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) = e−x − x with the starting value x₀ = 1. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 1
x2 =
2.Easy
xyx₀ = 4

Apply Newton–Raphson to f(x) = ln x + x − 4 with the starting value x₀ = 4. Find x2.

f(x) = ln x + x − 4f′(x) = 1/x + 1x₀ = 4
x2 =
3.Medium

The trapezoid rule is used for ∫13 1/x dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [1, 3]. Find the smallest number of strips n that guarantees an error below 0.001.

f(x) = 1/x[a, b] = [1, 3]ε = 0.001
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
00.050.10.150.20.50.60.7ty(0, 0.5)

Use Heun's method (improved Euler) with h = 0.1 for y′ = e−ty, y(0) = 0.5 to estimate y2 ≈ y(0.2).

y′ = e−tyy(0) = 0.5h = 0.1
y2 =
6.Hard
00.51123ty(0, 3)

Solve y′ = t − y, y(0) = 3 with Euler's method and step h = 0.25 up to t = 1. The exact solution is y = t − 1 + (y₀ + 1)e−t. Find the global error |E| = |y(1) − y4|.

y′ = t − yy(0) = 3h = 0.25
|E| =

Answer key

  1. 1. x2 = 0.567
  2. 2. x2 = 2.926
  3. 3. n = 37
  4. 4. |E| = 1
  5. 5. y2 = 0.6877
  6. 6. |E| = 0.2059

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