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 = 0.5

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0.5. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x1 =
2.Easy
xya = 1b = 2

Use the bisection method on f(x) = x³ + 2x − 5 with the starting interval [1, 2]. Find the midpoint c3 of step 3.

f(x) = x³ + 2x − 5[a, b] = [1, 2]n = 3
c3 =
3.Medium
t (s)05101520v (m/s)46121723

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
4.Medium
t (s)024681012v (m/s)371215201924

A vehicle's speed is measured every 2 s (table). Estimate the distance s travelled from t = 0 to t = 12 s with Simpson's rule.

h = 2 sn = 6
s = m
5.Hard
00.20.411.5ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = 1 + sin(ty), y(0) = 1. Find y₁ ≈ y(0.5).

y′ = 1 + sin(ty)y(0) = 1h = 0.5
y1 =
6.Hard
00.20.40.60.51ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.2 for y′ = 1 + sin(ty), y(0) = 0.5 to estimate y3 ≈ y(0.6).

y′ = 1 + sin(ty)y(0) = 0.5h = 0.2
y3 =

Answer key

  1. 1. x1 = 0.5663
  2. 2. c3 = 1.375
  3. 3. s = 242.5 m
  4. 4. s = 170 m
  5. 5. y1 = 1.667
  6. 6. y3 = 1.27

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