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 = 0b = 1

Use the bisection method on f(x) = e−x − x with the starting interval [0, 1]. Find the midpoint c3 of step 3.

f(x) = e−x − x[a, b] = [0, 1]n = 3
c3 =
2.Easy
xyx₀ = 1.5

Apply Newton–Raphson to f(x) = x³ + 2x − 7 with the starting value x₀ = 1.5. Find x1.

f(x) = x³ + 2x − 7f′(x) = 3x² + 2x₀ = 1.5
x1 =
3.Medium
xy00.33330.666711.3331.6672

Approximate ∫02 √x·e−x dx with the trapezoid rule using n = 6 strips. Give T.

f(x) = √x·e−x[a, b] = [0, 2]n = 6
T =
4.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 =
5.Hard
00.050.10.150.200.10.2ty(0, 0)

Take one step of the classical Runge–Kutta method (RK4) with h = 0.2 for y′ = e−ty, y(0) = 0. Find y₁ ≈ y(0.2).

y′ = e−tyy(0) = 0h = 0.2
y1 =
6.Hard
00.510.30.40.50.6ty(0, 0.5)

Solve y′ = t − y, y(0) = 0.5 with Euler's method and step h = 0.25 up to t = 1. The exact solution is y = t − 1 + (y₀ + 1)e−t. Find the global error |E| = |y(1) − y4|.

y′ = t − yy(0) = 0.5h = 0.25
|E| =

Answer key

  1. 1. c3 = 0.625
  2. 2. x1 = 1.571
  3. 3. T = 0.6149
  4. 4. n = 127
  5. 5. y1 = 0.1974
  6. 6. |E| = 0.07721

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