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 = 1.5

Apply Newton–Raphson to f(x) = x² − 2 with the starting value x₀ = 1.5. Find x1.

f(x) = x² − 2f′(x) = 2xx₀ = 1.5
x1 =
2.Easy
xya = 1b = 3

Use the bisection method on f(x) = x³ − x − 5 with the starting interval [1, 3]. Find the midpoint c4 of step 4.

f(x) = x³ − x − 5[a, b] = [1, 3]n = 4
c4 =
3.Medium
xy00.511.52

Approximate ∫02 1/(1 + x³) dx with the trapezoid rule using n = 4 strips. Give T.

f(x) = 1/(1 + x³)[a, b] = [0, 2]n = 4
T =
4.Medium
t (s)01234v (m/s)410151311

A vehicle's speed is measured every 1 s (table). Estimate the distance s travelled from t = 0 to t = 4 s with the trapezoid rule.

h = 1 sn = 4
s = m
5.Hard
00.10.20.30.40.60.8ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.2 for y′ = e−ty, y(0) = 0.5 to estimate y2 ≈ y(0.4).

y′ = e−tyy(0) = 0.5h = 0.2
y2 =
6.Hard
00.050.11.91.952ty(0, 2)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.1 for y′ = cos t − y, y(0) = 2. Find y₁ ≈ y(0.1).

y′ = cos t − yy(0) = 2h = 0.1
y1 =

Answer key

  1. 1. x1 = 1.417
  2. 2. c4 = 1.875
  3. 3. T = 1.087
  4. 4. s = 45.5 m
  5. 5. y2 = 0.845
  6. 6. y1 = 1.905

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