# FE Exam Flashcards: Engineering Equations, Units & Handbook Skills ![Open notebook with a calculator, ruler, drafting compass, and geometric solids on a dark engineering desk\.]() Review common FE equations, variable meanings, units, assumptions, conversions, and searchable handbook navigation\. By author [Open Study Contributor]() [Study this deck for free]() Nibomo opens so you can start studying\. ## Deck details - Cards: 276 cards - Version: 1 - Published: September 3, 2026 - Last updated: September 10, 2026 - License: Common knowledge · CC0 1\.0 - Audience languages: [en]() - Subject: Engineering - Level: FE Exam - Tags: fe, units, math, core\-equations, economics, handbook, statistics, ethics ## About this deck Build quick recall for broad topic families found across all seven FE discipline specifications: mathematics, probability and statistics, engineering economics, and ethics and professional practice. Individual subtopics vary by discipline. The deck also covers unit discipline, handbook navigation, and selected equations useful across multiple disciplines. The 276 original cards practice equation → use, variable → meaning or unit, condition → applicable relationship, quantity → unit family, conversion setup → result, short scenario → handbook search cue, and equation behavior → interpretation. Answers come first, followed by the assumption, sign convention, unit condition, or practical check that keeps the relationship honest. Related variants are separated in a deterministic mixed sequence, while prerequisites stay ahead of dependent uses. This is a common recall layer, not a compressed copy of the reference handbook. It deliberately leaves out mechanical reversals, exhaustive symbol permutations, long worked problems, handbook page numbers, discipline-specific design procedures, and claims that every equation appears in every FE discipline. Use it alongside your current [FE discipline specification](https://ncees.org/exams/fe-exam/), the current reference handbook available through MyNCEES, and timed discipline-specific problem practice. The [current NCEES Examinee Guide](https://ncees.org/examinee-guide/) explains how the onscreen reference material is supplied and searched. Independent and unofficial. Not affiliated with or endorsed by NCEES. No NCEES exam questions, answer keys, practice questions, handbook tables or prose, or exam-specification text were copied. ## Similar decks - [SI Units Flashcards: 7 Base Units & 22 Named Derived Units]() - [MCAT Physics & Chemistry Equation Flashcards: Formulas, Units & When to Use Them]() - [California Permit Test Flashcards: DMV Handbook Rules & Signs]() - [AP Statistics Flashcards: Complete 5\-Unit Course Review]() - [Digital SAT Flashcards: Math, Grammar & Words in Context]() - [CompTIA A\+ 220\-1201 & 220\-1202 Flashcards: Core 1 and Core 2]() ## Cards in this deck ### Card 1 **Question** SI base unit of length? **Answer** The metre, symbol $m$. ### Card 2 **Question** Slope between two points? **Answer** $$ m=\frac{y_2-y_1}{x_2-x_1} $$ This requires $x_2\ne x_1$; slope is the change in the dependent variable per unit change in the independent variable. ### Card 3 **Question** Net force relationship for constant mass in an inertial frame? **Answer** $$ \sum\mathbf{F}=m\mathbf{a} $$ Use a coherent unit system and include only external forces on the chosen body. ### Card 4 **Question** Cash-flow sign convention from the decision maker's viewpoint? **Answer** Receipts are positive and disbursements are negative. Choose one viewpoint and keep it consistent across the cash-flow diagram and equations. ### Card 5 **Question** Where should an FE candidate get the current reference handbook for practice? **Answer** From the candidate's MyNCEES account. Use the current supplied file rather than an old indexed copy or a third-party mirror. ### Card 6 **Question** Probability of the complement of event $A$? **Answer** $$ P(A^c)=1-P(A) $$ ### Card 7 **Question** An engineering judgment conflicts with pressure to protect schedule or cost. First duty? **Answer** Protect public health, safety, and welfare. Document the technical concern and escalate through appropriate channels. ### Card 8 **Question** SI base unit of mass? **Answer** The kilogram, symbol $kg$. It is a mass unit, not a force unit. ### Card 9 **Question** Linear equation from slope and intercept? **Answer** $$ y=mx+b $$ Here $m$ is slope and $b$ is the value of $y$ where $x=0$. ### Card 10 **Question** Velocity after time $t$ under constant acceleration? **Answer** $$ v=v_0+at $$ This one-dimensional form requires constant $a$ and a consistent positive direction. ### Card 11 **Question** Where is an end-of-period cash flow placed? **Answer** At the boundary marking the end of that period. Interest factors assume consistent period lengths and cash-flow timing. ### Card 12 **Question** How is the FE reference handbook available during the exam? **Answer** It is supplied onscreen as a searchable PDF. Practice locating information in the same current reference rather than relying on memory of a printed layout. ### Card 13 **Question** Probability of $A$ or $B$ for possibly overlapping events? **Answer** $$ P(A\cup B)=P(A)+P(B)-P(A\cap B) $$ ### Card 14 **Question** SI base unit of time? **Answer** The second, symbol $s$. ### Card 15 **Question** Linear equation from one point and a slope? **Answer** $$ y-y_1=m(x-x_1) $$ Use it when the line passes through $(x_1,y_1)$ and has slope $m$. ### Card 16 **Question** Position after time $t$ under constant acceleration? **Answer** $$ x=x_0+v_0t+\frac{1}{2}at^2 $$ The acceleration is constant and all signed components use one coordinate direction. ### Card 17 **Question** A project assignment lies outside your competence. Proper response? **Answer** Accept only the parts you are qualified to perform, or obtain qualified supervision or support. Do not present unfamiliar work as your own competent judgment. ### Card 18 **Question** SI base unit of electric current? **Answer** The ampere, symbol $A$. ### Card 19 **Question** Future amount under simple interest? **Answer** $$ F=P(1+in) $$ Here $P$ is principal, $i$ is interest per period, and $n$ is the number of periods; interest is earned only on principal. ### Card 20 **Question** Distance between two Cartesian points? **Answer** $$ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} $$ The coordinate components must use the same length unit. ### Card 21 **Question** Constant-acceleration relation without elapsed time? **Answer** $$ v^2=v_0^2+2a(x-x_0) $$ Use signed one-dimensional components and constant $a$. ### Card 22 **Question** Where do you search the supplied reference material in the exam interface? **Answer** Use the exam software's search box beside the supplied reference material. ### Card 23 **Question** Conditional probability of $A$ given $B$? **Answer** $$ P(A\mid B)=\frac{P(A\cap B)}{P(B)} $$ This requires $P(B)>0$. ### Card 24 **Question** SI base unit of thermodynamic temperature? **Answer** The kelvin, symbol $K$. Kelvin is written without a degree sign. ### Card 25 **Question** A client asks you to remove an unfavorable test result from a report. Proper response? **Answer** Keep the material result and explain it accurately. Engineering statements should be objective, complete enough to avoid deception, and supported by evidence. ### Card 26 **Question** Midpoint of two Cartesian points? **Answer** $$ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) $$ Average corresponding coordinates. ### Card 27 **Question** Linear momentum of a particle? **Answer** $$ \mathbf{p}=m\mathbf{v} $$ For constant mass, SI units are $kg\cdot m/s$. ### Card 28 **Question** Future amount under compound interest? **Answer** $$ F=P(1+i)^n $$ The rate period must match the period used for $n$. ### Card 29 **Question** Should an FE search workflow depend on Ctrl+F? **Answer** No. The May 2026 NCEES Examinee Guide says Ctrl+F isn't available; practice with the exam interface's own search box. ### Card 30 **Question** SI base unit of amount of substance? **Answer** The mole, symbol $mol$. ### Card 31 **Question** Joint probability expressed with a conditional probability? **Answer** $$ P(A\cap B)=P(A\mid B)P(B) $$ This elementary formula requires $P(B)>0$. ### Card 32 **Question** Standard equation of a circle centered at $(h,k)$? **Answer** $$ (x-h)^2+(y-k)^2=r^2 $$ The coordinate units and radius unit must match. ### Card 33 **Question** Impulse-momentum relationship? **Answer** $$ \int_{t_1}^{t_2}\sum\mathbf{F}\,dt=\mathbf{p}_2-\mathbf{p}_1 $$ The force is the net external force on the chosen system. ### Card 34 **Question** You discover a financial interest that could affect your recommendation. Proper response? **Answer** Disclose the conflict promptly to affected parties and obtain appropriate consent or withdraw. Hidden personal interests undermine independent judgment. ### Card 35 **Question** SI base unit of luminous intensity? **Answer** The candela, symbol $cd$. ### Card 36 **Question** Present worth of one future amount? **Answer** $$ P=\frac{F}{(1+i)^n} $$ Use an effective rate $i$ per period consistent with $n$. ### Card 37 **Question** Roots of $ax^2+bx+c=0$? **Answer** $$ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} $$ This form requires $a\ne0$. ### Card 38 **Question** When is total linear momentum conserved? **Answer** When net external impulse on the system is zero. Internal forces can redistribute momentum but do not change the system total. ### Card 39 **Question** Why avoid memorizing FE handbook page numbers? **Answer** Pagination and layout can change. Remember section names, concepts, and strong search terms instead. ### Card 40 **Question** Probability condition that defines independence of $A$ and $B$? **Answer** $$ P(A\cap B)=P(A)P(B) $$ Independence is not the same as mutual exclusivity. ### Card 41 **Question** SI unit of frequency in base-unit form? **Answer** The hertz, $Hz=s^{-1}$, applies to periodic phenomena. One hertz is one cycle per second. ### Card 42 **Question** Present worth of an end-of-period uniform series $A$? **Answer** $$ P=A\frac{(1+i)^n-1}{i(1+i)^n} $$ This requires $i\ne0$; at $i=0$, $P=An$. ### Card 43 **Question** What does $b^2-4ac$ reveal for a real-coefficient quadratic? **Answer** It determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex-conjugate pair. ### Card 44 **Question** Work done by a constant force through displacement? **Answer** $$ W=\mathbf{F}\cdot\mathbf{s}=Fs\cos\theta $$ Use the angle between force and displacement; SI units are joules. ### Card 45 **Question** A vendor offers a valuable gift while you evaluate bids. Proper response? **Answer** Decline it and follow the organization's disclosure policy. Even an intended courtesy can create an actual or apparent conflict. ### Card 46 **Question** SI unit of force in base-unit form? **Answer** The newton, $N=kg\cdot m/s^2$. ### Card 47 **Question** First search cue for a named quantity such as coefficient of thermal expansion? **Answer** Search the exact technical noun phrase. Distinctive nouns usually narrow results better than a whole question sentence. ### Card 48 **Question** Bayes' rule for $P(A\mid B)$? **Answer** $$ P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)} $$ This requires $P(B)>0$. ### Card 49 **Question** Product of powers with the same positive base and real exponents? **Answer** $$ a^m a^n=a^{m+n} $$ This form assumes $a>0$ and real exponents $m$ and $n$. ### Card 50 **Question** Translational kinetic energy of a particle? **Answer** $$ K=\frac{1}{2}mv^2 $$ Mass and speed must use a coherent unit system. ### Card 51 **Question** SI unit of pressure in force-area form? **Answer** The pascal, $Pa=N/m^2$. It is the SI derived unit for pressure and stress. ### Card 52 **Question** Equivalent annual amount for present worth $P$? **Answer** $$ A=P\frac{i(1+i)^n}{(1+i)^n-1} $$ For the displayed formula, $i\ne0$; at $i=0$, $A=P/n$. Payments occur at period ends and $i$ matches that period. ### Card 53 **Question** May confidential client information be shared for personal benefit? **Answer** No. Protect it unless the client authorizes disclosure or law, safety, or professional duty requires it. ### Card 54 **Question** A handbook search term returns too many results. Best refinement? **Answer** Add one meaningful qualifier, such as the phenomenon, geometry, distribution, or material property you need. ### Card 55 **Question** Quotient of powers with the same positive base and real exponents? **Answer** $$ \frac{a^m}{a^n}=a^{m-n} $$ This form assumes $a>0$ and real exponents $m$ and $n$. ### Card 56 **Question** Net work-energy relationship for a particle? **Answer** $$ W_{net}=K_2-K_1 $$ The net work includes the work of all forces represented in the particle model. ### Card 57 **Question** Ordered selections of $r$ objects from $n$ distinct objects? **Answer** $$ P(n,r)=\frac{n!}{(n-r)!} $$ Use this when order matters and selection is without replacement. ### Card 58 **Question** Future worth of an end-of-period uniform series $A$? **Answer** $$ F=A\frac{(1+i)^n-1}{i} $$ This requires $i\ne0$; at $i=0$, $F=An$. ### Card 59 **Question** Power of a power for a positive base and real exponents? **Answer** $$ (a^m)^n=a^{mn} $$ This form assumes $a>0$ and real exponents $m$ and $n$. ### Card 60 **Question** SI unit of energy in force-distance form? **Answer** The joule, $J=N\cdot m$. ### Card 61 **Question** A problem gives only an unfamiliar symbol. Best search preparation? **Answer** Translate the symbol into its likely quantity name and discipline context before searching; symbol-only searches are often ambiguous. ### Card 62 **Question** Confidential information reveals an imminent public-safety risk. What comes first? **Answer** Take steps required to protect people, using appropriate legal and organizational channels. Share only what is necessary and document the basis for escalation. ### Card 63 **Question** Unordered selections of $r$ objects from $n$ distinct objects? **Answer** $$ C(n,r)=\frac{n!}{r!(n-r)!} $$ Use this when order does not matter and selection is without replacement. ### Card 64 **Question** Change in gravitational potential energy near Earth's surface? **Answer** $$ \Delta U_g=mg\Delta h $$ Use approximately constant $g$ and a clearly chosen height datum. ### Card 65 **Question** Negative-exponent rule for a positive base and real exponent? **Answer** $$ a^{-n}=\frac{1}{a^n} $$ This form assumes $a>0$ and real exponent $n$. ### Card 66 **Question** Uniform end-of-period deposit needed for future amount $F$? **Answer** $$ A=F\frac{i}{(1+i)^n-1} $$ For the displayed formula, $i\ne0$; at $i=0$, $A=F/n$. The first deposit is one period from now and the last occurs at period $n$. ### Card 67 **Question** A handbook search for an acronym fails. Next move? **Answer** Try the expanded term and a common alternate name. Acronyms vary by section and discipline. ### Card 68 **Question** SI unit of power in energy-time form? **Answer** The watt, $W=J/s$. ### Card 69 **Question** Logarithm of a product? **Answer** $$ \log_b(xy)=\log_b x+\log_b y $$ For real logarithms, $b>0$, $b\ne1$, and both arguments are positive. ### Card 70 **Question** Conditions for a binomial model? **Answer** A fixed number of independent trials, two outcomes per trial, and the same success probability on every trial. ### Card 71 **Question** Using another engineer's analysis in your deliverable? **Answer** Credit the source and verify the analysis before relying on it. Do not present someone else's work as your own. ### Card 72 **Question** SI unit of electric charge in current-time form? **Answer** The coulomb, $C=A\cdot s$. ### Card 73 **Question** Average power from work over elapsed time? **Answer** $$ P_{avg}=\frac{W}{\Delta t} $$ Use consistent energy and time units; watts are joules per second. ### Card 74 **Question** Annual equivalent of cost $P$ with salvage $S$ after $n$ periods? **Answer** $$ A=P\frac{i(1+i)^n}{(1+i)^n-1}-S\frac{i}{(1+i)^n-1} $$ The first factor converts present cost to an annual amount; the second converts future salvage. Use the same effective rate and period basis. At $i=0$, $A=(P-S)/n$. ### Card 75 **Question** Logarithm of a positive quantity raised to a power? **Answer** $$ \log_b(x^p)=p\log_b x $$ For real logarithms, $x>0$, $b>0$, and $b\ne1$. ### Card 76 **Question** SI unit of electric potential difference in energy-charge form? **Answer** The volt, $V=J/C$. ### Card 77 **Question** Instantaneous mechanical power delivered by a force? **Answer** $$ P=\mathbf{F}\cdot\mathbf{v} $$ Use the velocity of the force's point of application. ### Card 78 **Question** A precise handbook term gives no useful match. Next move? **Answer** Try a standard synonym or the broader parent concept, then narrow within the relevant section. ### Card 79 **Question** Mean of $X\sim\operatorname{Binomial}(n,p)$? **Answer** $$ E[X]=np $$ ### Card 80 **Question** SI unit of electrical resistance in voltage-current form? **Answer** The ohm, $\Omega=V/A$. ### Card 81 **Question** Moment of a force about a point? **Answer** $$ \mathbf{M}=\mathbf{r}\times\mathbf{F} $$ Vector $\mathbf{r}$ runs from the moment point to a point on the force's line of action. ### Card 82 **Question** Cash-flow pattern described by an arithmetic gradient $G$? **Answer** A series that changes by the same amount $G$ each period. In the usual convention, the first gradient increment appears after the first base payment. ### Card 83 **Question** Inverse relationship between the natural log and exponential? **Answer** $$ \ln(e^x)=x,\qquad e^{\ln x}=x $$ The second identity requires $x>0$. ### Card 84 **Question** SI prefix giga? **Answer** $G=10^9$. ### Card 85 **Question** Angular velocity after time $t$ under constant angular acceleration? **Answer** $$ \omega=\omega_0+\alpha t $$ Use consistent angular sign convention and radians for angular measures. ### Card 86 **Question** When should an engineer sign or seal technical work? **Answer** Only when authorized and when the work was prepared by the engineer or under the level of responsible charge required by the applicable jurisdiction. ### Card 87 **Question** Best memory target for returning to a handbook formula? **Answer** The section or concept name plus a distinctive search phrase. That survives layout changes better than a page number. ### Card 88 **Question** SI prefix mega? **Answer** $M=10^6$. Uppercase matters: lowercase $m$ means milli. ### Card 89 **Question** Convert degrees to radians? **Answer** $$ \theta_{rad}=\theta_{deg}\frac{\pi}{180} $$ Use radians in calculus formulas unless a formula explicitly says otherwise. ### Card 90 **Question** Variance of $X\sim\operatorname{Binomial}(n,p)$? **Answer** $$ \operatorname{Var}(X)=np(1-p) $$ ### Card 91 **Question** Cash-flow pattern described by a geometric gradient rate $g$? **Answer** A series in which each payment is the previous payment multiplied by $1+g$. Keep $g$ on the same period basis as the interest rate. ### Card 92 **Question** Where do licensure and sealing requirements come from? **Answer** The applicable jurisdiction's current laws and board rules. Confirm local requirements rather than assuming one state's rules apply everywhere. ### Card 93 **Question** Rotational kinetic energy about a fixed axis? **Answer** $$ K_{rot}=\frac{1}{2}I\omega^2 $$ $I$ is the mass moment of inertia about that axis. ### Card 94 **Question** SI prefix kilo? **Answer** $k=10^3$. ### Card 95 **Question** Right-triangle definition of sine? **Answer** Sine is opposite over hypotenuse. This definition applies to an acute angle in a right triangle. ### Card 96 **Question** After search lands near a result, what should you scan first? **Answer** The section heading and nearby subheading. They establish whether the result belongs to the right model and subject. ### Card 97 **Question** Rotational analogue of Newton's second law about a fixed principal axis? **Answer** $$ \sum M=I\alpha $$ This scalar form assumes the stated fixed-axis model and a consistent sign convention. ### Card 98 **Question** SI prefix milli? **Answer** $m=10^{-3}$. Context distinguishes the prefix from the metre symbol. ### Card 99 **Question** Meaning of a nominal annual rate compounded $m$ times per year? **Answer** It is a quoted annual rate equal to $m$ times the periodic rate. It is not itself the effective annual rate unless compounding is annual. ### Card 100 **Question** Standardize an observation $x$ from a distribution with mean $\mu$ and standard deviation $\sigma$? **Answer** $$ z=\frac{x-\mu}{\sigma} $$ This requires $\sigma>0$; using a normal probability table also assumes an appropriate normal model. ### Card 101 **Question** Right-triangle definition of cosine? **Answer** Cosine is adjacent over hypotenuse. This definition applies to an acute angle in a right triangle. ### Card 102 **Question** Force-displacement relation for an ideal linear spring? **Answer** $$ F=-kx $$ The minus sign shows restoring direction; the model applies within the spring's linear range. ### Card 103 **Question** May qualifications or project responsibility be exaggerated in a proposal? **Answer** No. State education, licensure, experience, and responsibility accurately, including the actual roles of collaborators. ### Card 104 **Question** SI prefix micro? **Answer** $\mu=10^{-6}$. ### Card 105 **Question** A handbook symbol conflicts with your usual notation. What controls? **Answer** The definition beside the selected relationship. Symbols can represent different quantities in different sections. ### Card 106 **Question** Periodic rate from nominal annual rate $r$ compounded $m$ times per year? **Answer** $$ i=\frac{r}{m} $$ Use consistent decimal units, and pair this rate with the number of compounding periods. ### Card 107 **Question** Right-triangle definition of tangent? **Answer** Tangent is opposite over adjacent, or $\sin\theta/\cos\theta$ where $\cos\theta\ne0$. ### Card 108 **Question** Average normal stress under centered axial load? **Answer** $$ \sigma=\frac{F}{A} $$ $F$ is normal to area $A$; the average ignores local stress concentrations. ### Card 109 **Question** Mean, median, and mode of a symmetric normal distribution? **Answer** They are equal and located at $\mu$. The distribution is symmetric about that center. ### Card 110 **Question** SI prefix nano? **Answer** $n=10^{-9}$. ### Card 111 **Question** Fundamental Pythagorean trigonometric identity? **Answer** $$ \sin^2\theta+\cos^2\theta=1 $$ ### Card 112 **Question** Engineering normal strain? **Answer** $$ \varepsilon=\frac{\Delta L}{L_0} $$ It is dimensionless and uses original gauge length $L_0$. ### Card 113 **Question** A candidate retrieves a formula from the handbook. What must be checked with it? **Answer** Its applicability assumptions and variable definitions. A familiar equation used under the wrong model is still wrong. ### Card 114 **Question** Technical duty to an employer or client? **Answer** Act as a faithful professional within lawful, ethical limits, communicate material risks, and preserve independent technical judgment. ### Card 115 **Question** How does a length conversion factor apply to area? **Answer** Square it. If $1\,u=a\,v$, then $1\,u^2=a^2\,v^2$. ### Card 116 **Question** Effective annual rate from nominal rate $r$ compounded $m$ times per year? **Answer** $$ i_{eff}=\left(1+\frac{r}{m}\right)^m-1 $$ ### Card 117 **Question** When is a Poisson count model reasonable? **Answer** For counts in a fixed interval when events occur independently at an approximately constant average rate and simultaneous events are negligible. ### Card 118 **Question** Relationship for the third side of a non-right triangle? **Answer** $$ c^2=a^2+b^2-2ab\cos C $$ Angle $C$ lies between sides $a$ and $b$, opposite side $c$. ### Card 119 **Question** Linear elastic stress-strain relation in uniaxial loading? **Answer** $$ \sigma=E\varepsilon $$ Use it in the material's linear elastic range; $E$ has stress units. ### Card 120 **Question** How does a length conversion factor apply to volume? **Answer** Cube it. If $1\,u=a\,v$, then $1\,u^3=a^3\,v^3$. ### Card 121 **Question** What should you record for every variable in a retrieved equation? **Answer** Its meaning and expected unit or dimension. This catches swapped symbols and incompatible substitutions. ### Card 122 **Question** Effective annual rate for continuous compounding at nominal rate $r$? **Answer** $$ i_{eff}=e^r-1 $$ Here $r$ is expressed as a decimal per year. ### Card 123 **Question** A supervisor directs an illegal engineering action. Proper response? **Answer** Refuse the illegal action, document the direction, and escalate through appropriate legal or organizational channels. ### Card 124 **Question** Magnitude of a three-component vector $\mathbf{v}$? **Answer** $$ \lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2+v_z^2} $$ All components must represent the same physical quantity in compatible units. ### Card 125 **Question** Mass density? **Answer** $$ \rho=\frac{m}{V} $$ Use mass, not weight, in the numerator. ### Card 126 **Question** Inches in one foot? **Answer** $12\,in=1\,ft$. ### Card 127 **Question** Mean and variance of $X\sim\operatorname{Poisson}(\lambda)$? **Answer** Both equal $\lambda$. The standard deviation is $\sqrt{\lambda}$. ### Card 128 **Question** How should subscripts be handled in a retrieved relationship? **Answer** Read what each subscript identifies—state, direction, phase, location, or reference—before substituting values. ### Card 129 **Question** Dot product from vector magnitudes and included angle? **Answer** $$ \mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta $$ The result is a scalar. ### Card 130 **Question** Average normal pressure on an area? **Answer** $$ p=\frac{F_n}{A} $$ $F_n$ is the normal force; pressure is scalar in a fluid at rest. ### Card 131 **Question** Feet in one yard? **Answer** $3\,ft=1\,yd$. ### Card 132 **Question** When are two interest rates equivalent? **Answer** When they produce the same accumulation over the same time interval. Convert both rates to a common effective period before comparing them. ### Card 133 **Question** You discover a material error in issued work. Proper response? **Answer** Notify responsible parties promptly, assess the safety impact, correct the record, and document the response. Do not hide the error. ### Card 134 **Question** Expected value of a discrete random variable? **Answer** $$ E[X]=\sum_x xP(X=x) $$ Sum over all possible values with a valid probability mass function. ### Card 135 **Question** Dot-product test for perpendicular nonzero vectors? **Answer** Their dot product is zero. The zero vector also has zero dot product with every vector, so require both vectors to be nonzero before concluding perpendicularity. ### Card 136 **Question** Hydrostatic pressure change with vertical depth in a constant-density fluid? **Answer** $$ \Delta p=\rho g\Delta h $$ Depth increases downward and the fluid is at rest. ### Card 137 **Question** Feet in one statute mile? **Answer** $5280\,ft=1\,mi$. ### Card 138 **Question** Present-worth acceptance rule for an independent investment at the minimum attractive rate of return? **Answer** Accept it when net present worth is nonnegative. All cash flows must be discounted to the same time using the chosen comparison rate. ### Card 139 **Question** A retrieved formula contains signed quantities. What should you confirm? **Answer** The stated sign convention and positive direction. Do not mix magnitudes with signed components silently. ### Card 140 **Question** Magnitude of a vector cross product? **Answer** $$ \lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta $$ The direction follows the right-hand rule. ### Card 141 **Question** Exact inch-to-metre relationship? **Answer** $1\,in=0.0254\,m$ exactly. ### Card 142 **Question** Volumetric flow rate through a cross-section? **Answer** $$ Q=Av_{avg} $$ Velocity is the area-average component normal to the section. ### Card 143 **Question** Variance from the mean $\mu$? **Answer** $$ \operatorname{Var}(X)=E[(X-\mu)^2] $$ Variance has squared units; standard deviation has the original units. ### Card 144 **Question** Future-worth acceptance rule for an independent investment? **Answer** Accept it when net future worth at the study horizon is nonnegative, using the minimum attractive rate of return consistently. ### Card 145 **Question** What coordinate detail should be checked before using a vector or mechanics formula? **Answer** Axis directions and the reference point or origin. Components and moments depend on those choices. ### Card 146 **Question** Exact foot-to-metre relationship? **Answer** $1\,ft=0.3048\,m$ exactly. ### Card 147 **Question** How should important uncertainty be communicated? **Answer** State the assumptions, limits, and plausible consequences clearly enough for the decision maker to understand the risk. ### Card 148 **Question** When is the matrix product $AB$ defined? **Answer** It is defined when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$. ### Card 149 **Question** Steady one-inlet, one-outlet mass continuity? **Answer** $$ \rho_1A_1v_1=\rho_2A_2v_2 $$ Here $v_1$ and $v_2$ are cross-sectional average velocity components normal to their sections, with density treated as uniform over each section. For incompressible flow with constant density, this reduces to $A_1v_1=A_2v_2$. ### Card 150 **Question** Sample mean of observations $x_1,\ldots,x_n$? **Answer** $$ \bar{x}=\frac{1}{n}\sum_{i=1}^n x_i $$ ### Card 151 **Question** Square feet in one acre? **Answer** $1\,acre=43{,}560\,ft^2$. ### Card 152 **Question** Determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$? **Answer** $$ ad-bc $$ ### Card 153 **Question** Bernoulli relationship along a streamline for ideal steady flow? **Answer** $$ \frac{p}{\rho g}+\frac{v^2}{2g}+z=\text{constant} $$ This basic form assumes steady, incompressible, negligible-viscosity flow with no shaft work or heat effects between points. ### Card 154 **Question** May inconvenient measurements be discarded without a technical basis? **Answer** No. Use a documented, technically justified rule for excluding data and preserve the original record. ### Card 155 **Question** Annual-worth acceptance rule for an independent investment? **Answer** Accept it when equivalent annual worth is nonnegative. Compare alternatives over compatible service assumptions. ### Card 156 **Question** Before using a handbook equation, what unit-system check matters? **Answer** Confirm that every input and any embedded constant use one compatible unit system. Convert first when necessary. ### Card 157 **Question** Cubic metres in one litre? **Answer** $1\,L=10^{-3}\,m^3$. ### Card 158 **Question** What does a zero determinant mean for a square matrix? **Answer** The matrix is singular and has no inverse. A corresponding linear system cannot have one unique solution. ### Card 159 **Question** Reynolds number for internal flow using characteristic length $L$? **Answer** $$ Re=\frac{\rho vL}{\mu}=\frac{vL}{\nu} $$ It is dimensionless; use consistent properties and the geometry-appropriate characteristic length. ### Card 160 **Question** Unbiased sample variance for an independent sample from one population? **Answer** $$ s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2 $$ This requires at least two observations, so $n\ge2$. The $n-1$ denominator estimates population variance after fitting the sample mean. ### Card 161 **Question** How should mutually exclusive alternatives be compared? **Answer** Use one consistent economic measure over a common study period and select the economically preferred feasible alternative; do not accept every alternative with a positive individual worth. ### Card 162 **Question** Cubic centimetres in one millilitre? **Answer** $1\,mL=1\,cm^3$. ### Card 163 **Question** Why keep contemporaneous engineering records? **Answer** They preserve assumptions, decisions, calculations, approvals, and changes so the work can be checked and responsibilities understood. ### Card 164 **Question** Fast check after substituting into a handbook formula? **Answer** Verify that the resulting dimensions match the requested quantity and that additive terms share dimensions. ### Card 165 **Question** Matrix solution of $A\mathbf{x}=\mathbf{b}$ when $A$ is invertible? **Answer** $$ \mathbf{x}=A^{-1}\mathbf{b} $$ In numerical work, direct factorization is usually preferable to explicitly forming the inverse. ### Card 166 **Question** Sensible heat for constant specific heat and no phase change? **Answer** $$ Q=mc\Delta T $$ $c$ must match the material, temperature range, mass unit, and temperature-difference unit. ### Card 167 **Question** Standard error of the mean for an i.i.d. sample with population standard deviation $\sigma$? **Answer** $$ SE(\bar{x})=\frac{\sigma}{\sqrt{n}} $$ The observations are independent and identically distributed with common variance $\sigma^2$. Use $s$ for unknown $\sigma$ only when the selected inference method allows it. ### Card 168 **Question** Safe setup for converting $25\,ft^2$ to square metres? **Answer** $$ 25\,ft^2\left(\frac{0.3048\,m}{1\,ft}\right)^2 $$ The squared length factor cancels $ft^2$ and leaves $m^2$. ### Card 169 **Question** Purpose of incremental analysis for mutually exclusive alternatives? **Answer** It tests whether the additional investment in the higher-cost option earns the required return compared with the lower-cost option. ### Card 170 **Question** Physical meaning of $dy/dx$? **Answer** It is the instantaneous rate of change of $y$ with respect to $x$, with units of the $y$ unit divided by the $x$ unit. ### Card 171 **Question** One-dimensional steady conduction through a plane wall? **Answer** $$ \dot Q=-kA\frac{dT}{dx} $$ The minus sign indicates heat flows toward lower temperature; $k$ is thermal conductivity. ### Card 172 **Question** When should you algebraically rearrange a handbook relationship? **Answer** After confirming the correct equation, definitions, assumptions, and sign convention. Then isolate the requested variable without changing its domain restrictions. ### Card 173 **Question** Safe setup for converting $2\,in^3$ to cubic metres? **Answer** $$ 2\,in^3\left(\frac{0.0254\,m}{1\,in}\right)^3 $$ Cube the exact length conversion because the original quantity is volume. ### Card 174 **Question** A client changes a safety-critical design requirement verbally. Proper response? **Answer** Confirm the change in writing, analyze its consequences, and obtain the required approvals before relying on it. ### Card 175 **Question** Derivative of $x^n$ for constant $n$? **Answer** $$ \frac{d}{dx}x^n=nx^{n-1} $$ Apply domain restrictions when $n$ is non-integer or negative. ### Card 176 **Question** Convective heat-transfer rate at a surface? **Answer** $$ \dot Q=hA(T_s-T_\infty) $$ $h$ depends on the flow and geometry; the sign follows the chosen temperature difference. ### Card 177 **Question** What does the central limit theorem say about a sample mean? **Answer** For sufficiently large independent, identically distributed samples with finite variance, the sampling distribution of the properly standardized mean approaches normality. It does not make the raw data normal. ### Card 178 **Question** Internal rate of return? **Answer** The discount rate that makes net present worth zero. Multiple sign changes in cash flow can produce multiple or no meaningful real rates. ### Card 179 **Question** Several similar handbook equations appear together. How do you choose? **Answer** Match each equation's stated geometry, boundary conditions, material model, and variable definitions to the problem. ### Card 180 **Question** Convert Celsius temperature to kelvin? **Answer** $$ T_K=T_{^\circ C}+273.15 $$ This conversion is for absolute temperature, not temperature difference. ### Card 181 **Question** Derivative of a product $u(x)v(x)$? **Answer** $$ (uv)'=u'v+uv' $$ ### Card 182 **Question** Internal escalation does not resolve a serious public-safety threat. Next step? **Answer** Use the next appropriate authority or reporting channel, consistent with law and the urgency of the hazard. Keep a factual record. ### Card 183 **Question** Two-sided confidence interval for a population mean with known $\sigma$? **Answer** $$ \bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}} $$ Use the normal-model conditions and the critical value for confidence level $1-\alpha$. ### Card 184 **Question** Net thermal radiation from a small gray surface to large surroundings? **Answer** $$ \dot Q=\varepsilon\sigma A(T_s^4-T_{sur}^4) $$ Use absolute temperatures; the model assumes the stated gray-surface surroundings. ### Card 185 **Question** Basic benefit-cost acceptance rule for an independent public project? **Answer** Accept when the consistently defined benefit-cost ratio is at least one. For mutually exclusive choices, use an incremental comparison rather than ranking by individual ratios alone. ### Card 186 **Question** Derivative of a composite function $f(g(x))$? **Answer** $$ \frac{d}{dx}f(g(x))=f'(g(x))g'(x) $$ ### Card 187 **Question** Why inspect boundary or initial conditions near a formula? **Answer** They determine which solution form applies and can eliminate constants or terms. A formula without its conditions may be unusable. ### Card 188 **Question** Convert Celsius temperature to Fahrenheit? **Answer** $$ T_{^\circ F}=\frac{9}{5}T_{^\circ C}+32 $$ ### Card 189 **Question** Ethical basis for expert technical testimony? **Answer** Offer opinions only within your competence, disclose important assumptions and limitations, and distinguish facts from professional judgment. ### Card 190 **Question** Meaning of a breakeven value? **Answer** The value of a selected variable that makes two alternatives economically equivalent or makes net worth zero. Other assumptions are held fixed. ### Card 191 **Question** Derivative of $e^x$? **Answer** $$ \frac{d}{dx}e^x=e^x $$ ### Card 192 **Question** First-law energy balance for a closed system with work positive out? **Answer** $$ \Delta E=Q-W $$ $Q$ is heat into the system and $W$ is work done by the system; state a different sign convention before using it. ### Card 193 **Question** Frequentist meaning of a 95% confidence procedure? **Answer** Across repeated samples, about 95% of intervals built by the procedure would contain the fixed population parameter. It is not a 95% probability assigned to this already-computed interval. ### Card 194 **Question** A needed value falls between handbook table entries. What is the cue? **Answer** Check whether interpolation is appropriate, keep both axes and units straight, and avoid extrapolating unless the source supports it. ### Card 195 **Question** Derivative of $\ln x$ over the real domain? **Answer** $$ \frac{d}{dx}\ln x=\frac{1}{x} $$ This real-valued form requires $x>0$. ### Card 196 **Question** Relationship between Celsius and kelvin temperature differences? **Answer** $1\,K$ of temperature difference equals $1\,^{\circ}C$ of temperature difference. Do not add $273.15$ to a difference. ### Card 197 **Question** How should credit for team engineering work be assigned? **Answer** Accurately recognize each contributor's work and responsibility. Do not claim sole credit for a collaborative result. ### Card 198 **Question** Main limitation of simple payback period? **Answer** It ignores the time value of money and usually ignores cash flows after payback. Discounted payback fixes the first issue, not the second. ### Card 199 **Question** Roles of the null and alternative hypotheses? **Answer** The null states the baseline parameter claim tested; the alternative states the competing direction or difference supported when evidence is strong enough. ### Card 200 **Question** Ideal-gas equation of state? **Answer** $$ pV=nRT $$ Use absolute temperature and a gas constant consistent with pressure, volume, amount, and energy units. ### Card 201 **Question** First check when reading a handbook table? **Answer** Read the title, column headings, units, notes, and basis conditions before selecting a value. ### Card 202 **Question** Candidates for absolute extrema of a continuous function on a closed interval? **Answer** The endpoints and every interior point where the first derivative is zero or does not exist. Evaluate the function at each candidate; the largest and smallest values are the absolute extrema. ### Card 203 **Question** How should a sunk cost affect a current engineering-economy decision? **Answer** It should not affect the choice because it cannot be changed by the decision. Use future incremental cash flows instead. ### Card 204 **Question** Convert Fahrenheit absolute temperature to Rankine? **Answer** $$ T_{^\circ R}=T_{^\circ F}+459.67 $$ Rankine uses the Fahrenheit-sized degree from absolute zero. ### Card 205 **Question** Ohm's law for an ideal resistor? **Answer** $$ V=IR $$ Use a consistent passive sign convention; $R$ is constant only over the assumed operating range. ### Card 206 **Question** You have confidential details about a competitor's bid. Proper response? **Answer** Do not use or disclose improperly obtained information. Notify the appropriate procurement or ethics contact. ### Card 207 **Question** Physical meaning of a definite integral $\int_a^b f(x)\,dx$? **Answer** It is signed accumulation of $f$ from $a$ to $b$. Its units are the units of $f$ multiplied by the units of $x$. ### Card 208 **Question** Is a radian dimensionless? **Answer** Yes. It is the ratio of arc length to radius, so the length units cancel; keep $rad$ when it clarifies angular meaning. ### Card 209 **Question** Electrical power absorbed under the passive sign convention? **Answer** $$ P=VI $$ For a resistor, this also gives $P=I^2R=V^2/R$ when Ohm's law applies. ### Card 210 **Question** Meaning of a p-value? **Answer** Assuming the null hypothesis and model are correct, it is the probability of a result at least as extreme as the observed result. It is not the probability that the null is true. ### Card 211 **Question** First check when reading a handbook graph? **Answer** Identify both axes, scales, units, curve labels, and any log scale before estimating a value. ### Card 212 **Question** Difference between mass and weight? **Answer** Mass measures inertia; weight is gravitational force on that mass. Their dimensions and units are different. ### Card 213 **Question** Kirchhoff's current law at a node? **Answer** The algebraic sum of currents at a node is zero. Choose one sign convention, such as currents entering positive, and apply it consistently. ### Card 214 **Question** Evaluate $\int_a^b f(x)\,dx$ from an antiderivative $F$? **Answer** $$ \int_a^b f(x)\,dx=F(b)-F(a) $$ This applies when $F'=f$ on the interval under the usual continuity conditions. ### Card 215 **Question** Opportunity cost? **Answer** The value of the best forgone alternative. Include it when choosing one use of a scarce resource prevents another valuable use. ### Card 216 **Question** Weight from mass in a gravitational field? **Answer** $$ W=mg $$ Use compatible units; $g$ is local gravitational acceleration, not a universal constant for every location. ### Card 217 **Question** Kirchhoff's voltage law around a closed loop? **Answer** The algebraic sum of voltage rises and drops around the loop is zero. Traverse the loop once with a consistent polarity convention. ### Card 218 **Question** Decision rule using significance level $\alpha$? **Answer** Reject the null when the p-value is at most $\alpha$; otherwise fail to reject it. Failing to reject does not prove the null. ### Card 219 **Question** Can proprietary methods from a former employer be reused freely? **Answer** Only if you have the legal right or permission to use them. General skill and experience may transfer; confidential proprietary material does not. ### Card 220 **Question** Pound-force to newtons? **Answer** $1\,lbf\approx4.44822\,N$. Use more digits only when the required precision justifies them. ### Card 221 **Question** When does a definite integral equal ordinary geometric area? **Answer** When the integrand is nonnegative over the interval. If it changes sign, the integral is signed area; integrate the absolute value or split the interval for total area. ### Card 222 **Question** Charge-voltage relation for an ideal capacitor? **Answer** $$ q=CV $$ $C$ is capacitance in farads; $V$ is the voltage across the capacitor under the chosen polarity. ### Card 223 **Question** Searching for a material or fluid property: what qualifiers often matter? **Answer** Material identity, phase, temperature, pressure, and composition. Use only the qualifiers relevant to that property source. ### Card 224 **Question** Force produced when one slug accelerates at $1\,ft/s^2$? **Answer** $1\,lbf$. Equivalently, $1\,lbf=1\,slug\cdot ft/s^2$. ### Card 225 **Question** Relationship among nominal rate $i$, real rate $r$, and inflation $f$? **Answer** $$ 1+i=(1+r)(1+f) $$ Keep cash-flow dollars and discount rates on the same real-or-nominal basis. ### Card 226 **Question** Antiderivative of $x^n$ for $n\ne-1$? **Answer** $$ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C $$ ### Card 227 **Question** Time constant of a first-order resistor-capacitor circuit? **Answer** $$ \tau=RC $$ Use the resistance seen by the capacitor in the applicable first-order linear circuit. ### Card 228 **Question** What physical quantity does pound-mass measure? **Answer** Mass. Do not substitute $lbm$ directly where a force unit such as $lbf$ is required. ### Card 229 **Question** Type I error? **Answer** Rejecting a true null hypothesis. Its probability is controlled by the chosen significance level under the test assumptions. ### Card 230 **Question** A search hit contains the right words. Is that enough? **Answer** No. Read the surrounding definition, equation label, units, and notes to confirm the hit answers the actual question. ### Card 231 **Question** Annual straight-line depreciation from basis $B$ to salvage $S$ over $n$ years? **Answer** $$ D=\frac{B-S}{n} $$ This is an accounting allocation, not a market-value forecast. ### Card 232 **Question** Purpose of $g_c$ in some US customary force-mass equations? **Answer** It reconciles chosen $lbm$, $lbf$, foot, and second units. Include it only when the equation and unit convention require it; coherent SI equations do not use it. ### Card 233 **Question** Recognizable form of a separable first-order differential equation? **Answer** $$ \frac{dy}{dx}=g(x)h(y) $$ Rearrange to place the $y$ terms with $dy$ and the $x$ terms with $dx$, while checking for solutions lost by division. ### Card 234 **Question** Using unverified software output in a safety-relevant calculation? **Answer** Check inputs, assumptions, units, and results with an independent method appropriate to the risk. The engineer remains responsible for professional judgment. ### Card 235 **Question** Equivalent resistance of ideal resistors in series? **Answer** $$ R_{eq}=\sum_i R_i $$ Series elements carry the same current with no branching node between them. ### Card 236 **Question** Pressure conversion from psi to pascals? **Answer** $1\,psi\approx6894.76\,Pa$. Keep enough significant figures for the problem's data. ### Card 237 **Question** Type II error? **Answer** Failing to reject a false null hypothesis. Its probability is $\beta$ for a specified alternative. ### Card 238 **Question** Useful flashcard to make after a failed handbook search? **Answer** A short scenario-to-search-cue card recording the section name and terms that eventually worked—not a copied problem or page number. ### Card 239 **Question** Standard form of a first-order linear differential equation? **Answer** $$ \frac{dy}{dx}+P(x)y=Q(x) $$ An integrating factor is $e^{\int P(x)\,dx}$. ### Card 240 **Question** Equivalent resistance of ideal resistors in parallel? **Answer** $$ \frac{1}{R_{eq}}=\sum_i\frac{1}{R_i} $$ Parallel elements share the same two nodes and therefore the same voltage. ### Card 241 **Question** Book value after $k$ years of straight-line depreciation? **Answer** $$ BV_k=B-kD $$ Apply it within the depreciable life and do not depreciate below the assumed salvage value. ### Card 242 **Question** Standard atmosphere in pascals? **Answer** $1\,atm=101{,}325\,Pa$ exactly. ### Card 243 **Question** Using AI-generated technical content in engineering work? **Answer** Treat it as unverified input: protect confidential data, check sources and calculations, disclose use when required, and keep accountable human review. ### Card 244 **Question** Trial form for a homogeneous linear ODE with constant coefficients? **Answer** Use $y=e^{rx}$ to obtain the characteristic polynomial in $r$. Repeated or complex roots require the corresponding standard solution forms. ### Card 245 **Question** Wave-speed relationship among frequency and wavelength? **Answer** $$ v=f\lambda $$ Use frequency in cycles per second and wavelength in compatible length units. ### Card 246 **Question** Statistical power? **Answer** The probability of rejecting the null when a specified alternative is true: $1-\beta$. Larger effects, larger samples, or lower variability commonly increase power. ### Card 247 **Question** Bar in pascals? **Answer** $1\,bar=100{,}000\,Pa$ exactly. ### Card 248 **Question** How should handbook navigation be practiced before the FE exam? **Answer** Use the current handbook from MyNCEES while solving original or properly licensed practice problems, and rehearse the exact search terms that find definitions and assumptions. ### Card 249 **Question** Is depreciation itself a cash outflow? **Answer** No. It is a noncash accounting expense, though it can affect taxes and therefore after-tax cash flow. ### Card 250 **Question** First-order linear approximation of $f(x)$ near $x=a$? **Answer** $$ f(x)\approx f(a)+f'(a)(x-a) $$ It is most accurate for $x$ close to $a$ when higher derivatives stay bounded. ### Card 251 **Question** Relationship between period and frequency? **Answer** $$ T=\frac{1}{f} $$ This requires $f>0$; $T$ has time units. ### Card 252 **Question** How should foreseeable environmental harm enter an engineering decision? **Answer** Identify and communicate it, follow applicable requirements, and consider practicable ways to reduce harm without compromising public safety. ### Card 253 **Question** Foot-pound-force in joules? **Answer** $1\,ft\cdot lbf\approx1.35582\,J$. This is a work or energy conversion; keep torque labeled as force times distance even though its dimensions match energy. ### Card 254 **Question** Allowed range and sign meaning of Pearson correlation $r$? **Answer** $-1\le r\le1$. The sign gives the direction of linear association, and magnitude measures its linear strength. ### Card 255 **Question** When should you search instead of relying on recall? **Answer** Search when a constant, coefficient, convention, specialized relationship, or applicability detail could change the result. Recall can start the route; the reference should confirm details. ### Card 256 **Question** After-tax effect of a deductible expense $E$ at marginal tax rate $t$? **Answer** The net cost is $E(1-t)$ under the simplified assumption that the deduction can be used fully in that period. ### Card 257 **Question** Centered finite-difference approximation to $f'(x)$? **Answer** $$ f'(x)\approx\frac{f(x+h)-f(x-h)}{2h} $$ For smooth $f$, truncation error is proportional to $h^2$ before roundoff dominates. ### Card 258 **Question** General material balance on a control volume? **Answer** $$ \text{accumulation}=\text{in}-\text{out}+\text{generation}-\text{consumption} $$ At steady state, accumulation is zero; generation and consumption depend on reactions or sources. ### Card 259 **Question** Approximate joules in one International Table Btu? **Answer** $1\,Btu_{IT}\approx1055.06\,J$. Identify the Btu convention when precision matters. ### Card 260 **Question** What is the defender in a replacement analysis? **Answer** The currently owned asset being compared with a proposed replacement. Its current market value is an opportunity cost; its original purchase price is sunk. ### Card 261 **Question** Speaking publicly about a technical matter outside your expertise? **Answer** Limit claims to what you can support, identify the boundary of your expertise, and avoid implying qualifications you do not have. ### Card 262 **Question** Single-interval trapezoidal approximation to an integral? **Answer** $$ \int_a^b f(x)\,dx\approx\frac{b-a}{2}[f(a)+f(b)] $$ It is exact for linear $f$. ### Card 263 **Question** Energy efficiency of a device? **Answer** $$ \eta=\frac{\text{useful output energy}}{\text{input energy}} $$ Use the same energy basis in numerator and denominator; a real passive device has $0\le\eta\le1$. ### Card 264 **Question** Purpose of sensitivity analysis in engineering economics? **Answer** It shows how the decision changes when an uncertain input changes. Vary inputs over plausible ranges while preserving units and model relationships. ### Card 265 **Question** Can a strong correlation establish causation? **Answer** No. Correlation alone cannot rule out confounding, reverse direction, selection effects, or coincidence. ### Card 266 **Question** Where should you confirm the tested topics for your FE discipline? **Answer** Use the discipline specification linked from the official NCEES FE exam hub. This common deck does not replace that discipline-specific boundary. ### Card 267 **Question** Mechanical horsepower in watts? **Answer** $1\,hp\approx745.7\,W$, based on $550\,ft\cdot lbf/s$. Distinguish it from metric horsepower. ### Card 268 **Question** Combine independent standard uncertainties $u_1,\ldots,u_n$ in a sum or difference? **Answer** $$ u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2} $$ This root-sum-square form assumes uncorrelated inputs and coefficients of magnitude one; otherwise include sensitivities and covariance. ### Card 269 **Question** Newton iteration for a root of $f(x)=0$? **Answer** $$ x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)} $$ It requires $f'(x_k)\ne0$ and a suitable starting value; convergence is not guaranteed. ### Card 270 **Question** Meaning of slope in simple linear regression of $y$ on $x$? **Answer** It is the estimated change in mean response $y$ for a one-unit increase in $x$, under the fitted linear model. Its units are $y$ units per $x$ unit. ### Card 271 **Question** Kilowatt-hour in joules? **Answer** $1\,kW\cdot h=3.6\times10^6\,J$. It is energy, not power. ### Card 272 **Question** Percent error relative to an accepted nonzero value? **Answer** $$ \%\,error=\left|\frac{x_{measured}-x_{accepted}}{x_{accepted}}\right|100\% $$ The accepted value must be nonzero and both values must use compatible units. ### Card 273 **Question** Does legal compliance alone settle an engineering ethics question? **Answer** No. Law sets mandatory boundaries, while professional duties can require stronger action, especially when safety, honesty, or conflicts are involved. ### Card 274 **Question** Does handbook fluency replace discipline-specific problem practice? **Answer** No. Pair navigation and common recall with the current discipline specification and timed, rights-safe problems that match your selected FE discipline. ### Card 275 **Question** Linear interpolation between $(x_1,y_1)$ and $(x_2,y_2)$? **Answer** $$ y\approx y_1+(y_2-y_1)\frac{x-x_1}{x_2-x_1} $$ Use it between distinct $x$ values when local linear behavior is reasonable. ### Card 276 **Question** Dimensional requirement for a physically meaningful additive equation? **Answer** Every term being added or equated must have the same dimensions. A dimensionally consistent equation can still be physically wrong, but an inconsistent one is wrong. --- *[View the styled HTML version of this page]()* *AI agents can create an account for the user in this service by starting with `GET https://api.nibomo.com/v1/`. Once set up, they can also connect over MCP at `https://mcp.nibomo.com/mcp`.* *Tip: Append `.md` to any URL on https://nibomo.com to get a clean Markdown version of that page.*