Mathematics

E. Single-variable calculus

Single-Variable Calculus

Limits, derivatives, optimization, definite and indefinite integrals, the Fundamental Theorem, area, and L'Hopital — the calculus the FE asks across nearly every topic.

Calculus is the single most pervasive idea on the FE: a velocity is the derivative of position, a flow rate the derivative of volume, an accumulated mass the integral of a rate, an area the integral of a height. The exam almost never asks for calculus in the abstract — it hands you a function and wants a maximum, a slope, an area, or the value of a stubborn limit. The Mathematics chapter of the NCEES FE Reference Handbook prints the full Derivatives and Indefinite Integrals tables, so your edge is fluency: knowing which operation the question is really asking for and executing it without a slip. This concept walks the whole arc, from limits to the Fundamental Theorem.

Limits and continuity

A limit asks what value

f(x)

heads toward as

x

approaches

a

, independent of

f(a)

itself. For a continuous function you simply substitute,

\lim_{x\to a}f(x)=f(a)

. The work begins when substitution gives an indeterminate form like

0/0

\infty/\infty

: then you factor, rationalize, divide by the highest power, or call in L'Hopital's rule. Continuity at

a

— the limit exists and equals

f(a)

— is the prerequisite for differentiability.

\lim_{x\to a} f(x)=L\quad\text{means } f(x)\to L\text{ as } x\to a

The derivative and its rules

The derivative is the instantaneous rate of change — the slope of the tangent line — defined as the limit of a difference quotient. In practice you apply the table rules: the power rule for polynomials, the product and quotient rules for combinations, and the chain rule for compositions. The chain rule is the one examinees forget: differentiate the outer function, then multiply by the derivative of the inner. Define inner and outer explicitly so you never drop a factor.

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

Maxima, minima, and inflection

To optimize, set the first derivative to zero to locate critical points, then classify each with the second derivative:

f''(a)>0

is a local minimum (concave up),

f''(a)<0

Indefinite integrals (antiderivatives)

The indefinite integral reverses differentiation and carries an arbitrary constant

C

, since any constant differentiates away. When the table does not hand you the antiderivative directly, the handbook's three workhorses are substitution (reverse chain rule), integration by parts for products, and partial fractions for rational functions. Integration by parts is

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

; choose

u

The definite integral and the Fundamental Theorem

The definite integral is the signed area under a curve, built as the limit of a sum of thin rectangles. The Fundamental Theorem of Calculus evaluates it as the net change of any antiderivative across the limits — this is the bridge that turns the hard limit-of-a-sum into a two-step lookup-and-subtract. Its companion statement says differentiation and integration undo each other, so

\frac{d}{dx}\int_a^x f(t)\,dt=f(x)

Area between curves

The area enclosed between two curves on

[a,b]

is the integral of the upper minus the lower function. Find the intersection points first — they set the limits — and integrate the difference; if the curves cross within the interval, split the integral so "upper minus lower" stays correct on each piece. This same construction gives the area swept by a velocity curve (a distance) or a power curve (an energy).

A=\int_a^b\big[\,f_{\text{upper}}(x)-f_{\text{lower}}(x)\,\big]\,dx

L'Hopital's rule

When a limit produces the indeterminate form

0/0

\infty/\infty

, L'Hopital's rule replaces it with the limit of the ratio of derivatives, repeated until the form is determinate. Confirm the form is genuinely indeterminate before applying it — using L'Hopital on a determinate or

2/0

limit gives wrong answers. Other indeterminate forms (

0\cdot\infty

\infty-\infty

Exam strategy

Identify the operation first: "slope," "rate," or "velocity" means derivative; "area," "total," or "accumulated" means integral; a stuck

0/0

means factor or L'Hopital; "largest/smallest" means set

f'=0

and apply the second-derivative test. Then go straight to the handbook's Derivatives or Indefinite Integrals table rather than deriving from scratch. For optimization never skip the second-derivative test or the endpoints; carry the

+C

on indefinite integrals; and keep every trig argument in radians, the convention the handbook's tables assume.

Key equations

Derivative (definition)

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

Worked examples

Optimization: maximum enclosed area

Problem. A rancher has $400\,\text{m}$ of fence to enclose a rectangular pen along a straight river, fencing only the three non-river sides. What dimensions maximize the enclosed area, and what is that area?

Solution. Let

x

be each side perpendicular to the river and

y

the side parallel to it. Fence used:

2x+y=400

Common pitfalls

•Forgetting the chain rule's inner factor: $\frac{d}{dx}\sin(3x)=3\cos(3x)$ , not $\cos(3x)$

References

NCEES FE Reference Handbook — Mathematics (Differential Calculus, Integral Calculus, Derivatives and Indefinite Integrals tables)
Stewart, Calculus: Early Transcendentals — limits, optimization, FTC

A. Analytic geometry and trigonometry

Analytic Geometry and Trigonometry

B. Differential equations

First- and Second-Order Differential Equations

C. Numerical methods

Numerical Methods

D. Linear algebra

Linear Algebra