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

Use the bisection method on f(x) = x³ − x − 5 with the starting interval [1, 3]. Find the midpoint c₃ of step 3.

f(x) = x³ − x − 5[a, b] = [1, 3]n = 3

c₃ =

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 0.005?

b − a = 4ε = 0.005

n =

3.Medium

Approximate ∫₀¹ √(1 + x³) dx with Simpson's rule using n = 2 strips. Give S.

f(x) = √(1 + x³)[a, b] = [0, 1]n = 2

S =

4.Medium

Approximate ∫₀^1.5 e^−x² dx with Simpson's rule using n = 4 strips. Give S.

f(x) = e^−x²[a, b] = [0, 1.5]n = 4

S =

5.Hard

Solve y′ = −2ty, y(0) = 0.5 with Euler's method and step h = 0.5 up to t = 1.5. The exact solution is y = y₀e^−t². Find the global error |E| = |y(1.5) − y₃|.

y′ = −2tyy(0) = 0.5h = 0.5

|E| =

6.Hard

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

y′ = e^−tyy(0) = 0h = 0.5

y₁ =

Answer key

1. c₃ = 1.75
2. n = 10
3. S = 1.109
4. S = 0.8561
5. |E| = 0.0527
6. y₁ = 0.4629

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