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 = 0b = 2

Use the bisection method on f(x) = x³ + 2x − 2 with the starting interval [0, 2]. Find the midpoint c4 of step 4.

f(x) = x³ + 2x − 2[a, b] = [0, 2]n = 4
c4 =
2.Easy
xyx₀ = 1

Apply Newton–Raphson to f(x) = cos x − x with the starting value x₀ = 1. Find x1.

f(x) = cos x − xf′(x) = −sin x − 1x₀ = 1
x1 =
3.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| =
4.Medium
xy012

Approximate ∫02 √x·e−x dx with Simpson's rule using n = 2 strips. Give S.

f(x) = √x·e−x[a, b] = [0, 2]n = 2
S =
5.Hard
00.10.20.30.42.62.83ty(0, 3)

Solve y′ = −2ty, y(0) = 3 with Euler's method and step h = 0.1 up to t = 0.4. The exact solution is y = y₀e−t². Find the global error |E| = |y(0.4) − y4|.

y′ = −2tyy(0) = 3h = 0.1
|E| =
6.Hard
00.511.520123ty(0, 0)

Use Euler's method with h = 0.5 for y′ = 1 + sin(ty), y(0) = 0 to estimate y4 ≈ y(2).

y′ = 1 + sin(ty)y(0) = 0h = 0.5
y4 =

Answer key

  1. 1. c4 = 0.875
  2. 2. x1 = 0.7504
  3. 3. |E| = 1
  4. 4. S = 0.5543
  5. 5. |E| = 0.09662
  6. 6. y4 = 2.589

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