Numerical Methods: practice problems and exam questions

Computing answers that have no formula: roots of equations with bisection and Newton–Raphson, interpolation in tables, numerical integration with the trapezoid and Simpson rules, and differential equations with Euler, Heun and Runge–Kutta. Exams ask for a few iterations in a table and for the size of the error.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Numerical methodsroots and interpolation · trapezoid and Simpson rules · Euler, Heun and Runge–Kuttacalculations
  2. Sequences & seriesarithmetic and geometric series · convergence and power series · Taylor and Maclaurin polynomialscalculations
  3. Matrices & determinants2×2 determinant · 3×3 and products · eigenvalues and systemscalculations
  4. Differential equationsexponential growth and decay · Newton's law of cooling · first-order linearcalculations

Exam tips for Numerical Methods

  • Keep at least six significant digits in every iterate and round only the final answer.
  • Write the weights of the integration rule under the table of values before you add anything up.
  • Use Taylor polynomials to explain the error: Euler’s error is proportional to h, Simpson’s to h⁴.

Worked examples

Numerical methodsEasy

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 0.0001?

b − a = 4ε = 0.0001

Find n

Show hints
  1. After n steps the interval has length (b − a)/2ⁿ.
  2. Solve 4/2ⁿ < 0.0001, so 2ⁿ > 40000.
  3. n > log₂(40000) = 15.29
Show the answer and the step-by-step solution

Root finding and interpolation: bisection keeps an interval [a, b] on which f changes sign; each step takes the midpoint c = (a + b)/2 and keeps the half where the sign still changes, so after n steps the interval is (b − a)/2ⁿ wide. Newton–Raphson follows the tangent line to the axis: xn+1 = xn − f(xn)/f′(xn); near a simple root it roughly doubles the number of correct digits per step. Linear interpolation between two table points: y = y₁ + (y₂ − y₁)(x − x₁)/(x₂ − x₁).

Condition

4/2ⁿ < 0.0001 → 2ⁿ > 40000

Logarithm

n > ln(40000)/ln 2 = 15.29
n = 16

Check

2¹⁵ = 32768 < 40000 < 2¹⁶ = 65536

Answer

n = 16
Sequences & seriesMedium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 12/(k(k + 3)).

ak = 12/(k(k + 3))

Find S

Show hints
  1. Partial fractions: 1/(k(k + 3)) = (1/3)·(1/k − 1/(k + 3)).
  2. Write out the first few terms: most of them cancel (a telescoping sum).
  3. What remains is (12/3)·(1 + … + 1/3).
Show the answer and the step-by-step solution

Convergence: if the terms do not tend to 0 the series diverges (n-th term test). Σ1/kᵖ converges only for p > 1, and Σrᵏ only for |r| < 1. Compare a series with one of these when its terms behave like them for large k. Ratio test: L = lim |ak+1/ak|; root test: L = lim |ak|1/k. L < 1 converges, L > 1 diverges, L = 1 decides nothing. A power series Σck(x − x₀)ᵏ converges for |x − x₀| < R with R = lim |ck/ck+1|; an alternating series with terms decreasing to 0 converges.

Partial fractions

12/(k(k + 3)) = (12/3)·(1/k − 1/(k + 3))

Cancel

S = (12/3)·(1 + 1/2 + 1/3) = 7.333

Answer

S = 7.333

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Matrix determinant and inverse calculatorEigenvalue and eigenvector calculatorHalf-life and exponential decay calculatorSystem of linear equations solver (2×2 and 3×3)

Related courses

Engineering Mathematics 2Engineering MathematicsDifferential EquationsCalculus 2AP Calculus practice problemsCalculus 1Introductory StatisticsStatistics 2: Statistical Inference
Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.