Engineering Mathematics: practice problems and exam questions

The mathematics that engineering courses build on: calculus, complex numbers, matrices, probability and statistics.

Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.

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

Topics in this course

  1. Limitsdirect substitution and factoring · limits at infinity · L'Hôpital's rulecalculations
  2. Derivativespower rule · chain and product rules · tangents and extremacalculations
  3. Integralspolynomials · exponentials, logs and substitution · areas between curvescalculations
  4. Complex numbersarithmetic in rectangular form · polar and exponential form · De Moivre, roots and equationscalculations
  5. Differential equationsexponential growth and decay · Newton's law of cooling · first-order linearcalculations
  6. Matrices & determinants2×2 determinant · 3×3 and products · eigenvalues and systemscalculations
  7. Vectors: dot & cross productmagnitude and components · dot product · cross product and momentscalculations
  8. Probabilityequally likely outcomes · counting and complements · binomial distributioncalculations
  9. Descriptive statisticsmean and median · standard deviation · normal distributioncalculations
  10. Normal distributionarea in one tail · area between two values · percentilescalculations

Exam tips for Engineering Mathematics

  • Check the dimensions of a matrix product before you multiply.
  • A determinant of zero means the matrix has no inverse.
  • Sketch a function before you differentiate or integrate it; the sketch catches sign errors.

Worked examples

LimitsEasy
xyx = 1

Find the limit L = limx→1 (x4 − 1)/(x − 1).

f(x) = (x4 − 1)/(x − 1)x → 1

Find L

Show hints
  1. Substitution gives 0/0. Factor the numerator.
  2. xⁿ − aⁿ = (x − a)(xⁿ⁻¹ + a·xⁿ⁻² + … + aⁿ⁻¹).
  3. x4 − 1 = (x − 1)(x3 + x2 + x + 1).
Show the answer and the step-by-step solution

Limits: first try direct substitution. If it gives 0/0, simplify first: factor and cancel the common factor, or multiply by the conjugate. As x → ∞, divide numerator and denominator by the highest power of x. L'Hôpital's rule: if f/g gives 0/0 or ∞/∞, then lim f/g = lim f′/g′, differentiating the numerator and the denominator separately (not with the quotient rule).

Try direct substitution

numerator → 0 and denominator → 0, so the form is 0/0.

Factor and cancel

(x4 − 1)/(x − 1) = x3 + x2 + x + 1

Substitute

L = 1·13 + 1·12 + 1·1 + 1·1 = 4

Answer

L = 4

This is the derivative of xⁿ at x = a: n·aⁿ⁻¹.

DerivativesMedium
xyx = 1

Let f(x) = x3·e−x. Find f′(1).

f(x) = x3·e−xx = 1

Find f′(1)

Show hints
  1. Product rule with u = x3 and v = e−x.
  2. u′ = 3x2, v′ = −1·e−x.
  3. f′(x) = e−x·(3x2 + (−1)·x3).
Show the answer and the step-by-step solution

Differentiation: the derivative f′(x) is the slope of the graph at x. Power rule: (xⁿ)′ = n·xⁿ⁻¹. Product rule: (uv)′ = u′v + uv′. Chain rule: (f(g(x)))′ = f′(g(x))·g′(x). Where f′(x) = 0 the graph has a horizontal tangent: a maximum, minimum or saddle point.

Product rule

f′(x) = u′v + uv′ = e−x·(3x2 + (−1)·x3)

Substitute

f′(1) = 0.3679·(3 + (−1)) = 0.7358

Answer

f′(1) = 0.7358

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

Z-score and normal probability calculatorMean, median and standard deviation calculatorMatrix determinant and inverse calculatorNormal distribution calculatorBinomial distribution calculatorPoisson distribution calculatorPermutation and combination calculator (nPr and nCr)Eigenvalue and eigenvector calculatorExpected value and variance calculatorHalf-life and exponential decay calculatorSignificant figures calculator (count, round and calculate)Percent error calculator (and percent difference)Quadratic equation solver (ax² + bx + c = 0)System of linear equations solver (2×2 and 3×3)Derivative calculator with stepsIntegral calculator (definite and indefinite)

Related courses

Engineering Mathematics 2AP Calculus practice problemsCalculus 1Introductory StatisticsDifferential EquationsAP Statistics practice problemsResearch Methods and Statistics for PsychologyCFA Level I quantitative methods and fixed income practice
Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.