FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

这套卡组中的卡片

1. 卡片 1

   问题

   SI base unit of length?

   答案

   The metre, symbol $m$.

2. 卡片 2

   问题

   Slope between two points?

   答案

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

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

3. 卡片 3

   问题

   Net force relationship for constant mass in an inertial frame?

   答案

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

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

4. 卡片 4

   问题

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

   答案

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

5. 卡片 5

   问题

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

   答案

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

6. 卡片 6

   问题

   Probability of the complement of event $A$?

   答案

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

7. 卡片 7

   问题

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

   答案

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

8. 卡片 8

   问题

   SI base unit of mass?

   答案

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

9. 卡片 9

   问题

   Linear equation from slope and intercept?

   答案

   $$y=mx+b$$

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

10. 卡片 10

    问题

    Velocity after time $t$ under constant acceleration?

    答案

    $$v=v_0+at$$

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

11. 卡片 11

    问题

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

    答案

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

12. 卡片 12

    问题

    How is the FE reference handbook available during the exam?

    答案

    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. 卡片 13

    问题

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

    答案

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

14. 卡片 14

    问题

    SI base unit of time?

    答案

    The second, symbol $s$.

15. 卡片 15

    问题

    Linear equation from one point and a slope?

    答案

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

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

16. 卡片 16

    问题

    Position after time $t$ under constant acceleration?

    答案

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

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

17. 卡片 17

    问题

    A project assignment lies outside your competence. Proper response?

    答案

    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. 卡片 18

    问题

    SI base unit of electric current?

    答案

    The ampere, symbol $A$.

19. 卡片 19

    问题

    Future amount under simple interest?

    答案

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

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

20. 卡片 20

    问题

    Distance between two Cartesian points?

    答案

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

    The coordinate components must use the same length unit.

21. 卡片 21

    问题

    Constant-acceleration relation without elapsed time?

    答案

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

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

22. 卡片 22

    问题

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

    答案

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

23. 卡片 23

    问题

    Conditional probability of $A$ given $B$?

    答案

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

    This requires $P(B)>0$.

24. 卡片 24

    问题

    SI base unit of thermodynamic temperature?

    答案

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

25. 卡片 25

    问题

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

    答案

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

26. 卡片 26

    问题

    Midpoint of two Cartesian points?

    答案

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

    Average corresponding coordinates.

27. 卡片 27

    问题

    Linear momentum of a particle?

    答案

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

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

28. 卡片 28

    问题

    Future amount under compound interest?

    答案

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

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

29. 卡片 29

    问题

    Should an FE search workflow depend on Ctrl+F?

    答案

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

30. 卡片 30

    问题

    SI base unit of amount of substance?

    答案

    The mole, symbol $mol$.

31. 卡片 31

    问题

    Joint probability expressed with a conditional probability?

    答案

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

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

32. 卡片 32

    问题

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

    答案

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

    The coordinate units and radius unit must match.

33. 卡片 33

    问题

    Impulse-momentum relationship?

    答案

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

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

34. 卡片 34

    问题

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

    答案

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

35. 卡片 35

    问题

    SI base unit of luminous intensity?

    答案

    The candela, symbol $cd$.

36. 卡片 36

    问题

    Present worth of one future amount?

    答案

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

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

37. 卡片 37

    问题

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

    答案

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

    This form requires $a\ne0$.

38. 卡片 38

    问题

    When is total linear momentum conserved?

    答案

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

39. 卡片 39

    问题

    Why avoid memorizing FE handbook page numbers?

    答案

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

40. 卡片 40

    问题

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

    答案

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

    Independence is not the same as mutual exclusivity.

41. 卡片 41

    问题

    SI unit of frequency in base-unit form?

    答案

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

42. 卡片 42

    问题

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

    答案

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

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

43. 卡片 43

    问题

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

    答案

    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. 卡片 44

    问题

    Work done by a constant force through displacement?

    答案

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

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

45. 卡片 45

    问题

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

    答案

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

46. 卡片 46

    问题

    SI unit of force in base-unit form?

    答案

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

47. 卡片 47

    问题

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

    答案

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

48. 卡片 48

    问题

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

    答案

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

    This requires $P(B)>0$.

49. 卡片 49

    问题

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

    答案

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

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

50. 卡片 50

    问题

    Translational kinetic energy of a particle?

    答案

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

    Mass and speed must use a coherent unit system.

51. 卡片 51

    问题

    SI unit of pressure in force-area form?

    答案

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

52. 卡片 52

    问题

    Equivalent annual amount for present worth $P$?

    答案

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

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

53. 卡片 53

    问题

    May confidential client information be shared for personal benefit?

    答案

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

54. 卡片 54

    问题

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

    答案

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

55. 卡片 55

    问题

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

    答案

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

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

56. 卡片 56

    问题

    Net work-energy relationship for a particle?

    答案

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

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

57. 卡片 57

    问题

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

    答案

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

    Use this when order matters and selection is without replacement.

58. 卡片 58

    问题

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

    答案

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

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

59. 卡片 59

    问题

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

    答案

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

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

60. 卡片 60

    问题

    SI unit of energy in force-distance form?

    答案

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

61. 卡片 61

    问题

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

    答案

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

62. 卡片 62

    问题

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

    答案

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

63. 卡片 63

    问题

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

    答案

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

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

64. 卡片 64

    问题

    Change in gravitational potential energy near Earth's surface?

    答案

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

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

65. 卡片 65

    问题

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

    答案

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

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

66. 卡片 66

    问题

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

    答案

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

    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. 卡片 67

    问题

    A handbook search for an acronym fails. Next move?

    答案

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

68. 卡片 68

    问题

    SI unit of power in energy-time form?

    答案

    The watt, $W=J/s$.

69. 卡片 69

    问题

    Logarithm of a product?

    答案

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

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

70. 卡片 70

    问题

    Conditions for a binomial model?

    答案

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

71. 卡片 71

    问题

    Using another engineer's analysis in your deliverable?

    答案

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

72. 卡片 72

    问题

    SI unit of electric charge in current-time form?

    答案

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

73. 卡片 73

    问题

    Average power from work over elapsed time?

    答案

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

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

74. 卡片 74

    问题

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

    答案

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

    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. 卡片 75

    问题

    Logarithm of a positive quantity raised to a power?

    答案

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

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

76. 卡片 76

    问题

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

    答案

    The volt, $V=J/C$.

77. 卡片 77

    问题

    Instantaneous mechanical power delivered by a force?

    答案

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

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

78. 卡片 78

    问题

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

    答案

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

79. 卡片 79

    问题

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

    答案

    $$E[X]=np$$

80. 卡片 80

    问题

    SI unit of electrical resistance in voltage-current form?

    答案

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

81. 卡片 81

    问题

    Moment of a force about a point?

    答案

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

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

82. 卡片 82

    问题

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

    答案

    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. 卡片 83

    问题

    Inverse relationship between the natural log and exponential?

    答案

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

    The second identity requires $x>0$.

84. 卡片 84

    问题

    SI prefix giga?

    答案

    $G=10^9$.

85. 卡片 85

    问题

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

    答案

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

    Use consistent angular sign convention and radians for angular measures.

86. 卡片 86

    问题

    When should an engineer sign or seal technical work?

    答案

    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. 卡片 87

    问题

    Best memory target for returning to a handbook formula?

    答案

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

88. 卡片 88

    问题

    SI prefix mega?

    答案

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

89. 卡片 89

    问题

    Convert degrees to radians?

    答案

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

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

90. 卡片 90

    问题

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

    答案

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

91. 卡片 91

    问题

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

    答案

    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. 卡片 92

    问题

    Where do licensure and sealing requirements come from?

    答案

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

93. 卡片 93

    问题

    Rotational kinetic energy about a fixed axis?

    答案

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

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

94. 卡片 94

    问题

    SI prefix kilo?

    答案

    $k=10^3$.

95. 卡片 95

    问题

    Right-triangle definition of sine?

    答案

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

96. 卡片 96

    问题

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

    答案

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

97. 卡片 97

    问题

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

    答案

    $$\sum M=I\alpha$$

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

98. 卡片 98

    问题

    SI prefix milli?

    答案

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

99. 卡片 99

    问题

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

    答案

    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. 卡片 100

     问题

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

     答案

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

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

101. 卡片 101

     问题

     Right-triangle definition of cosine?

     答案

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

102. 卡片 102

     问题

     Force-displacement relation for an ideal linear spring?

     答案

     $$F=-kx$$

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

103. 卡片 103

     问题

     May qualifications or project responsibility be exaggerated in a proposal?

     答案

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

104. 卡片 104

     问题

     SI prefix micro?

     答案

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

105. 卡片 105

     问题

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

     答案

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

106. 卡片 106

     问题

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

     答案

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

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

107. 卡片 107

     问题

     Right-triangle definition of tangent?

     答案

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

108. 卡片 108

     问题

     Average normal stress under centered axial load?

     答案

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

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

109. 卡片 109

     问题

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

     答案

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

110. 卡片 110

     问题

     SI prefix nano?

     答案

     $n=10^{-9}$.

111. 卡片 111

     问题

     Fundamental Pythagorean trigonometric identity?

     答案

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

112. 卡片 112

     问题

     Engineering normal strain?

     答案

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

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

113. 卡片 113

     问题

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

     答案

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

114. 卡片 114

     问题

     Technical duty to an employer or client?

     答案

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

115. 卡片 115

     问题

     How does a length conversion factor apply to area?

     答案

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

116. 卡片 116

     问题

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

     答案

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

117. 卡片 117

     问题

     When is a Poisson count model reasonable?

     答案

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

118. 卡片 118

     问题

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

     答案

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

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

119. 卡片 119

     问题

     Linear elastic stress-strain relation in uniaxial loading?

     答案

     $$\sigma=E\varepsilon$$

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

120. 卡片 120

     问题

     How does a length conversion factor apply to volume?

     答案

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

121. 卡片 121

     问题

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

     答案

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

122. 卡片 122

     问题

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

     答案

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

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

123. 卡片 123

     问题

     A supervisor directs an illegal engineering action. Proper response?

     答案

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

124. 卡片 124

     问题

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

     答案

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

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

125. 卡片 125

     问题

     Mass density?

     答案

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

     Use mass, not weight, in the numerator.

126. 卡片 126

     问题

     Inches in one foot?

     答案

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

127. 卡片 127

     问题

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

     答案

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

128. 卡片 128

     问题

     How should subscripts be handled in a retrieved relationship?

     答案

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

129. 卡片 129

     问题

     Dot product from vector magnitudes and included angle?

     答案

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

     The result is a scalar.

130. 卡片 130

     问题

     Average normal pressure on an area?

     答案

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

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

131. 卡片 131

     问题

     Feet in one yard?

     答案

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

132. 卡片 132

     问题

     When are two interest rates equivalent?

     答案

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

133. 卡片 133

     问题

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

     答案

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

134. 卡片 134

     问题

     Expected value of a discrete random variable?

     答案

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

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

135. 卡片 135

     问题

     Dot-product test for perpendicular nonzero vectors?

     答案

     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. 卡片 136

     问题

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

     答案

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

     Depth increases downward and the fluid is at rest.

137. 卡片 137

     问题

     Feet in one statute mile?

     答案

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

138. 卡片 138

     问题

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

     答案

     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. 卡片 139

     问题

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

     答案

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

140. 卡片 140

     问题

     Magnitude of a vector cross product?

     答案

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

     The direction follows the right-hand rule.

141. 卡片 141

     问题

     Exact inch-to-metre relationship?

     答案

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

142. 卡片 142

     问题

     Volumetric flow rate through a cross-section?

     答案

     $$Q=Av_{avg}$$

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

143. 卡片 143

     问题

     Variance from the mean $\mu$?

     答案

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

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

144. 卡片 144

     问题

     Future-worth acceptance rule for an independent investment?

     答案

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

145. 卡片 145

     问题

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

     答案

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

146. 卡片 146

     问题

     Exact foot-to-metre relationship?

     答案

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

147. 卡片 147

     问题

     How should important uncertainty be communicated?

     答案

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

148. 卡片 148

     问题

     When is the matrix product $AB$ defined?

     答案

     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. 卡片 149

     问题

     Steady one-inlet, one-outlet mass continuity?

     答案

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. 卡片 150

     问题

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

     答案

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

151. 卡片 151

     问题

     Square feet in one acre?

     答案

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

152. 卡片 152

     问题

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

     答案

     $$ad-bc$$

153. 卡片 153

     问题

     Bernoulli relationship along a streamline for ideal steady flow?

     答案

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

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

154. 卡片 154

     问题

     May inconvenient measurements be discarded without a technical basis?

     答案

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

155. 卡片 155

     问题

     Annual-worth acceptance rule for an independent investment?

     答案

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

156. 卡片 156

     问题

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

     答案

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

157. 卡片 157

     问题

     Cubic metres in one litre?

     答案

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

158. 卡片 158

     问题

     What does a zero determinant mean for a square matrix?

     答案

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

159. 卡片 159

     问题

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

     答案

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

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

160. 卡片 160

     问题

     Unbiased sample variance for an independent sample from one population?

     答案

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

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

161. 卡片 161

     问题

     How should mutually exclusive alternatives be compared?

     答案

     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. 卡片 162

     问题

     Cubic centimetres in one millilitre?

     答案

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

163. 卡片 163

     问题

     Why keep contemporaneous engineering records?

     答案

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

164. 卡片 164

     问题

     Fast check after substituting into a handbook formula?

     答案

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

165. 卡片 165

     问题

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

     答案

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

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

166. 卡片 166

     问题

     Sensible heat for constant specific heat and no phase change?

     答案

     $$Q=mc\Delta T$$

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

167. 卡片 167

     问题

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

     答案

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

     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. 卡片 168

     问题

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

     答案

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

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

169. 卡片 169

     问题

     Purpose of incremental analysis for mutually exclusive alternatives?

     答案

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

170. 卡片 170

     问题

     Physical meaning of $dy/dx$?

     答案

     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. 卡片 171

     问题

     One-dimensional steady conduction through a plane wall?

     答案

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

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

172. 卡片 172

     问题

     When should you algebraically rearrange a handbook relationship?

     答案

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

173. 卡片 173

     问题

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

     答案

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

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

174. 卡片 174

     问题

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

     答案

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

175. 卡片 175

     问题

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

     答案

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

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

176. 卡片 176

     问题

     Convective heat-transfer rate at a surface?

     答案

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

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

177. 卡片 177

     问题

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

     答案

     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. 卡片 178

     问题

     Internal rate of return?

     答案

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

179. 卡片 179

     问题

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

     答案

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

180. 卡片 180

     问题

     Convert Celsius temperature to kelvin?

     答案

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

     This conversion is for absolute temperature, not temperature difference.

181. 卡片 181

     问题

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

     答案

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

182. 卡片 182

     问题

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

     答案

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

183. 卡片 183

     问题

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

     答案

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

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

184. 卡片 184

     问题

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

     答案

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

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

185. 卡片 185

     问题

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

     答案

     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. 卡片 186

     问题

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

     答案

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

187. 卡片 187

     问题

     Why inspect boundary or initial conditions near a formula?

     答案

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

188. 卡片 188

     问题

     Convert Celsius temperature to Fahrenheit?

     答案

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

189. 卡片 189

     问题

     Ethical basis for expert technical testimony?

     答案

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

190. 卡片 190

     问题

     Meaning of a breakeven value?

     答案

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

191. 卡片 191

     问题

     Derivative of $e^x$?

     答案

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

192. 卡片 192

     问题

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

     答案

     $$\Delta E=Q-W$$

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

193. 卡片 193

     问题

     Frequentist meaning of a 95% confidence procedure?

     答案

     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. 卡片 194

     问题

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

     答案

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

195. 卡片 195

     问题

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

     答案

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

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

196. 卡片 196

     问题

     Relationship between Celsius and kelvin temperature differences?

     答案

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

197. 卡片 197

     问题

     How should credit for team engineering work be assigned?

     答案

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

198. 卡片 198

     问题

     Main limitation of simple payback period?

     答案

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

199. 卡片 199

     问题

     Roles of the null and alternative hypotheses?

     答案

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

200. 卡片 200

     问题

     Ideal-gas equation of state?

     答案

     $$pV=nRT$$

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

201. 卡片 201

     问题

     First check when reading a handbook table?

     答案

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

202. 卡片 202

     问题

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

     答案

     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. 卡片 203

     问题

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

     答案

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

204. 卡片 204

     问题

     Convert Fahrenheit absolute temperature to Rankine?

     答案

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. 卡片 205

     问题

     Ohm's law for an ideal resistor?

     答案

     $$V=IR$$

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

206. 卡片 206

     问题

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

     答案

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

207. 卡片 207

     问题

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

     答案

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

208. 卡片 208

     问题

     Is a radian dimensionless?

     答案

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

209. 卡片 209

     问题

     Electrical power absorbed under the passive sign convention?

     答案

     $$P=VI$$

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

210. 卡片 210

     问题

     Meaning of a p-value?

     答案

     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. 卡片 211

     问题

     First check when reading a handbook graph?

     答案

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

212. 卡片 212

     问题

     Difference between mass and weight?

     答案

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

213. 卡片 213

     问题

     Kirchhoff's current law at a node?

     答案

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

214. 卡片 214

     问题

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

     答案

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

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

215. 卡片 215

     问题

     Opportunity cost?

     答案

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

216. 卡片 216

     问题

     Weight from mass in a gravitational field?

     答案

     $$W=mg$$

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

217. 卡片 217

     问题

     Kirchhoff's voltage law around a closed loop?

     答案

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

218. 卡片 218

     问题

     Decision rule using significance level $\alpha$?

     答案

     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. 卡片 219

     问题

     Can proprietary methods from a former employer be reused freely?

     答案

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

220. 卡片 220

     问题

     Pound-force to newtons?

     答案

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

221. 卡片 221

     问题

     When does a definite integral equal ordinary geometric area?

     答案

     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. 卡片 222

     问题

     Charge-voltage relation for an ideal capacitor?

     答案

     $$q=CV$$

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

223. 卡片 223

     问题

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

     答案

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

224. 卡片 224

     问题

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

     答案

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

225. 卡片 225

     问题

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

     答案

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

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

226. 卡片 226

     问题

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

     答案

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

227. 卡片 227

     问题

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

     答案

     $$\tau=RC$$

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

228. 卡片 228

     问题

     What physical quantity does pound-mass measure?

     答案

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

229. 卡片 229

     问题

     Type I error?

     答案

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

230. 卡片 230

     问题

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

     答案

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

231. 卡片 231

     问题

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

     答案

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

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

232. 卡片 232

     问题

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

     答案

     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. 卡片 233

     问题

     Recognizable form of a separable first-order differential equation?

     答案

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

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

234. 卡片 234

     问题

     Using unverified software output in a safety-relevant calculation?

     答案

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

235. 卡片 235

     问题

     Equivalent resistance of ideal resistors in series?

     答案

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

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

236. 卡片 236

     问题

     Pressure conversion from psi to pascals?

     答案

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

237. 卡片 237

     问题

     Type II error?

     答案

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

238. 卡片 238

     问题

     Useful flashcard to make after a failed handbook search?

     答案

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

239. 卡片 239

     问题

     Standard form of a first-order linear differential equation?

     答案

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

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

240. 卡片 240

     问题

     Equivalent resistance of ideal resistors in parallel?

     答案

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

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

241. 卡片 241

     问题

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

     答案

     $$BV_k=B-kD$$

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

242. 卡片 242

     问题

     Standard atmosphere in pascals?

     答案

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

243. 卡片 243

     问题

     Using AI-generated technical content in engineering work?

     答案

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

244. 卡片 244

     问题

     Trial form for a homogeneous linear ODE with constant coefficients?

     答案

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

245. 卡片 245

     问题

     Wave-speed relationship among frequency and wavelength?

     答案

     $$v=f\lambda$$

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

246. 卡片 246

     问题

     Statistical power?

     答案

     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. 卡片 247

     问题

     Bar in pascals?

     答案

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

248. 卡片 248

     问题

     How should handbook navigation be practiced before the FE exam?

     答案

     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. 卡片 249

     问题

     Is depreciation itself a cash outflow?

     答案

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

250. 卡片 250

     问题

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

     答案

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

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

251. 卡片 251

     问题

     Relationship between period and frequency?

     答案

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

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

252. 卡片 252

     问题

     How should foreseeable environmental harm enter an engineering decision?

     答案

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

253. 卡片 253

     问题

     Foot-pound-force in joules?

     答案

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

254. 卡片 254

     问题

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

     答案

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

255. 卡片 255

     问题

     When should you search instead of relying on recall?

     答案

     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. 卡片 256

     问题

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

     答案

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

257. 卡片 257

     问题

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

     答案

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

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

258. 卡片 258

     问题

     General material balance on a control volume?

     答案

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

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

259. 卡片 259

     问题

     Approximate joules in one International Table Btu?

     答案

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

260. 卡片 260

     问题

     What is the defender in a replacement analysis?

     答案

     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. 卡片 261

     问题

     Speaking publicly about a technical matter outside your expertise?

     答案

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

262. 卡片 262

     问题

     Single-interval trapezoidal approximation to an integral?

     答案

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

     It is exact for linear $f$.

263. 卡片 263

     问题

     Energy efficiency of a device?

     答案

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

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

264. 卡片 264

     问题

     Purpose of sensitivity analysis in engineering economics?

     答案

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

265. 卡片 265

     问题

     Can a strong correlation establish causation?

     答案

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

266. 卡片 266

     问题

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

     答案

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

267. 卡片 267

     问题

     Mechanical horsepower in watts?

     答案

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

268. 卡片 268

     问题

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

     答案

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

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

269. 卡片 269

     问题

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

     答案

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

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

270. 卡片 270

     问题

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

     答案

     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. 卡片 271

     问题

     Kilowatt-hour in joules?

     答案

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

272. 卡片 272

     问题

     Percent error relative to an accepted nonzero value?

     答案

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

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

273. 卡片 273

     问题

     Does legal compliance alone settle an engineering ethics question?

     答案

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

274. 卡片 274

     问题

     Does handbook fluency replace discipline-specific problem practice?

     答案

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

275. 卡片 275

     问题

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

     答案

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

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

276. 卡片 276

     问题

     Dimensional requirement for a physically meaningful additive equation?

     答案

     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.