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 = 2

Apply Newton–Raphson to f(x) = x³ + 2x − 3 with the starting value x₀ = 2. Find x2.

f(x) = x³ + 2x − 3f′(x) = 3x² + 2x₀ = 2
x2 =
2.Easy
x00.511.522.5y0.69310.91631.09861.25281.38631.5041

Use linear interpolation in the table to estimate y at x = 0.875.

x = 0.875
y =
3.Medium
xy11.752.53.254

Approximate ∫14 1/x dx with the trapezoid rule, n = 4, and compare it with the exact value. Find the absolute error |E| = |T − I|.

f(x) = 1/x[a, b] = [1, 4]n = 4
|E| =
4.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 e−x² dx with Simpson's rule using n = 6 strips. Give S.

f(x) = e−x²[a, b] = [0, 1]n = 6
S =
5.Hard
00.20.40.50.60.7ty(0, 0.5)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = t² + y², y(0) = 0.5. Find y₁ ≈ y(0.5).

y′ = t² + y²y(0) = 0.5h = 0.5
y1 =
6.Hard
00.511234ty(0, 1)

Solve y′ = t + y, y(0) = 1 with Euler's method and step h = 0.2 up to t = 1. The exact solution is y = −t − 1 + (y₀ + 1)eᵗ. Find the global error |E| = |y(1) − y5|.

y′ = t + yy(0) = 1h = 0.2
|E| =

Answer key

  1. 1. x2 = 1.063
  2. 2. y = 1.053
  3. 3. |E| = 0.0418
  4. 4. S = 0.7468
  5. 5. y1 = 0.716
  6. 6. |E| = 0.4599

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