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) = cos x − x with the starting value x₀ = 1. Find x1.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 1
x1 =
2.Easy
xyx₀ = 0.5

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0.5. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x2 =
3.Medium
t (s)01234v (m/s)76799

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
4.Medium
xy11.52

Approximate ∫12 1/x dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.

f(x) = 1/x[a, b] = [1, 2]n = 2
|E| =
5.Hard
00.51123ty(0, 3)

Solve y′ = t − y, y(0) = 3 with Euler's method and step h = 0.2 up to t = 1. The exact solution is y = t − 1 + (y₀ + 1)e−t. Find the global error |E| = |y(1) − y5|.

y′ = t − yy(0) = 3h = 0.2
|E| =
6.Hard
00.050.100.050.1ty(0, 0)

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

y′ = e−tyy(0) = 0h = 0.1
y1 =

Answer key

  1. 1. x1 = 0.7504
  2. 2. x2 = 0.5671
  3. 3. s = 30 m
  4. 4. |E| = 0.001297
  5. 5. |E| = 0.1608
  6. 6. y1 = 0.09967

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