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) = x³ + 2x − 10 with the starting value x₀ = 1. Find x2.

f(x) = x³ + 2x − 10f′(x) = 3x² + 2x₀ = 1
x2 =
2.Easy

The bisection method starts on an interval of length 1. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.01?

b − a = 1ε = 0.01
n =
3.Medium

The trapezoid rule is used for ∫12 1/x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [1, 2]. Find the smallest number of strips n that guarantees an error below 0.0005.

f(x) = 1/x[a, b] = [1, 2]ε = 0.0005
n =
4.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| =
5.Hard
00.050.10.150.211.11.21.3ty(0, 1)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = t² + y², y(0) = 1. Find y₁ ≈ y(0.2).

y′ = t² + y²y(0) = 1h = 0.2
y1 =
6.Hard
00.10.20.30.40.450.5ty(0, 0.5)

Use Euler's method with h = 0.1 for y′ = −2ty, y(0) = 0.5 to estimate y4 ≈ y(0.4).

y′ = −2tyy(0) = 0.5h = 0.1
y4 =

Answer key

  1. 1. x2 = 1.953
  2. 2. n = 7
  3. 3. n = 19
  4. 4. |E| = 1
  5. 5. y1 = 1.253
  6. 6. y4 = 0.4422

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