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) = x² − 2 with the starting value x₀ = 1. Find x1.

f(x) = x² − 2f′(x) = 2xx₀ = 1
x1 =
2.Easy
xyx₀ = 1

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

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 1
x2 =
3.Medium

The trapezoid rule is used for ∫01 sin x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 1]. Find the smallest number of strips n that guarantees an error below 0.0005.

f(x) = sin x[a, b] = [0, 1]ε = 0.0005
n =
4.Medium
xy00.40.81.21.62

Approximate ∫02 e−x² dx with the trapezoid rule using n = 5 strips. Give T.

f(x) = e−x²[a, b] = [0, 2]n = 5
T =
5.Hard
00.20.40.60.511.5ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.25 for y′ = 1 + sin(ty), y(0) = 0.5 to estimate y3 ≈ y(0.75).

y′ = 1 + sin(ty)y(0) = 0.5h = 0.25
y3 =
6.Hard
00.20.40.60.81ty(0, 1)

Use Heun's method (improved Euler) with h = 0.2 for y′ = t − y, y(0) = 1 to estimate y3 ≈ y(0.6).

y′ = t − yy(0) = 1h = 0.2
y3 =

Answer key

  1. 1. x1 = 1.5
  2. 2. x2 = 0.7391
  3. 3. n = 12
  4. 4. T = 0.8811
  5. 5. y3 = 1.54
  6. 6. y3 = 0.7027

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