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) = x² − 2 with the starting value x₀ = 1. Find x1.

f(x) = x² − 2f′(x) = 2xx₀ = 1
x1 =
2.Easy
xya = 0b = 1

Use the bisection method on f(x) = e−x − x with the starting interval [0, 1]. Find the midpoint c4 of step 4.

f(x) = e−x − x[a, b] = [0, 1]n = 4
c4 =
3.Medium
xy00.33330.66671

Approximate ∫01 √x·e−x dx with the trapezoid rule using n = 3 strips. Give T.

f(x) = √x·e−x[a, b] = [0, 1]n = 3
T =
4.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 √x·e−x dx with the trapezoid rule using n = 6 strips. Give T.

f(x) = √x·e−x[a, b] = [0, 1]n = 6
T =
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′ = 1 + sin(ty), y(0) = 0.5 to estimate y2 ≈ y(0.2).

y′ = 1 + sin(ty)y(0) = 0.5h = 0.1
y2 =
6.Hard
00.51−1−0.5ty(0, −1)

Solve y′ = −2ty, y(0) = −1 with Euler's method and step h = 0.2 up to t = 1. The exact solution is y = y₀e−t². Find the global error |E| = |y(1) − y5|.

y′ = −2tyy(0) = −1h = 0.2
|E| =

Answer key

  1. 1. x1 = 1.5
  2. 2. c4 = 0.5625
  3. 3. T = 0.3389
  4. 4. T = 0.3647
  5. 5. y2 = 0.7131
  6. 6. |E| = 0.0315

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