Fundamentals of Engineering Exam (FE) Mathematics, Probability and Statistics Syllabus

Mathematics, Probability and Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics, Probability and Statistics in Fundamentals of Engineering Exam (FE), not a summary of it.

1. 1

   Analytic Geometry and Algebra

   5 topics

   • Algebra and Functions
     • Polynomial, rational, exponential and logarithmic functions
     • Roots, factoring and the quadratic formula
     • Partial fraction decomposition
   • Analytic Geometry
     • Lines, distance and midpoint formulas
     • Conic sections: circle, ellipse, parabola, hyperbola
   • Trigonometry
     • Identities, laws of sines and cosines
     • Radian measure and unit circle
   • Complex Numbers
     • Rectangular and polar (Euler) forms
     • De Moivre's theorem and roots
   • Vectors and Matrices
     • Dot and cross products
     • Matrix operations, determinants and inverses
     • Solving linear systems; eigenvalues and eigenvectors
2. 2

   Calculus

   4 topics

   • Differential Calculus
     • Limits, continuity and L'Hopital's rule
     • Derivative rules; implicit differentiation
     • Maxima, minima and inflection points
     • Partial derivatives and gradients
   • Integral Calculus
     • Indefinite and definite integrals
     • Techniques: substitution, parts, partial fractions
     • Applications: area, volume, centroids
   • Series and Sequences
     • Convergence tests
     • Taylor and Maclaurin series
   • Differential Equations
     • First-order linear and separable ODEs
     • Second-order linear constant-coefficient ODEs
     • Initial and boundary value problems
3. 3

   Probability and Statistics

   4 topics

   • Probability Fundamentals
     • Sample spaces, events and axioms
     • Conditional probability and Bayes' theorem
     • Combinations and permutations
   • Probability Distributions
     • Discrete: binomial, Poisson
     • Continuous: normal, exponential, uniform
     • Expected value and variance
   • Descriptive Statistics
     • Measures of central tendency and dispersion
     • Histograms and probability density functions
   • Inferential Statistics
     • Confidence intervals
     • Hypothesis testing
     • Linear regression and correlation
4. 4

   Numerical Methods

   3 topics

   • Root Finding
     • Bisection method
     • Newton-Raphson method
   • Numerical Integration
     • Trapezoidal rule
     • Simpson's rule
   • Numerical Solution of ODEs
     • Euler's method

Mathematics, Probability and Statistics flashcards for Fundamentals of Engineering Exam (FE)

25 of 58 cards from the Mathematics, Probability and Statistics deck — real questions with worked answers.

1. What is the quadratic formula for solving $ax^{2}+bx+c=0$?

   $x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

2. In the quadratic $ax^{2}+bx+c=0$, what does the discriminant $b^{2}-4ac$ tell you about the roots?

   If $b^{2}-4ac>0$ there are two distinct real roots; if $=0$ one repeated real root; if $<0$ two complex conjugate roots.

3. State the laws of exponents for $a^{m}\cdot a^{n}$ and $(a^{m})^{n}$.

   $a^{m}\cdot a^{n}=a^{m+n}$ and $(a^{m})^{n}=a^{mn}$.

4. What is the change-of-base formula for logarithms?

   $\log_{b} x = \dfrac{\ln x}{\ln b} = \dfrac{\log x}{\log b}$

5. Simplify $\log(MN)$ and $\log\!\left(\dfrac{M}{N}\right)$ using log properties.

   $\log(MN)=\log M + \log N$ and $\log\!\left(\dfrac{M}{N}\right)=\log M - \log N$.

6. What is the standard-form equation of a circle with center $(h,k)$ and radius $r$?

   $(x-h)^{2}+(y-k)^{2}=r^{2}$

7. Give the distance formula between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$.

   $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

8. What is the standard-form equation of an ellipse centered at the origin with semi-axes $a$ and $b$?

   $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$

9. State the slope-intercept and point-slope forms of a line.

   Slope-intercept: $y = mx + b$. Point-slope: $y - y_{1} = m(x - x_{1})$.

10. State the Pythagorean trig identity and its two derived forms.

    $\sin^{2}\theta + \cos^{2}\theta = 1$; dividing gives $1 + \tan^{2}\theta = \sec^{2}\theta$ and $1 + \cot^{2}\theta = \csc^{2}\theta$.

11. What are the double-angle formulas for $\sin 2\theta$ and $\cos 2\theta$?

    $\sin 2\theta = 2\sin\theta\cos\theta$; $\cos 2\theta = \cos^{2}\theta - \sin^{2}\theta = 1 - 2\sin^{2}\theta = 2\cos^{2}\theta - 1$.

12. State the Law of Cosines for a triangle with sides $a,b,c$ and angle $C$ opposite $c$.

    $c^{2} = a^{2} + b^{2} - 2ab\cos C$

13. State the Law of Sines.

    $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

14. Write a complex number in rectangular and polar (trigonometric) form.

    Rectangular: $z = a + bi$. Polar: $z = r(\cos\theta + i\sin\theta) = r e^{i\theta}$, where $r=\sqrt{a^{2}+b^{2}}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$.

15. State Euler's formula.

    $e^{i\theta} = \cos\theta + i\sin\theta$

16. State De Moivre's theorem for powers of a complex number.

    $\left[r(\cos\theta + i\sin\theta)\right]^{n} = r^{n}(\cos n\theta + i\sin n\theta)$

17. What is the complex conjugate of $z = a + bi$, and what is $z\bar{z}$?

    $\bar{z} = a - bi$, and $z\bar{z} = a^{2} + b^{2} = |z|^{2}$.

18. How do you compute the dot product of $\vec{a}=\langle a_{1},a_{2},a_{3}\rangle$ and $\vec{b}=\langle b_{1},b_{2},b_{3}\rangle$, and what does it give geometrically?

    $\vec{a}\cdot\vec{b} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} = |\vec{a}||\vec{b}|\cos\theta$; it yields a scalar and is zero when the vectors are perpendicular.

19. What does the cross product $\vec{a}\times\vec{b}$ produce and what is its magnitude?

    A vector perpendicular to both $\vec{a}$ and $\vec{b}$ with magnitude $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ (the area of the parallelogram they span).

20. How do you compute the determinant of a $2\times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

    $\det = ad - bc$

21. What condition on a square matrix's determinant guarantees it is invertible (nonsingular)?

    The matrix is invertible if and only if $\det A \neq 0$.

22. What is the inverse of a $2\times 2$ matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

    $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad-bc\neq 0$.

23. What defines an eigenvalue $\lambda$ and eigenvector $\vec{v}$ of a matrix $A$?

    They satisfy $A\vec{v} = \lambda\vec{v}$ with $\vec{v}\neq\vec{0}$; eigenvalues solve $\det(A - \lambda I) = 0$.

24. State the limit definition of the derivative.

    $f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$

25. State the product rule and quotient rule for derivatives.

    Product: $(uv)' = u'v + uv'$. Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^{2}}$.

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