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
xya = 0b = 1

Use the bisection method on f(x) = e−x − x with the starting interval [0, 1]. Find the midpoint c4 of step 4.

f(x) = e−x − x[a, b] = [0, 1]n = 4
c4 =
2.Easy
xyx₀ = 0

Apply Newton–Raphson to f(x) = x³ + 2x − 2 with the starting value x₀ = 0. Find x2.

f(x) = x³ + 2x − 2f′(x) = 3x² + 2x₀ = 0
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 1 × 10⁻⁵.

f(x) = sin x[a, b] = [0, 1]ε = 1 × 10⁻⁵
n =
4.Medium
xy00.3750.751.1251.5

Approximate ∫01.5 sin(x²) dx with the trapezoid rule using n = 4 strips. Give T.

f(x) = sin(x²)[a, b] = [0, 1.5]n = 4
T =
5.Hard
00.050.10.150.21.81.92ty(0, 2)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = cos t − y, y(0) = 2. Find y₁ ≈ y(0.2).

y′ = cos t − yy(0) = 2h = 0.2
y1 =
6.Hard
00.51246ty(0, 2)

Use Heun's method (improved Euler) with h = 0.5 for y′ = t + y, y(0) = 2 to estimate y2 ≈ y(1).

y′ = t + yy(0) = 2h = 0.5
y2 =

Answer key

  1. 1. c4 = 0.5625
  2. 2. x2 = 0.8
  3. 3. n = 84
  4. 4. T = 0.7561
  5. 5. y1 = 1.817
  6. 6. y2 = 5.922

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