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) = e−x − x with the starting value x₀ = 1. Find x1.

f(x) = e−x − xf′(x) = −e−x − 1x₀ = 1
x1 =
2.Easy

The bisection method starts on an interval of length 4. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.01?

b − a = 4ε = 0.01
n =
3.Medium
xy00.16670.33330.50.66670.83331

Approximate ∫01 e−x² dx with the trapezoid rule using n = 6 strips. Give T.

f(x) = e−x²[a, b] = [0, 1]n = 6
T =
4.Medium
xy00.51

Approximate ∫01 eˣ dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.

f(x) = eˣ[a, b] = [0, 1]n = 2
|E| =
5.Hard
00.10.20.3−1.5−1ty(0, −1)

Use Heun's method (improved Euler) with h = 0.1 for y′ = 1.5y, y(0) = −1 to estimate y3 ≈ y(0.3).

y′ = 1.5yy(0) = −1h = 0.1
y3 =
6.Hard
00.20.40.40.60.81ty(0, 0.5)

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

y′ = e−tyy(0) = 0.5h = 0.5
y1 =

Answer key

  1. 1. x1 = 0.5379
  2. 2. n = 9
  3. 3. T = 0.7451
  4. 4. |E| = 0.0005793
  5. 5. y3 = −1.566
  6. 6. y1 = 0.9138

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