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) = ln x + x − 2 with the starting value x₀ = 1.5. Find x1.

f(x) = ln x + x − 2f′(x) = 1/x + 1x₀ = 1.5
x1 =
2.Easy
xya = 0b = 1

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

f(x) = cos x − x[a, b] = [0, 1]n = 4
c4 =
3.Medium
xy00.51

Approximate ∫01 eˣ dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.

f(x) = eˣ[a, b] = [0, 1]n = 2
|E| =
4.Medium
t (s)05101520v (m/s)1015182225

A vehicle's speed is measured every 5 s (table). Estimate the distance s travelled from t = 0 to t = 20 s with the trapezoid rule.

h = 5 sn = 4
s = m
5.Hard
00.5111.52ty(0, 1)

Use Euler's method with h = 0.2 for y′ = y(2 − y), y(0) = 1 to estimate y5 ≈ y(1).

y′ = y(2 − y)y(0) = 1h = 0.2
y5 =
6.Hard
00.050.10.150.2−1−0.8−0.6ty(0, −1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = cos t − y, y(0) = −1. Find y₁ ≈ y(0.2).

y′ = cos t − yy(0) = −1h = 0.2
y1 =

Answer key

  1. 1. x1 = 1.557
  2. 2. c4 = 0.6875
  3. 3. |E| = 0.0005793
  4. 4. s = 362.5 m
  5. 5. y5 = 1.801
  6. 6. y1 = −0.6387

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