Single-variable calculus for the FE Civil: derivatives for rates, slopes and optimization; integrals for areas, volumes and centroids.
NCEES lists these Mathematics and Statistics topics for the FE Civil: analytic geometry; single-variable calculus; vector operations; statistics: distributions, mean, mode, standard deviation, confidence intervals, regression and curve fitting; probability: estimation and expected value. On Newton’s Notes this area spans 4 chapters: this one and Analytic Geometry and Trigonometry, Algebra and Linear Algebra and Probability and Statistics. The FE Civil asks single-variable calculus only, so 3.6 Partial Derivatives and Gradients is not on this page.
Choose any section below to get a free FE Civil practice problem. Work it with the FE Reference Handbook open, tap your answer, then check the solution.
A derivative is a slope: how fast one quantity changes as another changes.
What is the derivative of sin(4x2)?
Chain rule: derivative of the outside, cos(4x2), times derivative of the inside, 8x. Result: 8x cos(4x2).
Common trap: (D) forgets the chain rule.
Differential CalculusderivativeThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Start your 3-day free trial →
When plugging in gives 0/0 or ∞/∞, the limit is still defined, you just need another route.
Evaluate the limit of (3x2 + 2)/(5x2 − x) as x → ∞.
Highest powers match (x2), so the limit is the ratio of leading coefficients: 3/5 = 0.6.
Common trap: (C) flips the ratio.
L’HôpitallimitThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Start your 3-day free trial →
Set the first derivative to zero to find critical points.
Where is the inflection point of f(x) = x3 − 6x2 + 9x + 1?
f″ = 6x − 12 = 0 → x = 2, and f″ changes sign there.
maximaminimainflectionThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Start your 3-day free trial →
Integration undoes differentiation.
Evaluate the integral of 1/x dx from 1 to e.
[ln x] from 1 to e = ln e − ln 1 = 1.
Integral Calculusintegration by partsThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Start your 3-day free trial →
An integral adds up thin slices.
The region under y = √x from x = 0 to 4 is spun about the x-axis. What is the volume?
Disks: V = π∫y2dx = π∫x dx from 0 to 4 = π(8) = 25.1.
Common trap: (D) forgets the π.
area between curvesvolume of revolutioncentroidThat was 1 of 3 problems in this section. The full course has the lesson, a worked example and every problem. Start your 3-day free trial →
Newton’s Notes FE Civil has 18 chapters (11 shared with Mechanical, 7 Civil-only) with lessons, worked examples and 420+ FE-style practice problems with full solutions, 171 of them Civil-only. Pro adds a timed full-length FE Civil practice exam (110 problems), a custom test builder and Ask Newton, the AI tutor. Try it free for 3 days, no card to start.
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