FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

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1. Spjald 1

   Spurning

   SI base unit of length?

   Svar

   The metre, symbol $m$.

2. Spjald 2

   Spurning

   Slope between two points?

   Svar

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

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

3. Spjald 3

   Spurning

   Net force relationship for constant mass in an inertial frame?

   Svar

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

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

4. Spjald 4

   Spurning

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

   Svar

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

5. Spjald 5

   Spurning

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

   Svar

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

6. Spjald 6

   Spurning

   Probability of the complement of event $A$?

   Svar

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

7. Spjald 7

   Spurning

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

   Svar

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

8. Spjald 8

   Spurning

   SI base unit of mass?

   Svar

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

9. Spjald 9

   Spurning

   Linear equation from slope and intercept?

   Svar

   $$y=mx+b$$

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

10. Spjald 10

    Spurning

    Velocity after time $t$ under constant acceleration?

    Svar

    $$v=v_0+at$$

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

11. Spjald 11

    Spurning

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

    Svar

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

12. Spjald 12

    Spurning

    How is the FE reference handbook available during the exam?

    Svar

    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. Spjald 13

    Spurning

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

    Svar

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

14. Spjald 14

    Spurning

    SI base unit of time?

    Svar

    The second, symbol $s$.

15. Spjald 15

    Spurning

    Linear equation from one point and a slope?

    Svar

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

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

16. Spjald 16

    Spurning

    Position after time $t$ under constant acceleration?

    Svar

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

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

17. Spjald 17

    Spurning

    A project assignment lies outside your competence. Proper response?

    Svar

    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. Spjald 18

    Spurning

    SI base unit of electric current?

    Svar

    The ampere, symbol $A$.

19. Spjald 19

    Spurning

    Future amount under simple interest?

    Svar

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

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

20. Spjald 20

    Spurning

    Distance between two Cartesian points?

    Svar

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

    The coordinate components must use the same length unit.

21. Spjald 21

    Spurning

    Constant-acceleration relation without elapsed time?

    Svar

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

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

22. Spjald 22

    Spurning

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

    Svar

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

23. Spjald 23

    Spurning

    Conditional probability of $A$ given $B$?

    Svar

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

    This requires $P(B)>0$.

24. Spjald 24

    Spurning

    SI base unit of thermodynamic temperature?

    Svar

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

25. Spjald 25

    Spurning

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

    Svar

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

26. Spjald 26

    Spurning

    Midpoint of two Cartesian points?

    Svar

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

    Average corresponding coordinates.

27. Spjald 27

    Spurning

    Linear momentum of a particle?

    Svar

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

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

28. Spjald 28

    Spurning

    Future amount under compound interest?

    Svar

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

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

29. Spjald 29

    Spurning

    Should an FE search workflow depend on Ctrl+F?

    Svar

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

30. Spjald 30

    Spurning

    SI base unit of amount of substance?

    Svar

    The mole, symbol $mol$.

31. Spjald 31

    Spurning

    Joint probability expressed with a conditional probability?

    Svar

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

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

32. Spjald 32

    Spurning

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

    Svar

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

    The coordinate units and radius unit must match.

33. Spjald 33

    Spurning

    Impulse-momentum relationship?

    Svar

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

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

34. Spjald 34

    Spurning

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

    Svar

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

35. Spjald 35

    Spurning

    SI base unit of luminous intensity?

    Svar

    The candela, symbol $cd$.

36. Spjald 36

    Spurning

    Present worth of one future amount?

    Svar

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

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

37. Spjald 37

    Spurning

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

    Svar

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

    This form requires $a\ne0$.

38. Spjald 38

    Spurning

    When is total linear momentum conserved?

    Svar

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

39. Spjald 39

    Spurning

    Why avoid memorizing FE handbook page numbers?

    Svar

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

40. Spjald 40

    Spurning

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

    Svar

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

    Independence is not the same as mutual exclusivity.

41. Spjald 41

    Spurning

    SI unit of frequency in base-unit form?

    Svar

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

42. Spjald 42

    Spurning

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

    Svar

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

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

43. Spjald 43

    Spurning

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

    Svar

    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. Spjald 44

    Spurning

    Work done by a constant force through displacement?

    Svar

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

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

45. Spjald 45

    Spurning

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

    Svar

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

46. Spjald 46

    Spurning

    SI unit of force in base-unit form?

    Svar

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

47. Spjald 47

    Spurning

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

    Svar

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

48. Spjald 48

    Spurning

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

    Svar

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

    This requires $P(B)>0$.

49. Spjald 49

    Spurning

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

    Svar

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

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

50. Spjald 50

    Spurning

    Translational kinetic energy of a particle?

    Svar

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

    Mass and speed must use a coherent unit system.

51. Spjald 51

    Spurning

    SI unit of pressure in force-area form?

    Svar

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

52. Spjald 52

    Spurning

    Equivalent annual amount for present worth $P$?

    Svar

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

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

53. Spjald 53

    Spurning

    May confidential client information be shared for personal benefit?

    Svar

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

54. Spjald 54

    Spurning

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

    Svar

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

55. Spjald 55

    Spurning

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

    Svar

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

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

56. Spjald 56

    Spurning

    Net work-energy relationship for a particle?

    Svar

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

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

57. Spjald 57

    Spurning

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

    Svar

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

    Use this when order matters and selection is without replacement.

58. Spjald 58

    Spurning

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

    Svar

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

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

59. Spjald 59

    Spurning

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

    Svar

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

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

60. Spjald 60

    Spurning

    SI unit of energy in force-distance form?

    Svar

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

61. Spjald 61

    Spurning

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

    Svar

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

62. Spjald 62

    Spurning

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

    Svar

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

63. Spjald 63

    Spurning

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

    Svar

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

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

64. Spjald 64

    Spurning

    Change in gravitational potential energy near Earth's surface?

    Svar

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

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

65. Spjald 65

    Spurning

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

    Svar

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

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

66. Spjald 66

    Spurning

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

    Svar

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

    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. Spjald 67

    Spurning

    A handbook search for an acronym fails. Next move?

    Svar

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

68. Spjald 68

    Spurning

    SI unit of power in energy-time form?

    Svar

    The watt, $W=J/s$.

69. Spjald 69

    Spurning

    Logarithm of a product?

    Svar

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

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

70. Spjald 70

    Spurning

    Conditions for a binomial model?

    Svar

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

71. Spjald 71

    Spurning

    Using another engineer's analysis in your deliverable?

    Svar

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

72. Spjald 72

    Spurning

    SI unit of electric charge in current-time form?

    Svar

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

73. Spjald 73

    Spurning

    Average power from work over elapsed time?

    Svar

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

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

74. Spjald 74

    Spurning

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

    Svar

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

    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. Spjald 75

    Spurning

    Logarithm of a positive quantity raised to a power?

    Svar

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

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

76. Spjald 76

    Spurning

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

    Svar

    The volt, $V=J/C$.

77. Spjald 77

    Spurning

    Instantaneous mechanical power delivered by a force?

    Svar

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

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

78. Spjald 78

    Spurning

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

    Svar

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

79. Spjald 79

    Spurning

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

    Svar

    $$E[X]=np$$

80. Spjald 80

    Spurning

    SI unit of electrical resistance in voltage-current form?

    Svar

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

81. Spjald 81

    Spurning

    Moment of a force about a point?

    Svar

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

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

82. Spjald 82

    Spurning

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

    Svar

    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. Spjald 83

    Spurning

    Inverse relationship between the natural log and exponential?

    Svar

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

    The second identity requires $x>0$.

84. Spjald 84

    Spurning

    SI prefix giga?

    Svar

    $G=10^9$.

85. Spjald 85

    Spurning

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

    Svar

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

    Use consistent angular sign convention and radians for angular measures.

86. Spjald 86

    Spurning

    When should an engineer sign or seal technical work?

    Svar

    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. Spjald 87

    Spurning

    Best memory target for returning to a handbook formula?

    Svar

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

88. Spjald 88

    Spurning

    SI prefix mega?

    Svar

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

89. Spjald 89

    Spurning

    Convert degrees to radians?

    Svar

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

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

90. Spjald 90

    Spurning

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

    Svar

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

91. Spjald 91

    Spurning

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

    Svar

    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. Spjald 92

    Spurning

    Where do licensure and sealing requirements come from?

    Svar

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

93. Spjald 93

    Spurning

    Rotational kinetic energy about a fixed axis?

    Svar

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

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

94. Spjald 94

    Spurning

    SI prefix kilo?

    Svar

    $k=10^3$.

95. Spjald 95

    Spurning

    Right-triangle definition of sine?

    Svar

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

96. Spjald 96

    Spurning

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

    Svar

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

97. Spjald 97

    Spurning

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

    Svar

    $$\sum M=I\alpha$$

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

98. Spjald 98

    Spurning

    SI prefix milli?

    Svar

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

99. Spjald 99

    Spurning

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

    Svar

    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. Spjald 100

     Spurning

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

     Svar

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

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

101. Spjald 101

     Spurning

     Right-triangle definition of cosine?

     Svar

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

102. Spjald 102

     Spurning

     Force-displacement relation for an ideal linear spring?

     Svar

     $$F=-kx$$

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

103. Spjald 103

     Spurning

     May qualifications or project responsibility be exaggerated in a proposal?

     Svar

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

104. Spjald 104

     Spurning

     SI prefix micro?

     Svar

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

105. Spjald 105

     Spurning

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

     Svar

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

106. Spjald 106

     Spurning

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

     Svar

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

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

107. Spjald 107

     Spurning

     Right-triangle definition of tangent?

     Svar

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

108. Spjald 108

     Spurning

     Average normal stress under centered axial load?

     Svar

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

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

109. Spjald 109

     Spurning

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

     Svar

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

110. Spjald 110

     Spurning

     SI prefix nano?

     Svar

     $n=10^{-9}$.

111. Spjald 111

     Spurning

     Fundamental Pythagorean trigonometric identity?

     Svar

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

112. Spjald 112

     Spurning

     Engineering normal strain?

     Svar

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

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

113. Spjald 113

     Spurning

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

     Svar

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

114. Spjald 114

     Spurning

     Technical duty to an employer or client?

     Svar

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

115. Spjald 115

     Spurning

     How does a length conversion factor apply to area?

     Svar

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

116. Spjald 116

     Spurning

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

     Svar

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

117. Spjald 117

     Spurning

     When is a Poisson count model reasonable?

     Svar

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

118. Spjald 118

     Spurning

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

     Svar

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

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

119. Spjald 119

     Spurning

     Linear elastic stress-strain relation in uniaxial loading?

     Svar

     $$\sigma=E\varepsilon$$

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

120. Spjald 120

     Spurning

     How does a length conversion factor apply to volume?

     Svar

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

121. Spjald 121

     Spurning

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

     Svar

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

122. Spjald 122

     Spurning

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

     Svar

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

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

123. Spjald 123

     Spurning

     A supervisor directs an illegal engineering action. Proper response?

     Svar

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

124. Spjald 124

     Spurning

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

     Svar

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

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

125. Spjald 125

     Spurning

     Mass density?

     Svar

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

     Use mass, not weight, in the numerator.

126. Spjald 126

     Spurning

     Inches in one foot?

     Svar

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

127. Spjald 127

     Spurning

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

     Svar

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

128. Spjald 128

     Spurning

     How should subscripts be handled in a retrieved relationship?

     Svar

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

129. Spjald 129

     Spurning

     Dot product from vector magnitudes and included angle?

     Svar

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

     The result is a scalar.

130. Spjald 130

     Spurning

     Average normal pressure on an area?

     Svar

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

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

131. Spjald 131

     Spurning

     Feet in one yard?

     Svar

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

132. Spjald 132

     Spurning

     When are two interest rates equivalent?

     Svar

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

133. Spjald 133

     Spurning

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

     Svar

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

134. Spjald 134

     Spurning

     Expected value of a discrete random variable?

     Svar

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

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

135. Spjald 135

     Spurning

     Dot-product test for perpendicular nonzero vectors?

     Svar

     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. Spjald 136

     Spurning

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

     Svar

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

     Depth increases downward and the fluid is at rest.

137. Spjald 137

     Spurning

     Feet in one statute mile?

     Svar

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

138. Spjald 138

     Spurning

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

     Svar

     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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     FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

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139. Spjald 139

     Spurning

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

     Svar

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

140. Spjald 140

     Spurning

     Magnitude of a vector cross product?

     Svar

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

     The direction follows the right-hand rule.

141. Spjald 141

     Spurning

     Exact inch-to-metre relationship?

     Svar

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

142. Spjald 142

     Spurning

     Volumetric flow rate through a cross-section?

     Svar

     $$Q=Av_{avg}$$

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

143. Spjald 143

     Spurning

     Variance from the mean $\mu$?

     Svar

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

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

144. Spjald 144

     Spurning

     Future-worth acceptance rule for an independent investment?

     Svar

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

145. Spjald 145

     Spurning

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

     Svar

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

146. Spjald 146

     Spurning

     Exact foot-to-metre relationship?

     Svar

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

147. Spjald 147

     Spurning

     How should important uncertainty be communicated?

     Svar

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

148. Spjald 148

     Spurning

     When is the matrix product $AB$ defined?

     Svar

     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. Spjald 149

     Spurning

     Steady one-inlet, one-outlet mass continuity?

     Svar

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Spjald 150

     Spurning

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

     Svar

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

151. Spjald 151

     Spurning

     Square feet in one acre?

     Svar

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

152. Spjald 152

     Spurning

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

     Svar

     $$ad-bc$$

153. Spjald 153

     Spurning

     Bernoulli relationship along a streamline for ideal steady flow?

     Svar

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

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

154. Spjald 154

     Spurning

     May inconvenient measurements be discarded without a technical basis?

     Svar

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

155. Spjald 155

     Spurning

     Annual-worth acceptance rule for an independent investment?

     Svar

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

156. Spjald 156

     Spurning

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

     Svar

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

157. Spjald 157

     Spurning

     Cubic metres in one litre?

     Svar

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

158. Spjald 158

     Spurning

     What does a zero determinant mean for a square matrix?

     Svar

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

159. Spjald 159

     Spurning

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

     Svar

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

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

160. Spjald 160

     Spurning

     Unbiased sample variance for an independent sample from one population?

     Svar

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

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

161. Spjald 161

     Spurning

     How should mutually exclusive alternatives be compared?

     Svar

     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. Spjald 162

     Spurning

     Cubic centimetres in one millilitre?

     Svar

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

163. Spjald 163

     Spurning

     Why keep contemporaneous engineering records?

     Svar

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

164. Spjald 164

     Spurning

     Fast check after substituting into a handbook formula?

     Svar

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

165. Spjald 165

     Spurning

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

     Svar

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

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

166. Spjald 166

     Spurning

     Sensible heat for constant specific heat and no phase change?

     Svar

     $$Q=mc\Delta T$$

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

167. Spjald 167

     Spurning

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

     Svar

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

     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. Spjald 168

     Spurning

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

     Svar

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

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

169. Spjald 169

     Spurning

     Purpose of incremental analysis for mutually exclusive alternatives?

     Svar

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

170. Spjald 170

     Spurning

     Physical meaning of $dy/dx$?

     Svar

     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. Spjald 171

     Spurning

     One-dimensional steady conduction through a plane wall?

     Svar

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

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

172. Spjald 172

     Spurning

     When should you algebraically rearrange a handbook relationship?

     Svar

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

173. Spjald 173

     Spurning

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

     Svar

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

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

174. Spjald 174

     Spurning

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

     Svar

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

175. Spjald 175

     Spurning

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

     Svar

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

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

176. Spjald 176

     Spurning

     Convective heat-transfer rate at a surface?

     Svar

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

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

177. Spjald 177

     Spurning

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

     Svar

     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. Spjald 178

     Spurning

     Internal rate of return?

     Svar

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

179. Spjald 179

     Spurning

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

     Svar

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

180. Spjald 180

     Spurning

     Convert Celsius temperature to kelvin?

     Svar

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

     This conversion is for absolute temperature, not temperature difference.

181. Spjald 181

     Spurning

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

     Svar

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

182. Spjald 182

     Spurning

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

     Svar

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

183. Spjald 183

     Spurning

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

     Svar

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

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

184. Spjald 184

     Spurning

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

     Svar

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

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

185. Spjald 185

     Spurning

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

     Svar

     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. Spjald 186

     Spurning

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

     Svar

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

187. Spjald 187

     Spurning

     Why inspect boundary or initial conditions near a formula?

     Svar

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

188. Spjald 188

     Spurning

     Convert Celsius temperature to Fahrenheit?

     Svar

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

189. Spjald 189

     Spurning

     Ethical basis for expert technical testimony?

     Svar

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

190. Spjald 190

     Spurning

     Meaning of a breakeven value?

     Svar

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

191. Spjald 191

     Spurning

     Derivative of $e^x$?

     Svar

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

192. Spjald 192

     Spurning

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

     Svar

     $$\Delta E=Q-W$$

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

193. Spjald 193

     Spurning

     Frequentist meaning of a 95% confidence procedure?

     Svar

     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. Spjald 194

     Spurning

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

     Svar

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

195. Spjald 195

     Spurning

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

     Svar

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

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

196. Spjald 196

     Spurning

     Relationship between Celsius and kelvin temperature differences?

     Svar

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

197. Spjald 197

     Spurning

     How should credit for team engineering work be assigned?

     Svar

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

198. Spjald 198

     Spurning

     Main limitation of simple payback period?

     Svar

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

199. Spjald 199

     Spurning

     Roles of the null and alternative hypotheses?

     Svar

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

200. Spjald 200

     Spurning

     Ideal-gas equation of state?

     Svar

     $$pV=nRT$$

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

201. Spjald 201

     Spurning

     First check when reading a handbook table?

     Svar

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

202. Spjald 202

     Spurning

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

     Svar

     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. Spjald 203

     Spurning

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

     Svar

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

204. Spjald 204

     Spurning

     Convert Fahrenheit absolute temperature to Rankine?

     Svar

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Spjald 205

     Spurning

     Ohm's law for an ideal resistor?

     Svar

     $$V=IR$$

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

206. Spjald 206

     Spurning

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

     Svar

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

207. Spjald 207

     Spurning

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

     Svar

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

208. Spjald 208

     Spurning

     Is a radian dimensionless?

     Svar

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

209. Spjald 209

     Spurning

     Electrical power absorbed under the passive sign convention?

     Svar

     $$P=VI$$

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

210. Spjald 210

     Spurning

     Meaning of a p-value?

     Svar

     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. Spjald 211

     Spurning

     First check when reading a handbook graph?

     Svar

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

212. Spjald 212

     Spurning

     Difference between mass and weight?

     Svar

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

213. Spjald 213

     Spurning

     Kirchhoff's current law at a node?

     Svar

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

214. Spjald 214

     Spurning

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

     Svar

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

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

215. Spjald 215

     Spurning

     Opportunity cost?

     Svar

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

216. Spjald 216

     Spurning

     Weight from mass in a gravitational field?

     Svar

     $$W=mg$$

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

217. Spjald 217

     Spurning

     Kirchhoff's voltage law around a closed loop?

     Svar

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

218. Spjald 218

     Spurning

     Decision rule using significance level $\alpha$?

     Svar

     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. Spjald 219

     Spurning

     Can proprietary methods from a former employer be reused freely?

     Svar

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

220. Spjald 220

     Spurning

     Pound-force to newtons?

     Svar

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

221. Spjald 221

     Spurning

     When does a definite integral equal ordinary geometric area?

     Svar

     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. Spjald 222

     Spurning

     Charge-voltage relation for an ideal capacitor?

     Svar

     $$q=CV$$

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

223. Spjald 223

     Spurning

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

     Svar

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

224. Spjald 224

     Spurning

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

     Svar

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

225. Spjald 225

     Spurning

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

     Svar

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

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

226. Spjald 226

     Spurning

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

     Svar

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

227. Spjald 227

     Spurning

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

     Svar

     $$\tau=RC$$

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

228. Spjald 228

     Spurning

     What physical quantity does pound-mass measure?

     Svar

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

229. Spjald 229

     Spurning

     Type I error?

     Svar

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

230. Spjald 230

     Spurning

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

     Svar

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

231. Spjald 231

     Spurning

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

     Svar

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

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

232. Spjald 232

     Spurning

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

     Svar

     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. Spjald 233

     Spurning

     Recognizable form of a separable first-order differential equation?

     Svar

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

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

234. Spjald 234

     Spurning

     Using unverified software output in a safety-relevant calculation?

     Svar

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

235. Spjald 235

     Spurning

     Equivalent resistance of ideal resistors in series?

     Svar

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

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

236. Spjald 236

     Spurning

     Pressure conversion from psi to pascals?

     Svar

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

237. Spjald 237

     Spurning

     Type II error?

     Svar

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

238. Spjald 238

     Spurning

     Useful flashcard to make after a failed handbook search?

     Svar

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

239. Spjald 239

     Spurning

     Standard form of a first-order linear differential equation?

     Svar

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

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

240. Spjald 240

     Spurning

     Equivalent resistance of ideal resistors in parallel?

     Svar

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

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

241. Spjald 241

     Spurning

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

     Svar

     $$BV_k=B-kD$$

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

242. Spjald 242

     Spurning

     Standard atmosphere in pascals?

     Svar

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

243. Spjald 243

     Spurning

     Using AI-generated technical content in engineering work?

     Svar

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

244. Spjald 244

     Spurning

     Trial form for a homogeneous linear ODE with constant coefficients?

     Svar

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

245. Spjald 245

     Spurning

     Wave-speed relationship among frequency and wavelength?

     Svar

     $$v=f\lambda$$

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

246. Spjald 246

     Spurning

     Statistical power?

     Svar

     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. Spjald 247

     Spurning

     Bar in pascals?

     Svar

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

248. Spjald 248

     Spurning

     How should handbook navigation be practiced before the FE exam?

     Svar

     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. Spjald 249

     Spurning

     Is depreciation itself a cash outflow?

     Svar

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

250. Spjald 250

     Spurning

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

     Svar

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

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

251. Spjald 251

     Spurning

     Relationship between period and frequency?

     Svar

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

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

252. Spjald 252

     Spurning

     How should foreseeable environmental harm enter an engineering decision?

     Svar

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

253. Spjald 253

     Spurning

     Foot-pound-force in joules?

     Svar

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

254. Spjald 254

     Spurning

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

     Svar

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

255. Spjald 255

     Spurning

     When should you search instead of relying on recall?

     Svar

     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. Spjald 256

     Spurning

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

     Svar

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

257. Spjald 257

     Spurning

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

     Svar

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

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

258. Spjald 258

     Spurning

     General material balance on a control volume?

     Svar

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

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

259. Spjald 259

     Spurning

     Approximate joules in one International Table Btu?

     Svar

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

260. Spjald 260

     Spurning

     What is the defender in a replacement analysis?

     Svar

     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. Spjald 261

     Spurning

     Speaking publicly about a technical matter outside your expertise?

     Svar

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

262. Spjald 262

     Spurning

     Single-interval trapezoidal approximation to an integral?

     Svar

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

     It is exact for linear $f$.

263. Spjald 263

     Spurning

     Energy efficiency of a device?

     Svar

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

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

264. Spjald 264

     Spurning

     Purpose of sensitivity analysis in engineering economics?

     Svar

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

265. Spjald 265

     Spurning

     Can a strong correlation establish causation?

     Svar

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

266. Spjald 266

     Spurning

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

     Svar

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

267. Spjald 267

     Spurning

     Mechanical horsepower in watts?

     Svar

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

268. Spjald 268

     Spurning

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

     Svar

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

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

269. Spjald 269

     Spurning

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

     Svar

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

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

270. Spjald 270

     Spurning

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

     Svar

     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. Spjald 271

     Spurning

     Kilowatt-hour in joules?

     Svar

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

272. Spjald 272

     Spurning

     Percent error relative to an accepted nonzero value?

     Svar

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

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

273. Spjald 273

     Spurning

     Does legal compliance alone settle an engineering ethics question?

     Svar

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

274. Spjald 274

     Spurning

     Does handbook fluency replace discipline-specific problem practice?

     Svar

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

275. Spjald 275

     Spurning

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

     Svar

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

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

276. Spjald 276

     Spurning

     Dimensional requirement for a physically meaningful additive equation?

     Svar

     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.