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

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

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

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 $ad-bc\neq 0$.

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 $\det(A - \lambda I) = 0$.

State the limit definition of the derivative.

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

State the product rule and quotient rule for derivatives.

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

State the chain rule for $\dfrac{d}{dx}f(g(x))$.

$\dfrac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$

What are the derivatives of $\sin x$, $\cos x$, $e^{x}$, and $\ln x$?

$\frac{d}{dx}\sin x = \cos x$; $\frac{d}{dx}\cos x = -\sin x$; $\frac{d}{dx}e^{x} = e^{x}$; $\frac{d}{dx}\ln x = \frac{1}{x}$.

How do you classify a critical point using the second derivative test?

At a critical point where $f'(x)=0$: if $f''(x)>0$ it is a local minimum; if $f''(x)<0$ a local maximum; if $f''(x)=0$ the test is inconclusive.

State L'Hopital's rule for an indeterminate limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

$\lim_{x\to a}\dfrac{f(x)}{g(x)} = \lim_{x\to a}\dfrac{f'(x)}{g'(x)}$, provided the latter limit exists.

State the power rule for integration.

$\displaystyle\int x^{n}\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

State the Fundamental Theorem of Calculus (evaluation form).

$\displaystyle\int_{a}^{b} f(x)\,dx = F(b) - F(a)$, where $F'(x) = f(x)$.

State the integration by parts formula.

$\displaystyle\int u\,dv = uv - \int v\,du$

What integral gives the average value of $f(x)$ over $[a,b]$?

$f_{\text{avg}} = \dfrac{1}{b-a}\displaystyle\int_{a}^{b} f(x)\,dx$

State the formula for the sum of a finite arithmetic series of $n$ terms.

$S_{n} = \dfrac{n}{2}(a_{1} + a_{n}) = \dfrac{n}{2}\left[2a_{1} + (n-1)d\right]$

State the sum of a finite geometric series and the condition for the infinite sum to converge.

$S_{n} = a_{1}\dfrac{1-r^{n}}{1-r}$; the infinite series converges to $\dfrac{a_{1}}{1-r}$ when $|r| < 1$.

Write the Taylor series expansion of $f(x)$ about $x=a$.

$f(x) = \displaystyle\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x-a)^{n}$

What distinguishes a first-order linear ODE, and what is its integrating factor?

It has the form $\dfrac{dy}{dx} + P(x)y = Q(x)$; the integrating factor is $\mu(x) = e^{\int P(x)\,dx}$.

For the second-order linear homogeneous ODE $ay''+by'+cy=0$, what is the characteristic equation?

$ar^{2} + br + c = 0$; its roots determine the form of the general solution.

For $ay''+by'+cy=0$ with distinct real roots $r_{1}, r_{2}$, what is the general solution?

$y = C_{1}e^{r_{1}x} + C_{2}e^{r_{2}x}$

How do you separate variables to solve $\dfrac{dy}{dx} = g(x)h(y)$?

Rewrite as $\dfrac{dy}{h(y)} = g(x)\,dx$, then integrate both sides: $\displaystyle\int \dfrac{dy}{h(y)} = \int g(x)\,dx + C$.

State the addition rule of probability for two events $A$ and $B$.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$; for mutually exclusive events the last term is $0$.

Define conditional probability and the multiplication rule.

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, so $P(A \cap B) = P(A \mid B)\,P(B)$. If independent, $P(A\cap B)=P(A)P(B)$.

State Bayes' theorem.

$P(A \mid B) = \dfrac{P(B \mid A)\,P(A)}{P(B)}$

Give the formulas for permutations and combinations of $n$ items taken $r$ at a time.

Permutations: $P(n,r) = \dfrac{n!}{(n-r)!}$. Combinations: $C(n,r) = \dfrac{n!}{r!\,(n-r)!}$.

State the probability mass function, mean, and variance of the binomial distribution.

$P(X=k) = \binom{n}{k}p^{k}(1-p)^{n-k}$, with mean $\mu = np$ and variance $\sigma^{2} = np(1-p)$.

State the mean and variance of a Poisson distribution with parameter $\lambda$.

Both equal $\lambda$: $\mu = \lambda$ and $\sigma^{2} = \lambda$, with $P(X=k)=\dfrac{\lambda^{k}e^{-\lambda}}{k!}$.

How is a value $x$ converted to a standard normal z-score?

$z = \dfrac{x - \mu}{\sigma}$

Define sample mean and sample variance for data $x_{1},\dots,x_{n}$.

$\bar{x} = \dfrac{1}{n}\displaystyle\sum_{i=1}^{n} x_{i}$ and $s^{2} = \dfrac{1}{n-1}\displaystyle\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$.

Compare the mean, median, and mode as measures of central tendency.

The mean is the arithmetic average (sensitive to outliers); the median is the middle value (robust to outliers); the mode is the most frequent value.

What does the Central Limit Theorem state about the sampling distribution of the sample mean?

For large $n$, the sample mean $\bar{x}$ is approximately normally distributed with mean $\mu$ and standard error $\dfrac{\sigma}{\sqrt{n}}$, regardless of the population's distribution.

Write the general form of a confidence interval for a population mean (known $\sigma$).

$\bar{x} \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$

In hypothesis testing, what are Type I and Type II errors?

A Type I error rejects a true null hypothesis (probability $\alpha$); a Type II error fails to reject a false null hypothesis (probability $\beta$).

State the Newton-Raphson iteration formula for root finding.

$x_{n+1} = x_{n} - \dfrac{f(x_{n})}{f'(x_{n})}$

Describe the bisection method for finding a root.

Given $[a,b]$ with $f(a)f(b)<0$, repeatedly evaluate the midpoint $c=\frac{a+b}{2}$ and replace whichever endpoint keeps the sign change bracketing the root, halving the interval each step.

State the trapezoidal rule for approximating $\displaystyle\int_{a}^{b} f(x)\,dx$ with step $h$.

$\displaystyle\int_{a}^{b} f(x)\,dx \approx \dfrac{h}{2}\left[f(x_{0}) + 2\sum_{i=1}^{n-1}f(x_{i}) + f(x_{n})\right]$

State Simpson's $\frac{1}{3}$ rule for numerical integration (requires an even number of intervals).

$\displaystyle\int_{a}^{b} f(x)\,dx \approx \dfrac{h}{3}\left[f(x_{0}) + 4\sum_{\text{odd}}f(x_{i}) + 2\sum_{\text{even}}f(x_{i}) + f(x_{n})\right]$

State Euler's method for numerically solving $y' = f(x,y)$ with step size $h$.

$y_{n+1} = y_{n} + h\,f(x_{n}, y_{n})$

Compare Euler's method and the fourth-order Runge-Kutta (RK4) method for solving ODEs.

Euler's method is first-order accurate (local error $\sim h^{2}$) using one slope evaluation per step; RK4 is fourth-order accurate using a weighted average of four slope evaluations per step, giving far greater accuracy for the same step size.