Mathematics

6% of exam

Analytic geometry, logarithms, and trigonometry, single-variable and integral calculus, ordinary and partial differential equations and Laplace transforms, matrix algebra and systems of equations, and numerical methods including error propagation, Taylor series, curve fitting, and Newton-Raphson.

4 concepts

Exam areas

A. Analytic geometry, logarithms, and trigonometry

Analytic Geometry, Logarithms and Trigonometry

Lines and conics, logarithm and exponential identities behind semilog plots and Arrhenius behavior, and the trig identities and unit circle used in process calculations.

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B. Calculus

Calculus, ODEs and Laplace Transforms

Derivatives as rates and integrals as accumulation, optimization, first- and second-order linear ODEs via the characteristic equation, and the Laplace method.

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D. Numerical methods

Numerical Methods, Error and Significant Figures

Newton-Raphson roots, trapezoidal and Simpson integration, Taylor truncation, relative-error propagation, and significant-figure discipline for FE-Chem.

Most chemical-engineering equations worth solving are not solvable by hand: a cubic equation of state for a molar volume, a Colebrook friction factor, a reactor design integral with no closed form, an enthalpy that lives inside a heat-capacity polynomial. Numerical methods are how you get a number out of those, and the FE-Chem exam tests whether you can drive one Newton step, lay down a Simpson rule, propagate an uncertainty, and report the answer to a defensible number of significant figures. The relevant relations sit in the FE Reference Handbook — Mathematics (Numerical Methods and Taylor's Series), with the significant-figure rules given in the handbook's Units and Conversion Factors front-matter section, and the equations are supplied on exam day — what is tested is whether you can set them up correctly and not lose a digit.

Newton-Raphson root finding

To solve

f(x)=0

when no formula exists, Newton-Raphson replaces the curve at your current guess with its tangent line and takes the next estimate where that tangent crosses zero. Each iteration is the current point minus the function divided by its slope; geometrically you slide down the tangent to the axis. Convergence is quadratic once you are close — the number of correct digits roughly doubles each step — but the method needs a decent starting guess and a non-zero derivative, and it can diverge if

f'(x)

is near zero or the function is badly behaved. For an equation of state, the ideal-gas value is almost always a good seed.

x_{j+1} = x_j - \dfrac{f(x_j)}{f'(x_j)}

Taylor series and truncation error

Newton-Raphson, and almost every numerical method, is a truncated Taylor series in disguise. Expanding

f

about a point

a

writes the function as its value plus successive derivative terms weighted by powers of

(x-a)

over factorials; keeping only the first two terms (the local tangent) is exactly the linearization Newton uses. Whatever you drop is the truncation error, and because the first neglected term scales as the step raised to a power, halving the step shrinks the error predictably — a first-order method improves linearly, Simpson's rule improves with the fourth power of the spacing. Setting

a=0

gives the Maclaurin series.

f(x) = f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2!}(x-a)^2 + \cdots

Numerical integration: trapezoidal and Simpson

When a definite integral has no antiderivative — a PFR design integral

\int dX/(-r_A)

, an enthalpy

\int C_p\,dT

from tabulated data, an area under an experimental curve — you approximate it on

n

equal strips of width

\Delta x

. The trapezoidal rule connects the data with straight chords; Simpson's 1/3 rule fits parabolas through pairs of strips and is dramatically more accurate, exact for any cubic. The Simpson weighting pattern is the thing students drop: ends count once, interior odd-indexed points count four, interior even-indexed points count two, all scaled by

\Delta x/3

. Simpson requires an even number of intervals (an odd number of points).

\int_a^b f(x)\,dx \approx \dfrac{\Delta x}{3}\Big[f_0 + 4f_1 + 2f_2 + 4f_3 + \cdots + 4f_{n-1} + f_n\Big]

Propagation of error

A computed result is only as good as the measurements feeding it, and uncertainties combine in a specific way. For a quantity built from a product or quotient of powers — Reynolds number

Re=\rho V D/\mu

, a rate from

-r_A = kC^n

— the fractional (relative) uncertainties add, each weighted by its exponent. The conservative worst-case adds the magnitudes linearly; if the input errors are independent and random, they combine in quadrature (root-sum-square), which is smaller. The handbook's own Measurement Uncertainty relation (FE Reference Handbook — Engineering Probability and Statistics) is the RSS (Kline-McClintock) form, not the linear sum; the worst-case linear-sum-by-exponent shortcut here is a textbook convenience. Use the linear sum when a problem asks for the maximum possible error and the RSS form when it asks for the expected or probable error.

\left|\dfrac{\Delta y}{y}\right| = \sum_i |a_i|\left|\dfrac{\Delta x_i}{x_i}\right| \quad\text{for}\quad y = x_1^{a_1}x_2^{a_2}\cdots

Significant figures and round-off

Significant figures communicate precision, and the FE handbook codifies the rules: non-zero digits always count, trapped zeros count, and in multiplication or division the result carries as many significant figures as the least-precise input. In addition and subtraction it is the decimal places that govern, not the count of digits — and subtracting two nearly equal large numbers (catastrophic cancellation) can destroy precision even when each input looks accurate. Carry full precision through intermediate steps and round only the final answer, customarily to three or four significant figures.

Exam strategy

Read which method the problem names and match the template. Newton-Raphson: identify

f

and

f'

, take the seed (ideal gas for an EOS), and do one or two iterations by hand — they rarely ask for more. Integration: count your strips, confirm Simpson needs an even

n

, and write the

1,4,2,4,\dots,1

pattern explicitly before plugging numbers (a misweighted node is the classic error). Error propagation: convert every uncertainty to a fraction first, weight by the exponent, and decide linear-sum versus RSS from the wording. Finally, report three significant figures with units and sanity-check the magnitude — a compressibility factor near 1 for a mild gas, an integral that brackets the trapezoidal and exact values.

Key equations

Newton-Raphson iteration

x_{j+1} = x_j - \dfrac{f(x_j)}{f'(x_j)}

Next root estimate; $f'$ is the derivative at the current guess. Quadratic convergence near a simple root; needs $f'\neq 0$ and a good seed.

Taylor series about a

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

Basis of truncation-error analysis. Truncating after the linear term gives the Newton tangent. $a=0$ is the Maclaurin series.

Trapezoidal rule (n strips)

\int_a^b f\,dx \approx \dfrac{\Delta x}{2}\Big[f_0 + 2f_1 + 2f_2 + \cdots + 2f_{n-1} + f_n\Big]

Straight-chord approximation; error $\propto \Delta x^2$ . $\Delta x=(b-a)/n$ .

Simpson's 1/3 rule

\int_a^b f\,dx \approx \dfrac{\Delta x}{3}\Big[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{n-1} + f_n\Big]

Parabolic fit; exact for cubics, error $\propto \Delta x^4$ . $n$ must be even (odd number of points).

Step width

\Delta x = \dfrac{b-a}{n}

Uniform spacing for $n$ subintervals between limits $a$ and $b$ .

Relative error

\varepsilon_{rel} = \dfrac{|x_{approx}-x_{true}|}{|x_{true}|}

Dimensionless; multiply by 100 for percent. Use the previous iterate as the reference when the true value is unknown.

Error propagation (products)

\left|\dfrac{\Delta y}{y}\right| = \sum_i |a_i|\left|\dfrac{\Delta x_i}{x_i}\right|

Worst-case relative error for $y=\prod x_i^{a_i}$ ; fractional errors add, weighted by exponents.

Error propagation (RSS)

\dfrac{\Delta y}{y} = \sqrt{\sum_i a_i^2\left(\dfrac{\Delta x_i}{x_i}\right)^2}

Probable error when inputs are independent and random; always smaller than the linear sum.

Error propagation (sums)

y = \sum_i c_i x_i \;\Rightarrow\; \Delta y = \sqrt{\sum_i c_i^2\,(\Delta x_i)^2}

For additive combinations the absolute errors (not relative) combine.

Worked examples

Newton-Raphson on the van der Waals equation

Problem. Find the molar volume of carbon dioxide at $T=300\,\text{K}$ and $P=2.00\,\text{MPa}$ using the van der Waals equation $P = \dfrac{RT}{V-b} - \dfrac{a}{V^2}$ with $a=0.3640\,\text{Pa}\,\text{m}^6/\text{mol}^2$ , $b=4.267\times10^{-5}\,\text{m}^3/\text{mol}$ , $R=8.314\,\text{J/(mol·K)}$ . Start from the ideal-gas estimate.

Solution. Write

f(V) = P - \dfrac{RT}{V-b} + \dfrac{a}{V^2}

, so

f'(V) = \dfrac{RT}{(V-b)^2} - \dfrac{2a}{V^3}

. Seed:

V_0 = RT/P = (8.314)(300)/2.00\times10^{6} = 1.2471\times10^{-3}\,\text{m}^3/\text{mol}

. Iteration 1:

f(V_0)=+1.632\times10^{5}

f'(V_0)=1.344\times10^{9}

, giving

V_1 = 1.2471\times10^{-3} - 1.632\times10^{5}/1.344\times10^{9} = 1.1257\times10^{-3}

. Iteration 2:

V_2 = 1.1354\times10^{-3}

. Iteration 3:

V_3 = 1.1355\times10^{-3}

(relative change

7\times10^{-5}

, converged). Final:

V = 1.14\times10^{-3}\,\text{m}^3/\text{mol}

. Sanity check:

Z = PV/RT = (2.00\times10^{6})(1.1355\times10^{-3})/[(8.314)(300)] = 0.910

— a real gas modestly below ideal (

Z<1

, attraction-dominated) at moderate pressure, as expected.

V_{j+1} = V_j - \dfrac{P - \frac{RT}{V_j-b} + \frac{a}{V_j^2}}{\frac{RT}{(V_j-b)^2} - \frac{2a}{V_j^3}}

Trapezoidal vs Simpson on a reactor design integral

Problem. A second-order liquid reaction needs $I = \int_0^{0.6}\dfrac{dX}{(1-X)^2}$ evaluated for a PFR sizing. Estimate $I$ with both the trapezoidal and Simpson 1/3 rules using $n=4$ intervals, and compare to the exact value.

Solution. Step width

\Delta x = 0.6/4 = 0.15

. Nodes at

X = 0,\,0.15,\,0.30,\,0.45,\,0.60

give

f = 1.0000,\,1.3841,\,2.0408,\,3.3058,\,6.2500

. Trapezoidal:

I \approx \dfrac{0.15}{2}[1.0000 + 2(1.3841+2.0408+3.3058) + 6.2500] = 1.553

. Simpson 1/3:

I \approx \dfrac{0.15}{3}[1.0000 + 4(1.3841)+2(2.0408)+4(3.3058) + 6.2500] = 1.505

. Exact:

\int_0^{0.6}(1-X)^{-2}dX = \big[\tfrac{1}{1-X}\big]_0^{0.6} = \tfrac{1}{0.4}-1 = 1.500

. Final: Simpson gives

1.50

(0.30% high); the trapezoidal estimate is

1.55

(3.6% high). Sanity check: both overestimate because the integrand is convex (curves upward), and Simpson — which fits parabolas — is an order of magnitude closer, as theory predicts.

I_{Simpson} = \dfrac{\Delta x}{3}\big[f_0 + 4f_1 + 2f_2 + 4f_3 + f_4\big]

Propagating measurement error into a Reynolds number

Problem. A Reynolds number $Re = \rho V D/\mu$ is computed from measurements with relative uncertainties: density $\pm 1.0\%$ , velocity $\pm 2.0\%$ , diameter $\pm 1.5\%$ , viscosity $\pm 2.5\%$ . What is the worst-case and the probable (RSS) relative uncertainty in $Re$ ?

Solution. Every exponent in

Re=\rho^{1}V^{1}D^{1}\mu^{-1}

has magnitude 1, so the fractional errors simply combine. Worst-case (linear sum):

\left|\Delta Re/Re\right| = 1.0 + 2.0 + 1.5 + 2.5 = 7.0\%

. Probable (RSS):

\sqrt{1.0^2 + 2.0^2 + 1.5^2 + 2.5^2} = \sqrt{1+4+2.25+6.25} = \sqrt{13.5} = 3.67\%

. Final:

Re

is uncertain by about

7\%

worst-case,

3.7\%

probable. Sanity check: the RSS value is always smaller than the linear sum and is dominated by the largest single contributor (viscosity,

2.5\%

), which is the term to tighten first.

\dfrac{\Delta Re}{Re} = \sqrt{\left(\tfrac{\Delta\rho}{\rho}\right)^2 + \left(\tfrac{\Delta V}{V}\right)^2 + \left(\tfrac{\Delta D}{D}\right)^2 + \left(\tfrac{\Delta\mu}{\mu}\right)^2}

Common pitfalls

•Misweighting Simpson nodes: the pattern is $1,4,2,4,\dots,4,1$ — interior points alternate 4 (odd index) and 2 (even index), never all 4. Write the weights out before substituting.
•Using Simpson 1/3 with an odd number of intervals. It requires an even $n$ (odd number of points); with odd $n$ use the trapezoidal rule or Simpson 3/8 on the odd piece.
•Adding absolute uncertainties for a product. For $y=\prod x_i^{a_i}$ the RELATIVE (fractional) errors add, weighted by exponents — not the absolute ones. Absolute errors add only for sums and differences.
•Confusing the worst-case linear sum with the RSS combination. Linear sum (larger) is the maximum possible error; RSS (smaller) is the probable error for independent inputs — read which the problem wants.
•Rounding intermediate results, then propagating the round-off. Carry full precision and round once at the end; subtracting nearly equal numbers (catastrophic cancellation) is especially fragile.
•Reporting more significant figures than the least-precise input justifies. A rate from a 2-sig-fig concentration cannot legitimately be quoted to 5 figures.
•Starting Newton-Raphson from a guess where $f'(x)\approx 0$ (a turning point) or far from the root — the step blows up or the iteration diverges. Seed an EOS with the ideal-gas volume.

References

NCEES FE Reference Handbook — Mathematics (Numerical Methods, Numerical Integration, Taylor's Series)
NCEES FE Reference Handbook — Units and Conversion Factors (Significant Figures, p.2)
NCEES FE Reference Handbook — Engineering Probability and Statistics (Measurement Uncertainty / Propagation of Error, RSS form)
Chapra & Canale, Numerical Methods for Engineers — Newton-Raphson, Simpson, and error-analysis background

E. Algebra

Algebra, Matrices and Systems of Equations

Solving simultaneous linear equations by Gauss elimination and Cramer's rule, matrix operations and the inverse, determinants, and matrix-form multi-unit material balances.

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