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 = 0.5

Apply Newton–Raphson to f(x) = e−x − x with the starting value x₀ = 0.5. Find x2.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 0.5
x2 =
2.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 =
3.Medium
xy00.751.5

Approximate ∫01.5 √(1 + x³) dx with Simpson's rule using n = 2 strips. Give S.

f(x) = √(1 + x³)[a, b] = [0, 1.5]n = 2
S =
4.Medium
xy11.52

Approximate ∫12 1/x dx with the trapezoid rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = 1/x[a, b] = [1, 2]n = 2
|E| =
5.Hard
01200.511.5ty(0, 0)

Use Euler's method with h = 0.5 for y′ = e−ty, y(0) = 0 to estimate y5 ≈ y(2.5).

y′ = e−tyy(0) = 0h = 0.5
y5 =
6.Hard
00.050.10.150.200.10.2ty(0, 0)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = e−ty, y(0) = 0. Find y₁ ≈ y(0.2).

y′ = e−tyy(0) = 0h = 0.2
y1 =

Answer key

  1. 1. x2 = 0.5671
  2. 2. x1 = 1.5
  3. 3. S = 1.965
  4. 4. |E| = 0.01519
  5. 5. y5 = 1.238
  6. 6. y1 = 0.1974

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