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 − x − 5 with the starting value x₀ = 1. Find x1.

f(x) = x³ − x − 5f′(x) = 3x² − 1x₀ = 1
x1 =
2.Easy
xya = 0b = 1

Use the bisection method on f(x) = cos x − x with the starting interval [0, 1]. Find the midpoint c3 of step 3.

f(x) = cos x − x[a, b] = [0, 1]n = 3
c3 =
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 1 × 10⁻⁵.

f(x) = 1/x[a, b] = [1, 3]ε = 1 × 10⁻⁵
n =
4.Medium
t (s)02468v (m/s)364812

A vehicle's speed is measured every 2 s (table). Estimate the distance s travelled from t = 0 to t = 8 s with Simpson's rule.

h = 2 sn = 4
s = m
5.Hard
00.20.411.52ty(0, 2)

Use Euler's method with h = 0.1 for y′ = −y, y(0) = 2 to estimate y5 ≈ y(0.5).

y′ = −yy(0) = 2h = 0.1
y5 =
6.Hard
00.511.5−10ty(0, −1)

Use Euler's method with h = 0.5 for y′ = t − y, y(0) = −1 to estimate y3 ≈ y(1.5).

y′ = t − yy(0) = −1h = 0.5
y3 =

Answer key

  1. 1. x1 = 3.5
  2. 2. c3 = 0.625
  3. 3. n = 366
  4. 4. s = 52.67 m
  5. 5. y5 = 1.181
  6. 6. y3 = 0.5

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