 2. Find x2.

f(x) = x² − 3f′(x) = 2xx₀ = 2
x2 =
3.Medium

The trapezoid rule is used for ∫02 eˣ 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) = eˣ[a, b] = [0, 2]ε = 0.001
n =
4.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 0.0005.

f(x) = 1/x[a, b] = [1, 3]ε = 0.0005
n =
5.Hard
00.050.10.150.20.50.550.6ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.1 for y′ = y − t², y(0) = 0.5 to estimate y2 ≈ y(0.2).

y′ = y − t²y(0) = 0.5h = 0.1
y2 =
6.Hard
00.20.40.600.020.04ty(0, 0)

Use Euler's method with h = 0.2 for y′ = t² + y², y(0) = 0 to estimate y3 ≈ y(0.6).

y′ = t² + y²y(0) = 0h = 0.2
y3 =

Answer key

  1. 1. n = 12
  2. 2. x2 = 1.732
  3. 3. n = 71
  4. 4. n = 52
  5. 5. y2 = 0.6074
  6. 6. y3 = 0.04001

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