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
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 =
2.Easy
xya = 1b = 4

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

f(x) = ln x + x − 4[a, b] = [1, 4]n = 4
c4 =
3.Medium
xy00.20.40.60.81

Approximate ∫01 sin(x²) dx with the trapezoid rule using n = 5 strips. Give T.

f(x) = sin(x²)[a, b] = [0, 1]n = 5
T =
4.Medium

The trapezoid rule is used for ∫02 eˣ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 2]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = eˣ[a, b] = [0, 2]ε = 0.001
n =
5.Hard
00.050.10.150.211.11.2ty(0, 1)

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

y′ = e−tyy(0) = 1h = 0.2
y1 =
6.Hard
00.050.111.051.1ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.1 for y′ = 1 + sin(ty), y(0) = 1. Find y₁ ≈ y(0.1).

y′ = 1 + sin(ty)y(0) = 1h = 0.1
y1 =

Answer key

  1. 1. c3 = 1.375
  2. 2. c4 = 3.063
  3. 3. T = 0.3139
  4. 4. n = 71
  5. 5. y1 = 1.179
  6. 6. y1 = 1.105

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