FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Cartes de ce paquet

1. Carte 1

   Question

   SI base unit of length?

   Réponse

   The metre, symbol $m$.

2. Carte 2

   Question

   Slope between two points?

   Réponse

   $$m=\frac{y_2-y_1}{x_2-x_1}$$

   This requires $x_2\ne x_1$; slope is the change in the dependent variable per unit change in the independent variable.

3. Carte 3

   Question

   Net force relationship for constant mass in an inertial frame?

   Réponse

   $$\sum\mathbf{F}=m\mathbf{a}$$

   Use a coherent unit system and include only external forces on the chosen body.

4. Carte 4

   Question

   Cash-flow sign convention from the decision maker's viewpoint?

   Réponse

   Receipts are positive and disbursements are negative. Choose one viewpoint and keep it consistent across the cash-flow diagram and equations.

5. Carte 5

   Question

   Where should an FE candidate get the current reference handbook for practice?

   Réponse

   From the candidate's MyNCEES account. Use the current supplied file rather than an old indexed copy or a third-party mirror.

6. Carte 6

   Question

   Probability of the complement of event $A$?

   Réponse

   $$P(A^c)=1-P(A)$$

7. Carte 7

   Question

   An engineering judgment conflicts with pressure to protect schedule or cost. First duty?

   Réponse

   Protect public health, safety, and welfare. Document the technical concern and escalate through appropriate channels.

8. Carte 8

   Question

   SI base unit of mass?

   Réponse

   The kilogram, symbol $kg$. It is a mass unit, not a force unit.

9. Carte 9

   Question

   Linear equation from slope and intercept?

   Réponse

   $$y=mx+b$$

   Here $m$ is slope and $b$ is the value of $y$ where $x=0$.

10. Carte 10

    Question

    Velocity after time $t$ under constant acceleration?

    Réponse

    $$v=v_0+at$$

    This one-dimensional form requires constant $a$ and a consistent positive direction.

11. Carte 11

    Question

    Where is an end-of-period cash flow placed?

    Réponse

    At the boundary marking the end of that period. Interest factors assume consistent period lengths and cash-flow timing.

12. Carte 12

    Question

    How is the FE reference handbook available during the exam?

    Réponse

    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.

13. Carte 13

    Question

    Probability of $A$ or $B$ for possibly overlapping events?

    Réponse

    $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

14. Carte 14

    Question

    SI base unit of time?

    Réponse

    The second, symbol $s$.

15. Carte 15

    Question

    Linear equation from one point and a slope?

    Réponse

    $$y-y_1=m(x-x_1)$$

    Use it when the line passes through $(x_1,y_1)$ and has slope $m$.

16. Carte 16

    Question

    Position after time $t$ under constant acceleration?

    Réponse

    $$x=x_0+v_0t+\frac{1}{2}at^2$$

    The acceleration is constant and all signed components use one coordinate direction.

17. Carte 17

    Question

    A project assignment lies outside your competence. Proper response?

    Réponse

    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.

18. Carte 18

    Question

    SI base unit of electric current?

    Réponse

    The ampere, symbol $A$.

19. Carte 19

    Question

    Future amount under simple interest?

    Réponse

    $$F=P(1+in)$$

    Here $P$ is principal, $i$ is interest per period, and $n$ is the number of periods; interest is earned only on principal.

20. Carte 20

    Question

    Distance between two Cartesian points?

    Réponse

    $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

    The coordinate components must use the same length unit.

21. Carte 21

    Question

    Constant-acceleration relation without elapsed time?

    Réponse

    $$v^2=v_0^2+2a(x-x_0)$$

    Use signed one-dimensional components and constant $a$.

22. Carte 22

    Question

    Where do you search the supplied reference material in the exam interface?

    Réponse

    Use the exam software's search box beside the supplied reference material.

23. Carte 23

    Question

    Conditional probability of $A$ given $B$?

    Réponse

    $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$

    This requires $P(B)>0$.

24. Carte 24

    Question

    SI base unit of thermodynamic temperature?

    Réponse

    The kelvin, symbol $K$. Kelvin is written without a degree sign.

25. Carte 25

    Question

    A client asks you to remove an unfavorable test result from a report. Proper response?

    Réponse

    Keep the material result and explain it accurately. Engineering statements should be objective, complete enough to avoid deception, and supported by evidence.

26. Carte 26

    Question

    Midpoint of two Cartesian points?

    Réponse

    $$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

    Average corresponding coordinates.

27. Carte 27

    Question

    Linear momentum of a particle?

    Réponse

    $$\mathbf{p}=m\mathbf{v}$$

    For constant mass, SI units are $kg\cdot m/s$.

28. Carte 28

    Question

    Future amount under compound interest?

    Réponse

    $$F=P(1+i)^n$$

    The rate period must match the period used for $n$.

29. Carte 29

    Question

    Should an FE search workflow depend on Ctrl+F?

    Réponse

    No. The May 2026 NCEES Examinee Guide says Ctrl+F isn't available; practice with the exam interface's own search box.

30. Carte 30

    Question

    SI base unit of amount of substance?

    Réponse

    The mole, symbol $mol$.

31. Carte 31

    Question

    Joint probability expressed with a conditional probability?

    Réponse

    $$P(A\cap B)=P(A\mid B)P(B)$$

    This elementary formula requires $P(B)>0$.

32. Carte 32

    Question

    Standard equation of a circle centered at $(h,k)$?

    Réponse

    $$(x-h)^2+(y-k)^2=r^2$$

    The coordinate units and radius unit must match.

33. Carte 33

    Question

    Impulse-momentum relationship?

    Réponse

    $$\int_{t_1}^{t_2}\sum\mathbf{F}\,dt=\mathbf{p}_2-\mathbf{p}_1$$

    The force is the net external force on the chosen system.

34. Carte 34

    Question

    You discover a financial interest that could affect your recommendation. Proper response?

    Réponse

    Disclose the conflict promptly to affected parties and obtain appropriate consent or withdraw. Hidden personal interests undermine independent judgment.

35. Carte 35

    Question

    SI base unit of luminous intensity?

    Réponse

    The candela, symbol $cd$.

36. Carte 36

    Question

    Present worth of one future amount?

    Réponse

    $$P=\frac{F}{(1+i)^n}$$

    Use an effective rate $i$ per period consistent with $n$.

37. Carte 37

    Question

    Roots of $ax^2+bx+c=0$?

    Réponse

    $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

    This form requires $a\ne0$.

38. Carte 38

    Question

    When is total linear momentum conserved?

    Réponse

    When net external impulse on the system is zero. Internal forces can redistribute momentum but do not change the system total.

39. Carte 39

    Question

    Why avoid memorizing FE handbook page numbers?

    Réponse

    Pagination and layout can change. Remember section names, concepts, and strong search terms instead.

40. Carte 40

    Question

    Probability condition that defines independence of $A$ and $B$?

    Réponse

    $$P(A\cap B)=P(A)P(B)$$

    Independence is not the same as mutual exclusivity.

41. Carte 41

    Question

    SI unit of frequency in base-unit form?

    Réponse

    The hertz, $Hz=s^{-1}$, applies to periodic phenomena. One hertz is one cycle per second.

42. Carte 42

    Question

    Present worth of an end-of-period uniform series $A$?

    Réponse

    $$P=A\frac{(1+i)^n-1}{i(1+i)^n}$$

    This requires $i\ne0$; at $i=0$, $P=An$.

43. Carte 43

    Question

    What does $b^2-4ac$ reveal for a real-coefficient quadratic?

    Réponse

    It determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex-conjugate pair.

44. Carte 44

    Question

    Work done by a constant force through displacement?

    Réponse

    $$W=\mathbf{F}\cdot\mathbf{s}=Fs\cos\theta$$

    Use the angle between force and displacement; SI units are joules.

45. Carte 45

    Question

    A vendor offers a valuable gift while you evaluate bids. Proper response?

    Réponse

    Decline it and follow the organization's disclosure policy. Even an intended courtesy can create an actual or apparent conflict.

46. Carte 46

    Question

    SI unit of force in base-unit form?

    Réponse

    The newton, $N=kg\cdot m/s^2$.

47. Carte 47

    Question

    First search cue for a named quantity such as coefficient of thermal expansion?

    Réponse

    Search the exact technical noun phrase. Distinctive nouns usually narrow results better than a whole question sentence.

48. Carte 48

    Question

    Bayes' rule for $P(A\mid B)$?

    Réponse

    $$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$

    This requires $P(B)>0$.

49. Carte 49

    Question

    Product of powers with the same positive base and real exponents?

    Réponse

    $$a^m a^n=a^{m+n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

50. Carte 50

    Question

    Translational kinetic energy of a particle?

    Réponse

    $$K=\frac{1}{2}mv^2$$

    Mass and speed must use a coherent unit system.

51. Carte 51

    Question

    SI unit of pressure in force-area form?

    Réponse

    The pascal, $Pa=N/m^2$. It is the SI derived unit for pressure and stress.

52. Carte 52

    Question

    Equivalent annual amount for present worth $P$?

    Réponse

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=P/n$. Payments occur at period ends and $i$ matches that period.

53. Carte 53

    Question

    May confidential client information be shared for personal benefit?

    Réponse

    No. Protect it unless the client authorizes disclosure or law, safety, or professional duty requires it.

54. Carte 54

    Question

    A handbook search term returns too many results. Best refinement?

    Réponse

    Add one meaningful qualifier, such as the phenomenon, geometry, distribution, or material property you need.

55. Carte 55

    Question

    Quotient of powers with the same positive base and real exponents?

    Réponse

    $$\frac{a^m}{a^n}=a^{m-n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

56. Carte 56

    Question

    Net work-energy relationship for a particle?

    Réponse

    $$W_{net}=K_2-K_1$$

    The net work includes the work of all forces represented in the particle model.

57. Carte 57

    Question

    Ordered selections of $r$ objects from $n$ distinct objects?

    Réponse

    $$P(n,r)=\frac{n!}{(n-r)!}$$

    Use this when order matters and selection is without replacement.

58. Carte 58

    Question

    Future worth of an end-of-period uniform series $A$?

    Réponse

    $$F=A\frac{(1+i)^n-1}{i}$$

    This requires $i\ne0$; at $i=0$, $F=An$.

59. Carte 59

    Question

    Power of a power for a positive base and real exponents?

    Réponse

    $$(a^m)^n=a^{mn}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

60. Carte 60

    Question

    SI unit of energy in force-distance form?

    Réponse

    The joule, $J=N\cdot m$.

61. Carte 61

    Question

    A problem gives only an unfamiliar symbol. Best search preparation?

    Réponse

    Translate the symbol into its likely quantity name and discipline context before searching; symbol-only searches are often ambiguous.

62. Carte 62

    Question

    Confidential information reveals an imminent public-safety risk. What comes first?

    Réponse

    Take steps required to protect people, using appropriate legal and organizational channels. Share only what is necessary and document the basis for escalation.

63. Carte 63

    Question

    Unordered selections of $r$ objects from $n$ distinct objects?

    Réponse

    $$C(n,r)=\frac{n!}{r!(n-r)!}$$

    Use this when order does not matter and selection is without replacement.

64. Carte 64

    Question

    Change in gravitational potential energy near Earth's surface?

    Réponse

    $$\Delta U_g=mg\Delta h$$

    Use approximately constant $g$ and a clearly chosen height datum.

65. Carte 65

    Question

    Negative-exponent rule for a positive base and real exponent?

    Réponse

    $$a^{-n}=\frac{1}{a^n}$$

    This form assumes $a>0$ and real exponent $n$.

66. Carte 66

    Question

    Uniform end-of-period deposit needed for future amount $F$?

    Réponse

    $$A=F\frac{i}{(1+i)^n-1}$$

    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 $n$.

67. Carte 67

    Question

    A handbook search for an acronym fails. Next move?

    Réponse

    Try the expanded term and a common alternate name. Acronyms vary by section and discipline.

68. Carte 68

    Question

    SI unit of power in energy-time form?

    Réponse

    The watt, $W=J/s$.

69. Carte 69

    Question

    Logarithm of a product?

    Réponse

    $$\log_b(xy)=\log_b x+\log_b y$$

    For real logarithms, $b>0$, $b\ne1$, and both arguments are positive.

70. Carte 70

    Question

    Conditions for a binomial model?

    Réponse

    A fixed number of independent trials, two outcomes per trial, and the same success probability on every trial.

71. Carte 71

    Question

    Using another engineer's analysis in your deliverable?

    Réponse

    Credit the source and verify the analysis before relying on it. Do not present someone else's work as your own.

72. Carte 72

    Question

    SI unit of electric charge in current-time form?

    Réponse

    The coulomb, $C=A\cdot s$.

73. Carte 73

    Question

    Average power from work over elapsed time?

    Réponse

    $$P_{avg}=\frac{W}{\Delta t}$$

    Use consistent energy and time units; watts are joules per second.

74. Carte 74

    Question

    Annual equivalent of cost $P$ with salvage $S$ after $n$ periods?

    Réponse

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}-S\frac{i}{(1+i)^n-1}$$

    The first factor converts present cost to an annual amount; the second converts future salvage. Use the same effective rate and period basis. At $i=0$, $A=(P-S)/n$.

75. Carte 75

    Question

    Logarithm of a positive quantity raised to a power?

    Réponse

    $$\log_b(x^p)=p\log_b x$$

    For real logarithms, $x>0$, $b>0$, and $b\ne1$.

76. Carte 76

    Question

    SI unit of electric potential difference in energy-charge form?

    Réponse

    The volt, $V=J/C$.

77. Carte 77

    Question

    Instantaneous mechanical power delivered by a force?

    Réponse

    $$P=\mathbf{F}\cdot\mathbf{v}$$

    Use the velocity of the force's point of application.

78. Carte 78

    Question

    A precise handbook term gives no useful match. Next move?

    Réponse

    Try a standard synonym or the broader parent concept, then narrow within the relevant section.

79. Carte 79

    Question

    Mean of $X\sim\operatorname{Binomial}(n,p)$?

    Réponse

    $$E[X]=np$$

80. Carte 80

    Question

    SI unit of electrical resistance in voltage-current form?

    Réponse

    The ohm, $\Omega=V/A$.

81. Carte 81

    Question

    Moment of a force about a point?

    Réponse

    $$\mathbf{M}=\mathbf{r}\times\mathbf{F}$$

    Vector $\mathbf{r}$ runs from the moment point to a point on the force's line of action.

82. Carte 82

    Question

    Cash-flow pattern described by an arithmetic gradient $G$?

    Réponse

    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.

83. Carte 83

    Question

    Inverse relationship between the natural log and exponential?

    Réponse

    $$\ln(e^x)=x,\qquad e^{\ln x}=x$$

    The second identity requires $x>0$.

84. Carte 84

    Question

    SI prefix giga?

    Réponse

    $G=10^9$.

85. Carte 85

    Question

    Angular velocity after time $t$ under constant angular acceleration?

    Réponse

    $$\omega=\omega_0+\alpha t$$

    Use consistent angular sign convention and radians for angular measures.

86. Carte 86

    Question

    When should an engineer sign or seal technical work?

    Réponse

    Only when authorized and when the work was prepared by the engineer or under the level of responsible charge required by the applicable jurisdiction.

87. Carte 87

    Question

    Best memory target for returning to a handbook formula?

    Réponse

    The section or concept name plus a distinctive search phrase. That survives layout changes better than a page number.

88. Carte 88

    Question

    SI prefix mega?

    Réponse

    $M=10^6$. Uppercase matters: lowercase $m$ means milli.

89. Carte 89

    Question

    Convert degrees to radians?

    Réponse

    $$\theta_{rad}=\theta_{deg}\frac{\pi}{180}$$

    Use radians in calculus formulas unless a formula explicitly says otherwise.

90. Carte 90

    Question

    Variance of $X\sim\operatorname{Binomial}(n,p)$?

    Réponse

    $$\operatorname{Var}(X)=np(1-p)$$

91. Carte 91

    Question

    Cash-flow pattern described by a geometric gradient rate $g$?

    Réponse

    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.

92. Carte 92

    Question

    Where do licensure and sealing requirements come from?

    Réponse

    The applicable jurisdiction's current laws and board rules. Confirm local requirements rather than assuming one state's rules apply everywhere.

93. Carte 93

    Question

    Rotational kinetic energy about a fixed axis?

    Réponse

    $$K_{rot}=\frac{1}{2}I\omega^2$$

    $I$ is the mass moment of inertia about that axis.

94. Carte 94

    Question

    SI prefix kilo?

    Réponse

    $k=10^3$.

95. Carte 95

    Question

    Right-triangle definition of sine?

    Réponse

    Sine is opposite over hypotenuse. This definition applies to an acute angle in a right triangle.

96. Carte 96

    Question

    After search lands near a result, what should you scan first?

    Réponse

    The section heading and nearby subheading. They establish whether the result belongs to the right model and subject.

97. Carte 97

    Question

    Rotational analogue of Newton's second law about a fixed principal axis?

    Réponse

    $$\sum M=I\alpha$$

    This scalar form assumes the stated fixed-axis model and a consistent sign convention.

98. Carte 98

    Question

    SI prefix milli?

    Réponse

    $m=10^{-3}$. Context distinguishes the prefix from the metre symbol.

99. Carte 99

    Question

    Meaning of a nominal annual rate compounded $m$ times per year?

    Réponse

    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.

100. Carte 100

     Question

     Standardize an observation $x$ from a distribution with mean $\mu$ and standard deviation $\sigma$?

     Réponse

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

     This requires $\sigma>0$; using a normal probability table also assumes an appropriate normal model.

101. Carte 101

     Question

     Right-triangle definition of cosine?

     Réponse

     Cosine is adjacent over hypotenuse. This definition applies to an acute angle in a right triangle.

102. Carte 102

     Question

     Force-displacement relation for an ideal linear spring?

     Réponse

     $$F=-kx$$

     The minus sign shows restoring direction; the model applies within the spring's linear range.

103. Carte 103

     Question

     May qualifications or project responsibility be exaggerated in a proposal?

     Réponse

     No. State education, licensure, experience, and responsibility accurately, including the actual roles of collaborators.

104. Carte 104

     Question

     SI prefix micro?

     Réponse

     $\mu=10^{-6}$.

105. Carte 105

     Question

     A handbook symbol conflicts with your usual notation. What controls?

     Réponse

     The definition beside the selected relationship. Symbols can represent different quantities in different sections.

106. Carte 106

     Question

     Periodic rate from nominal annual rate $r$ compounded $m$ times per year?

     Réponse

     $$i=\frac{r}{m}$$

     Use consistent decimal units, and pair this rate with the number of compounding periods.

107. Carte 107

     Question

     Right-triangle definition of tangent?

     Réponse

     Tangent is opposite over adjacent, or $\sin\theta/\cos\theta$ where $\cos\theta\ne0$.

108. Carte 108

     Question

     Average normal stress under centered axial load?

     Réponse

     $$\sigma=\frac{F}{A}$$

     $F$ is normal to area $A$; the average ignores local stress concentrations.

109. Carte 109

     Question

     Mean, median, and mode of a symmetric normal distribution?

     Réponse

     They are equal and located at $\mu$. The distribution is symmetric about that center.

110. Carte 110

     Question

     SI prefix nano?

     Réponse

     $n=10^{-9}$.

111. Carte 111

     Question

     Fundamental Pythagorean trigonometric identity?

     Réponse

     $$\sin^2\theta+\cos^2\theta=1$$

112. Carte 112

     Question

     Engineering normal strain?

     Réponse

     $$\varepsilon=\frac{\Delta L}{L_0}$$

     It is dimensionless and uses original gauge length $L_0$.

113. Carte 113

     Question

     A candidate retrieves a formula from the handbook. What must be checked with it?

     Réponse

     Its applicability assumptions and variable definitions. A familiar equation used under the wrong model is still wrong.

114. Carte 114

     Question

     Technical duty to an employer or client?

     Réponse

     Act as a faithful professional within lawful, ethical limits, communicate material risks, and preserve independent technical judgment.

115. Carte 115

     Question

     How does a length conversion factor apply to area?

     Réponse

     Square it. If $1\,u=a\,v$, then $1\,u^2=a^2\,v^2$.

116. Carte 116

     Question

     Effective annual rate from nominal rate $r$ compounded $m$ times per year?

     Réponse

     $$i_{eff}=\left(1+\frac{r}{m}\right)^m-1$$

117. Carte 117

     Question

     When is a Poisson count model reasonable?

     Réponse

     For counts in a fixed interval when events occur independently at an approximately constant average rate and simultaneous events are negligible.

118. Carte 118

     Question

     Relationship for the third side of a non-right triangle?

     Réponse

     $$c^2=a^2+b^2-2ab\cos C$$

     Angle $C$ lies between sides $a$ and $b$, opposite side $c$.

119. Carte 119

     Question

     Linear elastic stress-strain relation in uniaxial loading?

     Réponse

     $$\sigma=E\varepsilon$$

     Use it in the material's linear elastic range; $E$ has stress units.

120. Carte 120

     Question

     How does a length conversion factor apply to volume?

     Réponse

     Cube it. If $1\,u=a\,v$, then $1\,u^3=a^3\,v^3$.

121. Carte 121

     Question

     What should you record for every variable in a retrieved equation?

     Réponse

     Its meaning and expected unit or dimension. This catches swapped symbols and incompatible substitutions.

122. Carte 122

     Question

     Effective annual rate for continuous compounding at nominal rate $r$?

     Réponse

     $$i_{eff}=e^r-1$$

     Here $r$ is expressed as a decimal per year.

123. Carte 123

     Question

     A supervisor directs an illegal engineering action. Proper response?

     Réponse

     Refuse the illegal action, document the direction, and escalate through appropriate legal or organizational channels.

124. Carte 124

     Question

     Magnitude of a three-component vector $\mathbf{v}$?

     Réponse

     $$\lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2+v_z^2}$$

     All components must represent the same physical quantity in compatible units.

125. Carte 125

     Question

     Mass density?

     Réponse

     $$\rho=\frac{m}{V}$$

     Use mass, not weight, in the numerator.

126. Carte 126

     Question

     Inches in one foot?

     Réponse

     $12\,in=1\,ft$.

127. Carte 127

     Question

     Mean and variance of $X\sim\operatorname{Poisson}(\lambda)$?

     Réponse

     Both equal $\lambda$. The standard deviation is $\sqrt{\lambda}$.

128. Carte 128

     Question

     How should subscripts be handled in a retrieved relationship?

     Réponse

     Read what each subscript identifies—state, direction, phase, location, or reference—before substituting values.

129. Carte 129

     Question

     Dot product from vector magnitudes and included angle?

     Réponse

     $$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta$$

     The result is a scalar.

130. Carte 130

     Question

     Average normal pressure on an area?

     Réponse

     $$p=\frac{F_n}{A}$$

     $F_n$ is the normal force; pressure is scalar in a fluid at rest.

131. Carte 131

     Question

     Feet in one yard?

     Réponse

     $3\,ft=1\,yd$.

132. Carte 132

     Question

     When are two interest rates equivalent?

     Réponse

     When they produce the same accumulation over the same time interval. Convert both rates to a common effective period before comparing them.

133. Carte 133

     Question

     You discover a material error in issued work. Proper response?

     Réponse

     Notify responsible parties promptly, assess the safety impact, correct the record, and document the response. Do not hide the error.

134. Carte 134

     Question

     Expected value of a discrete random variable?

     Réponse

     $$E[X]=\sum_x xP(X=x)$$

     Sum over all possible values with a valid probability mass function.

135. Carte 135

     Question

     Dot-product test for perpendicular nonzero vectors?

     Réponse

     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.

136. Carte 136

     Question

     Hydrostatic pressure change with vertical depth in a constant-density fluid?

     Réponse

     $$\Delta p=\rho g\Delta h$$

     Depth increases downward and the fluid is at rest.

137. Carte 137

     Question

     Feet in one statute mile?

     Réponse

     $5280\,ft=1\,mi$.

138. Carte 138

     Question

     Present-worth acceptance rule for an independent investment at the minimum attractive rate of return?

     Réponse

     Accept it when net present worth is nonnegative. All cash flows must be discounted to the same time using the chosen comparison rate.

     

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139. Carte 139

     Question

     A retrieved formula contains signed quantities. What should you confirm?

     Réponse

     The stated sign convention and positive direction. Do not mix magnitudes with signed components silently.

140. Carte 140

     Question

     Magnitude of a vector cross product?

     Réponse

     $$\lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta$$

     The direction follows the right-hand rule.

141. Carte 141

     Question

     Exact inch-to-metre relationship?

     Réponse

     $1\,in=0.0254\,m$ exactly.

142. Carte 142

     Question

     Volumetric flow rate through a cross-section?

     Réponse

     $$Q=Av_{avg}$$

     Velocity is the area-average component normal to the section.

143. Carte 143

     Question

     Variance from the mean $\mu$?

     Réponse

     $$\operatorname{Var}(X)=E[(X-\mu)^2]$$

     Variance has squared units; standard deviation has the original units.

144. Carte 144

     Question

     Future-worth acceptance rule for an independent investment?

     Réponse

     Accept it when net future worth at the study horizon is nonnegative, using the minimum attractive rate of return consistently.

145. Carte 145

     Question

     What coordinate detail should be checked before using a vector or mechanics formula?

     Réponse

     Axis directions and the reference point or origin. Components and moments depend on those choices.

146. Carte 146

     Question

     Exact foot-to-metre relationship?

     Réponse

     $1\,ft=0.3048\,m$ exactly.

147. Carte 147

     Question

     How should important uncertainty be communicated?

     Réponse

     State the assumptions, limits, and plausible consequences clearly enough for the decision maker to understand the risk.

148. Carte 148

     Question

     When is the matrix product $AB$ defined?

     Réponse

     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 $m\times p$.

149. Carte 149

     Question

     Steady one-inlet, one-outlet mass continuity?

     Réponse

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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 $A_1v_1=A_2v_2$.

150. Carte 150

     Question

     Sample mean of observations $x_1,\ldots,x_n$?

     Réponse

     $$\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$$

151. Carte 151

     Question

     Square feet in one acre?

     Réponse

     $1\,acre=43{,}560\,ft^2$.

152. Carte 152

     Question

     Determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

     Réponse

     $$ad-bc$$

153. Carte 153

     Question

     Bernoulli relationship along a streamline for ideal steady flow?

     Réponse

     $$\frac{p}{\rho g}+\frac{v^2}{2g}+z=\text{constant}$$

     This basic form assumes steady, incompressible, negligible-viscosity flow with no shaft work or heat effects between points.

154. Carte 154

     Question

     May inconvenient measurements be discarded without a technical basis?

     Réponse

     No. Use a documented, technically justified rule for excluding data and preserve the original record.

155. Carte 155

     Question

     Annual-worth acceptance rule for an independent investment?

     Réponse

     Accept it when equivalent annual worth is nonnegative. Compare alternatives over compatible service assumptions.

156. Carte 156

     Question

     Before using a handbook equation, what unit-system check matters?

     Réponse

     Confirm that every input and any embedded constant use one compatible unit system. Convert first when necessary.

157. Carte 157

     Question

     Cubic metres in one litre?

     Réponse

     $1\,L=10^{-3}\,m^3$.

158. Carte 158

     Question

     What does a zero determinant mean for a square matrix?

     Réponse

     The matrix is singular and has no inverse. A corresponding linear system cannot have one unique solution.

159. Carte 159

     Question

     Reynolds number for internal flow using characteristic length $L$?

     Réponse

     $$Re=\frac{\rho vL}{\mu}=\frac{vL}{\nu}$$

     It is dimensionless; use consistent properties and the geometry-appropriate characteristic length.

160. Carte 160

     Question

     Unbiased sample variance for an independent sample from one population?

     Réponse

     $$s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2$$

     This requires at least two observations, so $n\ge2$. The $n-1$ denominator estimates population variance after fitting the sample mean.

161. Carte 161

     Question

     How should mutually exclusive alternatives be compared?

     Réponse

     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.

162. Carte 162

     Question

     Cubic centimetres in one millilitre?

     Réponse

     $1\,mL=1\,cm^3$.

163. Carte 163

     Question

     Why keep contemporaneous engineering records?

     Réponse

     They preserve assumptions, decisions, calculations, approvals, and changes so the work can be checked and responsibilities understood.

164. Carte 164

     Question

     Fast check after substituting into a handbook formula?

     Réponse

     Verify that the resulting dimensions match the requested quantity and that additive terms share dimensions.

165. Carte 165

     Question

     Matrix solution of $A\mathbf{x}=\mathbf{b}$ when $A$ is invertible?

     Réponse

     $$\mathbf{x}=A^{-1}\mathbf{b}$$

     In numerical work, direct factorization is usually preferable to explicitly forming the inverse.

166. Carte 166

     Question

     Sensible heat for constant specific heat and no phase change?

     Réponse

     $$Q=mc\Delta T$$

     $c$ must match the material, temperature range, mass unit, and temperature-difference unit.

167. Carte 167

     Question

     Standard error of the mean for an i.i.d. sample with population standard deviation $\sigma$?

     Réponse

     $$SE(\bar{x})=\frac{\sigma}{\sqrt{n}}$$

     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.

168. Carte 168

     Question

     Safe setup for converting $25\,ft^2$ to square metres?

     Réponse

     $$25\,ft^2\left(\frac{0.3048\,m}{1\,ft}\right)^2$$

     The squared length factor cancels $ft^2$ and leaves $m^2$.

169. Carte 169

     Question

     Purpose of incremental analysis for mutually exclusive alternatives?

     Réponse

     It tests whether the additional investment in the higher-cost option earns the required return compared with the lower-cost option.

170. Carte 170

     Question

     Physical meaning of $dy/dx$?

     Réponse

     It is the instantaneous rate of change of $y$ with respect to $x$, with units of the $y$ unit divided by the $x$ unit.

171. Carte 171

     Question

     One-dimensional steady conduction through a plane wall?

     Réponse

     $$\dot Q=-kA\frac{dT}{dx}$$

     The minus sign indicates heat flows toward lower temperature; $k$ is thermal conductivity.

172. Carte 172

     Question

     When should you algebraically rearrange a handbook relationship?

     Réponse

     After confirming the correct equation, definitions, assumptions, and sign convention. Then isolate the requested variable without changing its domain restrictions.

173. Carte 173

     Question

     Safe setup for converting $2\,in^3$ to cubic metres?

     Réponse

     $$2\,in^3\left(\frac{0.0254\,m}{1\,in}\right)^3$$

     Cube the exact length conversion because the original quantity is volume.

174. Carte 174

     Question

     A client changes a safety-critical design requirement verbally. Proper response?

     Réponse

     Confirm the change in writing, analyze its consequences, and obtain the required approvals before relying on it.

175. Carte 175

     Question

     Derivative of $x^n$ for constant $n$?

     Réponse

     $$\frac{d}{dx}x^n=nx^{n-1}$$

     Apply domain restrictions when $n$ is non-integer or negative.

176. Carte 176

     Question

     Convective heat-transfer rate at a surface?

     Réponse

     $$\dot Q=hA(T_s-T_\infty)$$

     $h$ depends on the flow and geometry; the sign follows the chosen temperature difference.

177. Carte 177

     Question

     What does the central limit theorem say about a sample mean?

     Réponse

     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.

178. Carte 178

     Question

     Internal rate of return?

     Réponse

     The discount rate that makes net present worth zero. Multiple sign changes in cash flow can produce multiple or no meaningful real rates.

179. Carte 179

     Question

     Several similar handbook equations appear together. How do you choose?

     Réponse

     Match each equation's stated geometry, boundary conditions, material model, and variable definitions to the problem.

180. Carte 180

     Question

     Convert Celsius temperature to kelvin?

     Réponse

     $$T_K=T_{^\circ C}+273.15$$

     This conversion is for absolute temperature, not temperature difference.

181. Carte 181

     Question

     Derivative of a product $u(x)v(x)$?

     Réponse

     $$(uv)'=u'v+uv'$$

182. Carte 182

     Question

     Internal escalation does not resolve a serious public-safety threat. Next step?

     Réponse

     Use the next appropriate authority or reporting channel, consistent with law and the urgency of the hazard. Keep a factual record.

183. Carte 183

     Question

     Two-sided confidence interval for a population mean with known $\sigma$?

     Réponse

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

     Use the normal-model conditions and the critical value for confidence level $1-\alpha$.

184. Carte 184

     Question

     Net thermal radiation from a small gray surface to large surroundings?

     Réponse

     $$\dot Q=\varepsilon\sigma A(T_s^4-T_{sur}^4)$$

     Use absolute temperatures; the model assumes the stated gray-surface surroundings.

185. Carte 185

     Question

     Basic benefit-cost acceptance rule for an independent public project?

     Réponse

     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.

186. Carte 186

     Question

     Derivative of a composite function $f(g(x))$?

     Réponse

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

187. Carte 187

     Question

     Why inspect boundary or initial conditions near a formula?

     Réponse

     They determine which solution form applies and can eliminate constants or terms. A formula without its conditions may be unusable.

188. Carte 188

     Question

     Convert Celsius temperature to Fahrenheit?

     Réponse

     $$T_{^\circ F}=\frac{9}{5}T_{^\circ C}+32$$

189. Carte 189

     Question

     Ethical basis for expert technical testimony?

     Réponse

     Offer opinions only within your competence, disclose important assumptions and limitations, and distinguish facts from professional judgment.

190. Carte 190

     Question

     Meaning of a breakeven value?

     Réponse

     The value of a selected variable that makes two alternatives economically equivalent or makes net worth zero. Other assumptions are held fixed.

191. Carte 191

     Question

     Derivative of $e^x$?

     Réponse

     $$\frac{d}{dx}e^x=e^x$$

192. Carte 192

     Question

     First-law energy balance for a closed system with work positive out?

     Réponse

     $$\Delta E=Q-W$$

     $Q$ is heat into the system and $W$ is work done by the system; state a different sign convention before using it.

193. Carte 193

     Question

     Frequentist meaning of a 95% confidence procedure?

     Réponse

     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.

194. Carte 194

     Question

     A needed value falls between handbook table entries. What is the cue?

     Réponse

     Check whether interpolation is appropriate, keep both axes and units straight, and avoid extrapolating unless the source supports it.

195. Carte 195

     Question

     Derivative of $\ln x$ over the real domain?

     Réponse

     $$\frac{d}{dx}\ln x=\frac{1}{x}$$

     This real-valued form requires $x>0$.

196. Carte 196

     Question

     Relationship between Celsius and kelvin temperature differences?

     Réponse

     $1\,K$ of temperature difference equals $1\,^{\circ}C$ of temperature difference. Do not add $273.15$ to a difference.

197. Carte 197

     Question

     How should credit for team engineering work be assigned?

     Réponse

     Accurately recognize each contributor's work and responsibility. Do not claim sole credit for a collaborative result.

198. Carte 198

     Question

     Main limitation of simple payback period?

     Réponse

     It ignores the time value of money and usually ignores cash flows after payback. Discounted payback fixes the first issue, not the second.

199. Carte 199

     Question

     Roles of the null and alternative hypotheses?

     Réponse

     The null states the baseline parameter claim tested; the alternative states the competing direction or difference supported when evidence is strong enough.

200. Carte 200

     Question

     Ideal-gas equation of state?

     Réponse

     $$pV=nRT$$

     Use absolute temperature and a gas constant consistent with pressure, volume, amount, and energy units.

201. Carte 201

     Question

     First check when reading a handbook table?

     Réponse

     Read the title, column headings, units, notes, and basis conditions before selecting a value.

202. Carte 202

     Question

     Candidates for absolute extrema of a continuous function on a closed interval?

     Réponse

     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.

203. Carte 203

     Question

     How should a sunk cost affect a current engineering-economy decision?

     Réponse

     It should not affect the choice because it cannot be changed by the decision. Use future incremental cash flows instead.

204. Carte 204

     Question

     Convert Fahrenheit absolute temperature to Rankine?

     Réponse

     $$T_{^\circ R}=T_{^\circ F}+459.67$$

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Carte 205

     Question

     Ohm's law for an ideal resistor?

     Réponse

     $$V=IR$$

     Use a consistent passive sign convention; $R$ is constant only over the assumed operating range.

206. Carte 206

     Question

     You have confidential details about a competitor's bid. Proper response?

     Réponse

     Do not use or disclose improperly obtained information. Notify the appropriate procurement or ethics contact.

207. Carte 207

     Question

     Physical meaning of a definite integral $\int_a^b f(x)\,dx$?

     Réponse

     It is signed accumulation of $f$ from $a$ to $b$. Its units are the units of $f$ multiplied by the units of $x$.

208. Carte 208

     Question

     Is a radian dimensionless?

     Réponse

     Yes. It is the ratio of arc length to radius, so the length units cancel; keep $rad$ when it clarifies angular meaning.

209. Carte 209

     Question

     Electrical power absorbed under the passive sign convention?

     Réponse

     $$P=VI$$

     For a resistor, this also gives $P=I^2R=V^2/R$ when Ohm's law applies.

210. Carte 210

     Question

     Meaning of a p-value?

     Réponse

     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.

211. Carte 211

     Question

     First check when reading a handbook graph?

     Réponse

     Identify both axes, scales, units, curve labels, and any log scale before estimating a value.

212. Carte 212

     Question

     Difference between mass and weight?

     Réponse

     Mass measures inertia; weight is gravitational force on that mass. Their dimensions and units are different.

213. Carte 213

     Question

     Kirchhoff's current law at a node?

     Réponse

     The algebraic sum of currents at a node is zero. Choose one sign convention, such as currents entering positive, and apply it consistently.

214. Carte 214

     Question

     Evaluate $\int_a^b f(x)\,dx$ from an antiderivative $F$?

     Réponse

     $$\int_a^b f(x)\,dx=F(b)-F(a)$$

     This applies when $F'=f$ on the interval under the usual continuity conditions.

215. Carte 215

     Question

     Opportunity cost?

     Réponse

     The value of the best forgone alternative. Include it when choosing one use of a scarce resource prevents another valuable use.

216. Carte 216

     Question

     Weight from mass in a gravitational field?

     Réponse

     $$W=mg$$

     Use compatible units; $g$ is local gravitational acceleration, not a universal constant for every location.

217. Carte 217

     Question

     Kirchhoff's voltage law around a closed loop?

     Réponse

     The algebraic sum of voltage rises and drops around the loop is zero. Traverse the loop once with a consistent polarity convention.

218. Carte 218

     Question

     Decision rule using significance level $\alpha$?

     Réponse

     Reject the null when the p-value is at most $\alpha$; otherwise fail to reject it. Failing to reject does not prove the null.

219. Carte 219

     Question

     Can proprietary methods from a former employer be reused freely?

     Réponse

     Only if you have the legal right or permission to use them. General skill and experience may transfer; confidential proprietary material does not.

220. Carte 220

     Question

     Pound-force to newtons?

     Réponse

     $1\,lbf\approx4.44822\,N$. Use more digits only when the required precision justifies them.

221. Carte 221

     Question

     When does a definite integral equal ordinary geometric area?

     Réponse

     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.

222. Carte 222

     Question

     Charge-voltage relation for an ideal capacitor?

     Réponse

     $$q=CV$$

     $C$ is capacitance in farads; $V$ is the voltage across the capacitor under the chosen polarity.

223. Carte 223

     Question

     Searching for a material or fluid property: what qualifiers often matter?

     Réponse

     Material identity, phase, temperature, pressure, and composition. Use only the qualifiers relevant to that property source.

224. Carte 224

     Question

     Force produced when one slug accelerates at $1\,ft/s^2$?

     Réponse

     $1\,lbf$. Equivalently, $1\,lbf=1\,slug\cdot ft/s^2$.

225. Carte 225

     Question

     Relationship among nominal rate $i$, real rate $r$, and inflation $f$?

     Réponse

     $$1+i=(1+r)(1+f)$$

     Keep cash-flow dollars and discount rates on the same real-or-nominal basis.

226. Carte 226

     Question

     Antiderivative of $x^n$ for $n\ne-1$?

     Réponse

     $$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$$

227. Carte 227

     Question

     Time constant of a first-order resistor-capacitor circuit?

     Réponse

     $$\tau=RC$$

     Use the resistance seen by the capacitor in the applicable first-order linear circuit.

228. Carte 228

     Question

     What physical quantity does pound-mass measure?

     Réponse

     Mass. Do not substitute $lbm$ directly where a force unit such as $lbf$ is required.

229. Carte 229

     Question

     Type I error?

     Réponse

     Rejecting a true null hypothesis. Its probability is controlled by the chosen significance level under the test assumptions.

230. Carte 230

     Question

     A search hit contains the right words. Is that enough?

     Réponse

     No. Read the surrounding definition, equation label, units, and notes to confirm the hit answers the actual question.

231. Carte 231

     Question

     Annual straight-line depreciation from basis $B$ to salvage $S$ over $n$ years?

     Réponse

     $$D=\frac{B-S}{n}$$

     This is an accounting allocation, not a market-value forecast.

232. Carte 232

     Question

     Purpose of $g_c$ in some US customary force-mass equations?

     Réponse

     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.

233. Carte 233

     Question

     Recognizable form of a separable first-order differential equation?

     Réponse

     $$\frac{dy}{dx}=g(x)h(y)$$

     Rearrange to place the $y$ terms with $dy$ and the $x$ terms with $dx$, while checking for solutions lost by division.

234. Carte 234

     Question

     Using unverified software output in a safety-relevant calculation?

     Réponse

     Check inputs, assumptions, units, and results with an independent method appropriate to the risk. The engineer remains responsible for professional judgment.

235. Carte 235

     Question

     Equivalent resistance of ideal resistors in series?

     Réponse

     $$R_{eq}=\sum_i R_i$$

     Series elements carry the same current with no branching node between them.

236. Carte 236

     Question

     Pressure conversion from psi to pascals?

     Réponse

     $1\,psi\approx6894.76\,Pa$. Keep enough significant figures for the problem's data.

237. Carte 237

     Question

     Type II error?

     Réponse

     Failing to reject a false null hypothesis. Its probability is $\beta$ for a specified alternative.

238. Carte 238

     Question

     Useful flashcard to make after a failed handbook search?

     Réponse

     A short scenario-to-search-cue card recording the section name and terms that eventually worked—not a copied problem or page number.

239. Carte 239

     Question

     Standard form of a first-order linear differential equation?

     Réponse

     $$\frac{dy}{dx}+P(x)y=Q(x)$$

     An integrating factor is $e^{\int P(x)\,dx}$.

240. Carte 240

     Question

     Equivalent resistance of ideal resistors in parallel?

     Réponse

     $$\frac{1}{R_{eq}}=\sum_i\frac{1}{R_i}$$

     Parallel elements share the same two nodes and therefore the same voltage.

241. Carte 241

     Question

     Book value after $k$ years of straight-line depreciation?

     Réponse

     $$BV_k=B-kD$$

     Apply it within the depreciable life and do not depreciate below the assumed salvage value.

242. Carte 242

     Question

     Standard atmosphere in pascals?

     Réponse

     $1\,atm=101{,}325\,Pa$ exactly.

243. Carte 243

     Question

     Using AI-generated technical content in engineering work?

     Réponse

     Treat it as unverified input: protect confidential data, check sources and calculations, disclose use when required, and keep accountable human review.

244. Carte 244

     Question

     Trial form for a homogeneous linear ODE with constant coefficients?

     Réponse

     Use $y=e^{rx}$ to obtain the characteristic polynomial in $r$. Repeated or complex roots require the corresponding standard solution forms.

245. Carte 245

     Question

     Wave-speed relationship among frequency and wavelength?

     Réponse

     $$v=f\lambda$$

     Use frequency in cycles per second and wavelength in compatible length units.

246. Carte 246

     Question

     Statistical power?

     Réponse

     The probability of rejecting the null when a specified alternative is true: $1-\beta$. Larger effects, larger samples, or lower variability commonly increase power.

247. Carte 247

     Question

     Bar in pascals?

     Réponse

     $1\,bar=100{,}000\,Pa$ exactly.

248. Carte 248

     Question

     How should handbook navigation be practiced before the FE exam?

     Réponse

     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.

249. Carte 249

     Question

     Is depreciation itself a cash outflow?

     Réponse

     No. It is a noncash accounting expense, though it can affect taxes and therefore after-tax cash flow.

250. Carte 250

     Question

     First-order linear approximation of $f(x)$ near $x=a$?

     Réponse

     $$f(x)\approx f(a)+f'(a)(x-a)$$

     It is most accurate for $x$ close to $a$ when higher derivatives stay bounded.

251. Carte 251

     Question

     Relationship between period and frequency?

     Réponse

     $$T=\frac{1}{f}$$

     This requires $f>0$; $T$ has time units.

252. Carte 252

     Question

     How should foreseeable environmental harm enter an engineering decision?

     Réponse

     Identify and communicate it, follow applicable requirements, and consider practicable ways to reduce harm without compromising public safety.

253. Carte 253

     Question

     Foot-pound-force in joules?

     Réponse

     $1\,ft\cdot lbf\approx1.35582\,J$. This is a work or energy conversion; keep torque labeled as force times distance even though its dimensions match energy.

254. Carte 254

     Question

     Allowed range and sign meaning of Pearson correlation $r$?

     Réponse

     $-1\le r\le1$. The sign gives the direction of linear association, and magnitude measures its linear strength.

255. Carte 255

     Question

     When should you search instead of relying on recall?

     Réponse

     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.

256. Carte 256

     Question

     After-tax effect of a deductible expense $E$ at marginal tax rate $t$?

     Réponse

     The net cost is $E(1-t)$ under the simplified assumption that the deduction can be used fully in that period.

257. Carte 257

     Question

     Centered finite-difference approximation to $f'(x)$?

     Réponse

     $$f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$

     For smooth $f$, truncation error is proportional to $h^2$ before roundoff dominates.

258. Carte 258

     Question

     General material balance on a control volume?

     Réponse

     $$\text{accumulation}=\text{in}-\text{out}+\text{generation}-\text{consumption}$$

     At steady state, accumulation is zero; generation and consumption depend on reactions or sources.

259. Carte 259

     Question

     Approximate joules in one International Table Btu?

     Réponse

     $1\,Btu_{IT}\approx1055.06\,J$. Identify the Btu convention when precision matters.

260. Carte 260

     Question

     What is the defender in a replacement analysis?

     Réponse

     The currently owned asset being compared with a proposed replacement. Its current market value is an opportunity cost; its original purchase price is sunk.

261. Carte 261

     Question

     Speaking publicly about a technical matter outside your expertise?

     Réponse

     Limit claims to what you can support, identify the boundary of your expertise, and avoid implying qualifications you do not have.

262. Carte 262

     Question

     Single-interval trapezoidal approximation to an integral?

     Réponse

     $$\int_a^b f(x)\,dx\approx\frac{b-a}{2}[f(a)+f(b)]$$

     It is exact for linear $f$.

263. Carte 263

     Question

     Energy efficiency of a device?

     Réponse

     $$\eta=\frac{\text{useful output energy}}{\text{input energy}}$$

     Use the same energy basis in numerator and denominator; a real passive device has $0\le\eta\le1$.

264. Carte 264

     Question

     Purpose of sensitivity analysis in engineering economics?

     Réponse

     It shows how the decision changes when an uncertain input changes. Vary inputs over plausible ranges while preserving units and model relationships.

265. Carte 265

     Question

     Can a strong correlation establish causation?

     Réponse

     No. Correlation alone cannot rule out confounding, reverse direction, selection effects, or coincidence.

266. Carte 266

     Question

     Where should you confirm the tested topics for your FE discipline?

     Réponse

     Use the discipline specification linked from the official NCEES FE exam hub. This common deck does not replace that discipline-specific boundary.

267. Carte 267

     Question

     Mechanical horsepower in watts?

     Réponse

     $1\,hp\approx745.7\,W$, based on $550\,ft\cdot lbf/s$. Distinguish it from metric horsepower.

268. Carte 268

     Question

     Combine independent standard uncertainties $u_1,\ldots,u_n$ in a sum or difference?

     Réponse

     $$u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}$$

     This root-sum-square form assumes uncorrelated inputs and coefficients of magnitude one; otherwise include sensitivities and covariance.

269. Carte 269

     Question

     Newton iteration for a root of $f(x)=0$?

     Réponse

     $$x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}$$

     It requires $f'(x_k)\ne0$ and a suitable starting value; convergence is not guaranteed.

270. Carte 270

     Question

     Meaning of slope in simple linear regression of $y$ on $x$?

     Réponse

     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.

271. Carte 271

     Question

     Kilowatt-hour in joules?

     Réponse

     $1\,kW\cdot h=3.6\times10^6\,J$. It is energy, not power.

272. Carte 272

     Question

     Percent error relative to an accepted nonzero value?

     Réponse

     $$\%\,error=\left|\frac{x_{measured}-x_{accepted}}{x_{accepted}}\right|100\%$$

     The accepted value must be nonzero and both values must use compatible units.

273. Carte 273

     Question

     Does legal compliance alone settle an engineering ethics question?

     Réponse

     No. Law sets mandatory boundaries, while professional duties can require stronger action, especially when safety, honesty, or conflicts are involved.

274. Carte 274

     Question

     Does handbook fluency replace discipline-specific problem practice?

     Réponse

     No. Pair navigation and common recall with the current discipline specification and timed, rights-safe problems that match your selected FE discipline.

275. Carte 275

     Question

     Linear interpolation between $(x_1,y_1)$ and $(x_2,y_2)$?

     Réponse

     $$y\approx y_1+(y_2-y_1)\frac{x-x_1}{x_2-x_1}$$

     Use it between distinct $x$ values when local linear behavior is reasonable.

276. Carte 276

     Question

     Dimensional requirement for a physically meaningful additive equation?

     Réponse

     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.