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

The bisection method starts on an interval of length 3. Every step halves the interval. How many steps n are needed at least before the interval is shorter than 0.0001?

b − a = 3ε = 0.0001

n =

2.Easy

Use linear interpolation in the table to estimate y at x = 1.65.

x = 1.65

y =

3.Medium

The trapezoid rule is used for ∫₀² x³ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 2]. Find the smallest number of strips n that guarantees an error below 0.0005.

f(x) = x³[a, b] = [0, 2]ε = 0.0005

n =

4.Medium

The trapezoid rule is used for ∫₀² eˣ dx. The error is at most (b − a)³·M/(12n²) with M = max|f″| on [0, 2]. Find the smallest number of strips n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣ[a, b] = [0, 2]ε = 1 × 10⁻⁵

n =

5.Hard

Take one step of the classical Runge–Kutta method (RK4) with h = 0.1 for y′ = 1 + sin(ty), y(0) = 1. Find y₁ ≈ y(0.1).

y′ = 1 + sin(ty)y(0) = 1h = 0.1

y₁ =

6.Hard

Use Euler's method with h = 0.25 for y′ = y − t², y(0) = 2 to estimate y₅ ≈ y(1.25).

y′ = y − t²y(0) = 2h = 0.25

y₅ =

Answer key

1. n = 15
2. y = 0.7296
3. n = 127
4. n = 702
5. y₁ = 1.105
6. y₅ = 5.55

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