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 x1.

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

Apply Newton–Raphson to f(x) = x³ − x − 3 with the starting value x₀ = 2. Find x2.

f(x) = x³ − x − 3f′(x) = 3x² − 1x₀ = 2
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 0.0001.

f(x) = sin x[a, b] = [0, 1]ε = 0.0001
n =
4.Medium
xy00.3750.751.1251.5

Approximate ∫01.5 e−x² dx with the trapezoid rule using n = 4 strips. Give T.

f(x) = e−x²[a, b] = [0, 1.5]n = 4
T =
5.Hard
00.10.20.30.411.2ty(0, 1)

Use Heun's method (improved Euler) with h = 0.2 for y′ = e−ty, y(0) = 1 to estimate y2 ≈ y(0.4).

y′ = e−tyy(0) = 1h = 0.2
y2 =
6.Hard
00.511.5123ty(0, 3)

Use Euler's method with h = 0.5 for y′ = cos t − y, y(0) = 3 to estimate y3 ≈ y(1.5).

y′ = cos t − yy(0) = 3h = 0.5
y3 =

Answer key

  1. 1. x1 = 1.538
  2. 2. x2 = 1.674
  3. 3. n = 27
  4. 4. T = 0.8525
  5. 5. y2 = 1.316
  6. 6. y3 = 0.9895

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