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

The bisection method starts on an interval of length 0.5. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.0001?

b − a = 0.5ε = 0.0001
n =
2.Easy
xyx₀ = 3

Apply Newton–Raphson to f(x) = ln x + x − 3 with the starting value x₀ = 3. Find x1.

f(x) = ln x + x − 3f′(x) = 1/x + 1x₀ = 3
x1 =
3.Medium
xy012

Approximate ∫02 x³ dx with the trapezoid rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = x³[a, b] = [0, 2]n = 2
|E| =
4.Medium
xy00.751.52.253

Approximate ∫03 √x·e−x dx with the trapezoid rule using n = 4 strips. Give T.

f(x) = √x·e−x[a, b] = [0, 3]n = 4
T =
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
00.10.20.30.60.8ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.1 for y′ = e−ty, y(0) = 0.5 to estimate y3 ≈ y(0.3).

y′ = e−tyy(0) = 0.5h = 0.1
y3 =

Answer key

  1. 1. n = 13
  2. 2. x1 = 2.176
  3. 3. |E| = 1
  4. 4. T = 0.6627
  5. 5. y1 = 1.667
  6. 6. y3 = 0.7709

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