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 = 2b = 3

Use the bisection method on f(x) = x² − 5 with the starting interval [2, 3]. Find the midpoint c3 of step 3.

f(x) = x² − 5[a, b] = [2, 3]n = 3
c3 =
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
xy00.33330.66671

Approximate ∫01 1/(1 + x³) dx with the trapezoid rule using n = 3 strips. Give T.

f(x) = 1/(1 + x³)[a, b] = [0, 1]n = 3
T =
4.Medium
xy00.250.50.751

Approximate ∫01 e−x² dx with Simpson's rule using n = 4 strips. Give S.

f(x) = e−x²[a, b] = [0, 1]n = 4
S =
5.Hard
00.20.400.5ty(0, 0)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.5 for y′ = 1 + sin(ty), y(0) = 0. Find y₁ ≈ y(0.5).

y′ = 1 + sin(ty)y(0) = 0h = 0.5
y1 =
6.Hard
00.511.50.40.60.81ty(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. c3 = 2.125
  2. 2. x1 = 1.5
  3. 3. T = 0.8286
  4. 4. S = 0.7469
  5. 5. y1 = 0.5434
  6. 6. y3 = 0.75

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