FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Картки в цій колоді

1. Картка 1

   Питання

   SI base unit of length?

   Відповідь

   The metre, symbol $m$.

2. Картка 2

   Питання

   Slope between two points?

   Відповідь

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

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

3. Картка 3

   Питання

   Net force relationship for constant mass in an inertial frame?

   Відповідь

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

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

4. Картка 4

   Питання

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

   Відповідь

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

5. Картка 5

   Питання

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

   Відповідь

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

6. Картка 6

   Питання

   Probability of the complement of event $A$?

   Відповідь

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

7. Картка 7

   Питання

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

   Відповідь

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

8. Картка 8

   Питання

   SI base unit of mass?

   Відповідь

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

9. Картка 9

   Питання

   Linear equation from slope and intercept?

   Відповідь

   $$y=mx+b$$

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

10. Картка 10

    Питання

    Velocity after time $t$ under constant acceleration?

    Відповідь

    $$v=v_0+at$$

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

11. Картка 11

    Питання

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

    Відповідь

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

12. Картка 12

    Питання

    How is the FE reference handbook available during the exam?

    Відповідь

    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. Картка 13

    Питання

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

    Відповідь

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

14. Картка 14

    Питання

    SI base unit of time?

    Відповідь

    The second, symbol $s$.

15. Картка 15

    Питання

    Linear equation from one point and a slope?

    Відповідь

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

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

16. Картка 16

    Питання

    Position after time $t$ under constant acceleration?

    Відповідь

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

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

17. Картка 17

    Питання

    A project assignment lies outside your competence. Proper response?

    Відповідь

    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. Картка 18

    Питання

    SI base unit of electric current?

    Відповідь

    The ampere, symbol $A$.

19. Картка 19

    Питання

    Future amount under simple interest?

    Відповідь

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

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

20. Картка 20

    Питання

    Distance between two Cartesian points?

    Відповідь

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

    The coordinate components must use the same length unit.

21. Картка 21

    Питання

    Constant-acceleration relation without elapsed time?

    Відповідь

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

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

22. Картка 22

    Питання

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

    Відповідь

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

23. Картка 23

    Питання

    Conditional probability of $A$ given $B$?

    Відповідь

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

    This requires $P(B)>0$.

24. Картка 24

    Питання

    SI base unit of thermodynamic temperature?

    Відповідь

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

25. Картка 25

    Питання

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

    Відповідь

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

26. Картка 26

    Питання

    Midpoint of two Cartesian points?

    Відповідь

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

    Average corresponding coordinates.

27. Картка 27

    Питання

    Linear momentum of a particle?

    Відповідь

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

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

28. Картка 28

    Питання

    Future amount under compound interest?

    Відповідь

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

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

29. Картка 29

    Питання

    Should an FE search workflow depend on Ctrl+F?

    Відповідь

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

30. Картка 30

    Питання

    SI base unit of amount of substance?

    Відповідь

    The mole, symbol $mol$.

31. Картка 31

    Питання

    Joint probability expressed with a conditional probability?

    Відповідь

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

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

32. Картка 32

    Питання

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

    Відповідь

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

    The coordinate units and radius unit must match.

33. Картка 33

    Питання

    Impulse-momentum relationship?

    Відповідь

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

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

34. Картка 34

    Питання

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

    Відповідь

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

35. Картка 35

    Питання

    SI base unit of luminous intensity?

    Відповідь

    The candela, symbol $cd$.

36. Картка 36

    Питання

    Present worth of one future amount?

    Відповідь

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

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

37. Картка 37

    Питання

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

    Відповідь

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

    This form requires $a\ne0$.

38. Картка 38

    Питання

    When is total linear momentum conserved?

    Відповідь

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

39. Картка 39

    Питання

    Why avoid memorizing FE handbook page numbers?

    Відповідь

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

40. Картка 40

    Питання

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

    Відповідь

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

    Independence is not the same as mutual exclusivity.

41. Картка 41

    Питання

    SI unit of frequency in base-unit form?

    Відповідь

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

42. Картка 42

    Питання

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

    Відповідь

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

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

43. Картка 43

    Питання

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

    Відповідь

    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. Картка 44

    Питання

    Work done by a constant force through displacement?

    Відповідь

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

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

45. Картка 45

    Питання

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

    Відповідь

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

46. Картка 46

    Питання

    SI unit of force in base-unit form?

    Відповідь

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

47. Картка 47

    Питання

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

    Відповідь

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

48. Картка 48

    Питання

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

    Відповідь

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

    This requires $P(B)>0$.

49. Картка 49

    Питання

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

    Відповідь

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

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

50. Картка 50

    Питання

    Translational kinetic energy of a particle?

    Відповідь

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

    Mass and speed must use a coherent unit system.

51. Картка 51

    Питання

    SI unit of pressure in force-area form?

    Відповідь

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

52. Картка 52

    Питання

    Equivalent annual amount for present worth $P$?

    Відповідь

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

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

53. Картка 53

    Питання

    May confidential client information be shared for personal benefit?

    Відповідь

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

54. Картка 54

    Питання

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

    Відповідь

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

55. Картка 55

    Питання

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

    Відповідь

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

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

56. Картка 56

    Питання

    Net work-energy relationship for a particle?

    Відповідь

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

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

57. Картка 57

    Питання

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

    Відповідь

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

    Use this when order matters and selection is without replacement.

58. Картка 58

    Питання

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

    Відповідь

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

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

59. Картка 59

    Питання

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

    Відповідь

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

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

60. Картка 60

    Питання

    SI unit of energy in force-distance form?

    Відповідь

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

61. Картка 61

    Питання

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

    Відповідь

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

62. Картка 62

    Питання

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

    Відповідь

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

63. Картка 63

    Питання

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

    Відповідь

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

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

64. Картка 64

    Питання

    Change in gravitational potential energy near Earth's surface?

    Відповідь

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

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

65. Картка 65

    Питання

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

    Відповідь

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

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

66. Картка 66

    Питання

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

    Відповідь

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

    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. Картка 67

    Питання

    A handbook search for an acronym fails. Next move?

    Відповідь

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

68. Картка 68

    Питання

    SI unit of power in energy-time form?

    Відповідь

    The watt, $W=J/s$.

69. Картка 69

    Питання

    Logarithm of a product?

    Відповідь

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

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

70. Картка 70

    Питання

    Conditions for a binomial model?

    Відповідь

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

71. Картка 71

    Питання

    Using another engineer's analysis in your deliverable?

    Відповідь

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

72. Картка 72

    Питання

    SI unit of electric charge in current-time form?

    Відповідь

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

73. Картка 73

    Питання

    Average power from work over elapsed time?

    Відповідь

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

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

74. Картка 74

    Питання

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

    Відповідь

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

    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. Картка 75

    Питання

    Logarithm of a positive quantity raised to a power?

    Відповідь

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

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

76. Картка 76

    Питання

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

    Відповідь

    The volt, $V=J/C$.

77. Картка 77

    Питання

    Instantaneous mechanical power delivered by a force?

    Відповідь

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

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

78. Картка 78

    Питання

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

    Відповідь

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

79. Картка 79

    Питання

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

    Відповідь

    $$E[X]=np$$

80. Картка 80

    Питання

    SI unit of electrical resistance in voltage-current form?

    Відповідь

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

81. Картка 81

    Питання

    Moment of a force about a point?

    Відповідь

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

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

82. Картка 82

    Питання

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

    Відповідь

    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. Картка 83

    Питання

    Inverse relationship between the natural log and exponential?

    Відповідь

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

    The second identity requires $x>0$.

84. Картка 84

    Питання

    SI prefix giga?

    Відповідь

    $G=10^9$.

85. Картка 85

    Питання

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

    Відповідь

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

    Use consistent angular sign convention and radians for angular measures.

86. Картка 86

    Питання

    When should an engineer sign or seal technical work?

    Відповідь

    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. Картка 87

    Питання

    Best memory target for returning to a handbook formula?

    Відповідь

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

88. Картка 88

    Питання

    SI prefix mega?

    Відповідь

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

89. Картка 89

    Питання

    Convert degrees to radians?

    Відповідь

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

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

90. Картка 90

    Питання

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

    Відповідь

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

91. Картка 91

    Питання

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

    Відповідь

    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. Картка 92

    Питання

    Where do licensure and sealing requirements come from?

    Відповідь

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

93. Картка 93

    Питання

    Rotational kinetic energy about a fixed axis?

    Відповідь

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

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

94. Картка 94

    Питання

    SI prefix kilo?

    Відповідь

    $k=10^3$.

95. Картка 95

    Питання

    Right-triangle definition of sine?

    Відповідь

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

96. Картка 96

    Питання

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

    Відповідь

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

97. Картка 97

    Питання

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

    Відповідь

    $$\sum M=I\alpha$$

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

98. Картка 98

    Питання

    SI prefix milli?

    Відповідь

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

99. Картка 99

    Питання

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

    Відповідь

    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. Картка 100

     Питання

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

     Відповідь

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

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

101. Картка 101

     Питання

     Right-triangle definition of cosine?

     Відповідь

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

102. Картка 102

     Питання

     Force-displacement relation for an ideal linear spring?

     Відповідь

     $$F=-kx$$

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

103. Картка 103

     Питання

     May qualifications or project responsibility be exaggerated in a proposal?

     Відповідь

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

104. Картка 104

     Питання

     SI prefix micro?

     Відповідь

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

105. Картка 105

     Питання

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

     Відповідь

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

106. Картка 106

     Питання

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

     Відповідь

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

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

107. Картка 107

     Питання

     Right-triangle definition of tangent?

     Відповідь

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

108. Картка 108

     Питання

     Average normal stress under centered axial load?

     Відповідь

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

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

109. Картка 109

     Питання

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

     Відповідь

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

110. Картка 110

     Питання

     SI prefix nano?

     Відповідь

     $n=10^{-9}$.

111. Картка 111

     Питання

     Fundamental Pythagorean trigonometric identity?

     Відповідь

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

112. Картка 112

     Питання

     Engineering normal strain?

     Відповідь

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

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

113. Картка 113

     Питання

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

     Відповідь

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

114. Картка 114

     Питання

     Technical duty to an employer or client?

     Відповідь

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

115. Картка 115

     Питання

     How does a length conversion factor apply to area?

     Відповідь

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

116. Картка 116

     Питання

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

     Відповідь

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

117. Картка 117

     Питання

     When is a Poisson count model reasonable?

     Відповідь

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

118. Картка 118

     Питання

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

     Відповідь

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

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

119. Картка 119

     Питання

     Linear elastic stress-strain relation in uniaxial loading?

     Відповідь

     $$\sigma=E\varepsilon$$

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

120. Картка 120

     Питання

     How does a length conversion factor apply to volume?

     Відповідь

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

121. Картка 121

     Питання

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

     Відповідь

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

122. Картка 122

     Питання

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

     Відповідь

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

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

123. Картка 123

     Питання

     A supervisor directs an illegal engineering action. Proper response?

     Відповідь

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

124. Картка 124

     Питання

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

     Відповідь

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

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

125. Картка 125

     Питання

     Mass density?

     Відповідь

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

     Use mass, not weight, in the numerator.

126. Картка 126

     Питання

     Inches in one foot?

     Відповідь

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

127. Картка 127

     Питання

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

     Відповідь

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

128. Картка 128

     Питання

     How should subscripts be handled in a retrieved relationship?

     Відповідь

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

129. Картка 129

     Питання

     Dot product from vector magnitudes and included angle?

     Відповідь

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

     The result is a scalar.

130. Картка 130

     Питання

     Average normal pressure on an area?

     Відповідь

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

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

131. Картка 131

     Питання

     Feet in one yard?

     Відповідь

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

132. Картка 132

     Питання

     When are two interest rates equivalent?

     Відповідь

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

133. Картка 133

     Питання

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

     Відповідь

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

134. Картка 134

     Питання

     Expected value of a discrete random variable?

     Відповідь

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

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

135. Картка 135

     Питання

     Dot-product test for perpendicular nonzero vectors?

     Відповідь

     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. Картка 136

     Питання

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

     Відповідь

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

     Depth increases downward and the fluid is at rest.

137. Картка 137

     Питання

     Feet in one statute mile?

     Відповідь

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

138. Картка 138

     Питання

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

     Відповідь

     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. Картка 139

     Питання

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

     Відповідь

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

140. Картка 140

     Питання

     Magnitude of a vector cross product?

     Відповідь

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

     The direction follows the right-hand rule.

141. Картка 141

     Питання

     Exact inch-to-metre relationship?

     Відповідь

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

142. Картка 142

     Питання

     Volumetric flow rate through a cross-section?

     Відповідь

     $$Q=Av_{avg}$$

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

143. Картка 143

     Питання

     Variance from the mean $\mu$?

     Відповідь

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

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

144. Картка 144

     Питання

     Future-worth acceptance rule for an independent investment?

     Відповідь

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

145. Картка 145

     Питання

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

     Відповідь

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

146. Картка 146

     Питання

     Exact foot-to-metre relationship?

     Відповідь

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

147. Картка 147

     Питання

     How should important uncertainty be communicated?

     Відповідь

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

148. Картка 148

     Питання

     When is the matrix product $AB$ defined?

     Відповідь

     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. Картка 149

     Питання

     Steady one-inlet, one-outlet mass continuity?

     Відповідь

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Картка 150

     Питання

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

     Відповідь

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

151. Картка 151

     Питання

     Square feet in one acre?

     Відповідь

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

152. Картка 152

     Питання

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

     Відповідь

     $$ad-bc$$

153. Картка 153

     Питання

     Bernoulli relationship along a streamline for ideal steady flow?

     Відповідь

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

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

154. Картка 154

     Питання

     May inconvenient measurements be discarded without a technical basis?

     Відповідь

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

155. Картка 155

     Питання

     Annual-worth acceptance rule for an independent investment?

     Відповідь

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

156. Картка 156

     Питання

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

     Відповідь

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

157. Картка 157

     Питання

     Cubic metres in one litre?

     Відповідь

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

158. Картка 158

     Питання

     What does a zero determinant mean for a square matrix?

     Відповідь

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

159. Картка 159

     Питання

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

     Відповідь

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

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

160. Картка 160

     Питання

     Unbiased sample variance for an independent sample from one population?

     Відповідь

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

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

161. Картка 161

     Питання

     How should mutually exclusive alternatives be compared?

     Відповідь

     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. Картка 162

     Питання

     Cubic centimetres in one millilitre?

     Відповідь

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

163. Картка 163

     Питання

     Why keep contemporaneous engineering records?

     Відповідь

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

164. Картка 164

     Питання

     Fast check after substituting into a handbook formula?

     Відповідь

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

165. Картка 165

     Питання

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

     Відповідь

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

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

166. Картка 166

     Питання

     Sensible heat for constant specific heat and no phase change?

     Відповідь

     $$Q=mc\Delta T$$

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

167. Картка 167

     Питання

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

     Відповідь

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

     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. Картка 168

     Питання

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

     Відповідь

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

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

169. Картка 169

     Питання

     Purpose of incremental analysis for mutually exclusive alternatives?

     Відповідь

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

170. Картка 170

     Питання

     Physical meaning of $dy/dx$?

     Відповідь

     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. Картка 171

     Питання

     One-dimensional steady conduction through a plane wall?

     Відповідь

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

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

172. Картка 172

     Питання

     When should you algebraically rearrange a handbook relationship?

     Відповідь

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

173. Картка 173

     Питання

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

     Відповідь

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

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

174. Картка 174

     Питання

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

     Відповідь

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

175. Картка 175

     Питання

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

     Відповідь

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

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

176. Картка 176

     Питання

     Convective heat-transfer rate at a surface?

     Відповідь

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

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

177. Картка 177

     Питання

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

     Відповідь

     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. Картка 178

     Питання

     Internal rate of return?

     Відповідь

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

179. Картка 179

     Питання

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

     Відповідь

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

180. Картка 180

     Питання

     Convert Celsius temperature to kelvin?

     Відповідь

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

     This conversion is for absolute temperature, not temperature difference.

181. Картка 181

     Питання

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

     Відповідь

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

182. Картка 182

     Питання

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

     Відповідь

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

183. Картка 183

     Питання

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

     Відповідь

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

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

184. Картка 184

     Питання

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

     Відповідь

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

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

185. Картка 185

     Питання

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

     Відповідь

     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. Картка 186

     Питання

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

     Відповідь

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

187. Картка 187

     Питання

     Why inspect boundary or initial conditions near a formula?

     Відповідь

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

188. Картка 188

     Питання

     Convert Celsius temperature to Fahrenheit?

     Відповідь

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

189. Картка 189

     Питання

     Ethical basis for expert technical testimony?

     Відповідь

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

190. Картка 190

     Питання

     Meaning of a breakeven value?

     Відповідь

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

191. Картка 191

     Питання

     Derivative of $e^x$?

     Відповідь

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

192. Картка 192

     Питання

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

     Відповідь

     $$\Delta E=Q-W$$

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

193. Картка 193

     Питання

     Frequentist meaning of a 95% confidence procedure?

     Відповідь

     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. Картка 194

     Питання

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

     Відповідь

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

195. Картка 195

     Питання

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

     Відповідь

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

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

196. Картка 196

     Питання

     Relationship between Celsius and kelvin temperature differences?

     Відповідь

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

197. Картка 197

     Питання

     How should credit for team engineering work be assigned?

     Відповідь

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

198. Картка 198

     Питання

     Main limitation of simple payback period?

     Відповідь

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

199. Картка 199

     Питання

     Roles of the null and alternative hypotheses?

     Відповідь

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

200. Картка 200

     Питання

     Ideal-gas equation of state?

     Відповідь

     $$pV=nRT$$

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

201. Картка 201

     Питання

     First check when reading a handbook table?

     Відповідь

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

202. Картка 202

     Питання

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

     Відповідь

     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. Картка 203

     Питання

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

     Відповідь

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

204. Картка 204

     Питання

     Convert Fahrenheit absolute temperature to Rankine?

     Відповідь

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Картка 205

     Питання

     Ohm's law for an ideal resistor?

     Відповідь

     $$V=IR$$

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

206. Картка 206

     Питання

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

     Відповідь

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

207. Картка 207

     Питання

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

     Відповідь

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

208. Картка 208

     Питання

     Is a radian dimensionless?

     Відповідь

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

209. Картка 209

     Питання

     Electrical power absorbed under the passive sign convention?

     Відповідь

     $$P=VI$$

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

210. Картка 210

     Питання

     Meaning of a p-value?

     Відповідь

     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. Картка 211

     Питання

     First check when reading a handbook graph?

     Відповідь

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

212. Картка 212

     Питання

     Difference between mass and weight?

     Відповідь

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

213. Картка 213

     Питання

     Kirchhoff's current law at a node?

     Відповідь

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

214. Картка 214

     Питання

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

     Відповідь

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

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

215. Картка 215

     Питання

     Opportunity cost?

     Відповідь

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

216. Картка 216

     Питання

     Weight from mass in a gravitational field?

     Відповідь

     $$W=mg$$

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

217. Картка 217

     Питання

     Kirchhoff's voltage law around a closed loop?

     Відповідь

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

218. Картка 218

     Питання

     Decision rule using significance level $\alpha$?

     Відповідь

     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. Картка 219

     Питання

     Can proprietary methods from a former employer be reused freely?

     Відповідь

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

220. Картка 220

     Питання

     Pound-force to newtons?

     Відповідь

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

221. Картка 221

     Питання

     When does a definite integral equal ordinary geometric area?

     Відповідь

     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. Картка 222

     Питання

     Charge-voltage relation for an ideal capacitor?

     Відповідь

     $$q=CV$$

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

223. Картка 223

     Питання

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

     Відповідь

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

224. Картка 224

     Питання

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

     Відповідь

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

225. Картка 225

     Питання

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

     Відповідь

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

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

226. Картка 226

     Питання

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

     Відповідь

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

227. Картка 227

     Питання

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

     Відповідь

     $$\tau=RC$$

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

228. Картка 228

     Питання

     What physical quantity does pound-mass measure?

     Відповідь

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

229. Картка 229

     Питання

     Type I error?

     Відповідь

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

230. Картка 230

     Питання

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

     Відповідь

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

231. Картка 231

     Питання

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

     Відповідь

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

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

232. Картка 232

     Питання

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

     Відповідь

     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. Картка 233

     Питання

     Recognizable form of a separable first-order differential equation?

     Відповідь

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

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

234. Картка 234

     Питання

     Using unverified software output in a safety-relevant calculation?

     Відповідь

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

235. Картка 235

     Питання

     Equivalent resistance of ideal resistors in series?

     Відповідь

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

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

236. Картка 236

     Питання

     Pressure conversion from psi to pascals?

     Відповідь

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

237. Картка 237

     Питання

     Type II error?

     Відповідь

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

238. Картка 238

     Питання

     Useful flashcard to make after a failed handbook search?

     Відповідь

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

239. Картка 239

     Питання

     Standard form of a first-order linear differential equation?

     Відповідь

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

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

240. Картка 240

     Питання

     Equivalent resistance of ideal resistors in parallel?

     Відповідь

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

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

241. Картка 241

     Питання

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

     Відповідь

     $$BV_k=B-kD$$

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

242. Картка 242

     Питання

     Standard atmosphere in pascals?

     Відповідь

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

243. Картка 243

     Питання

     Using AI-generated technical content in engineering work?

     Відповідь

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

244. Картка 244

     Питання

     Trial form for a homogeneous linear ODE with constant coefficients?

     Відповідь

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

245. Картка 245

     Питання

     Wave-speed relationship among frequency and wavelength?

     Відповідь

     $$v=f\lambda$$

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

246. Картка 246

     Питання

     Statistical power?

     Відповідь

     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. Картка 247

     Питання

     Bar in pascals?

     Відповідь

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

248. Картка 248

     Питання

     How should handbook navigation be practiced before the FE exam?

     Відповідь

     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. Картка 249

     Питання

     Is depreciation itself a cash outflow?

     Відповідь

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

250. Картка 250

     Питання

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

     Відповідь

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

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

251. Картка 251

     Питання

     Relationship between period and frequency?

     Відповідь

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

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

252. Картка 252

     Питання

     How should foreseeable environmental harm enter an engineering decision?

     Відповідь

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

253. Картка 253

     Питання

     Foot-pound-force in joules?

     Відповідь

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

254. Картка 254

     Питання

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

     Відповідь

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

255. Картка 255

     Питання

     When should you search instead of relying on recall?

     Відповідь

     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. Картка 256

     Питання

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

     Відповідь

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

257. Картка 257

     Питання

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

     Відповідь

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

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

258. Картка 258

     Питання

     General material balance on a control volume?

     Відповідь

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

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

259. Картка 259

     Питання

     Approximate joules in one International Table Btu?

     Відповідь

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

260. Картка 260

     Питання

     What is the defender in a replacement analysis?

     Відповідь

     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. Картка 261

     Питання

     Speaking publicly about a technical matter outside your expertise?

     Відповідь

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

262. Картка 262

     Питання

     Single-interval trapezoidal approximation to an integral?

     Відповідь

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

     It is exact for linear $f$.

263. Картка 263

     Питання

     Energy efficiency of a device?

     Відповідь

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

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

264. Картка 264

     Питання

     Purpose of sensitivity analysis in engineering economics?

     Відповідь

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

265. Картка 265

     Питання

     Can a strong correlation establish causation?

     Відповідь

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

266. Картка 266

     Питання

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

     Відповідь

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

267. Картка 267

     Питання

     Mechanical horsepower in watts?

     Відповідь

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

268. Картка 268

     Питання

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

     Відповідь

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

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

269. Картка 269

     Питання

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

     Відповідь

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

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

270. Картка 270

     Питання

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

     Відповідь

     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. Картка 271

     Питання

     Kilowatt-hour in joules?

     Відповідь

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

272. Картка 272

     Питання

     Percent error relative to an accepted nonzero value?

     Відповідь

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

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

273. Картка 273

     Питання

     Does legal compliance alone settle an engineering ethics question?

     Відповідь

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

274. Картка 274

     Питання

     Does handbook fluency replace discipline-specific problem practice?

     Відповідь

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

275. Картка 275

     Питання

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

     Відповідь

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

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

276. Картка 276

     Питання

     Dimensional requirement for a physically meaningful additive equation?

     Відповідь

     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.