FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Tämän pakan kortit

1. Kortti 1

   Kysymys

   SI base unit of length?

   Vastaus

   The metre, symbol $m$.

2. Kortti 2

   Kysymys

   Slope between two points?

   Vastaus

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

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

3. Kortti 3

   Kysymys

   Net force relationship for constant mass in an inertial frame?

   Vastaus

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

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

4. Kortti 4

   Kysymys

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

   Vastaus

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

5. Kortti 5

   Kysymys

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

   Vastaus

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

6. Kortti 6

   Kysymys

   Probability of the complement of event $A$?

   Vastaus

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

7. Kortti 7

   Kysymys

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

   Vastaus

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

8. Kortti 8

   Kysymys

   SI base unit of mass?

   Vastaus

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

9. Kortti 9

   Kysymys

   Linear equation from slope and intercept?

   Vastaus

   $$y=mx+b$$

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

10. Kortti 10

    Kysymys

    Velocity after time $t$ under constant acceleration?

    Vastaus

    $$v=v_0+at$$

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

11. Kortti 11

    Kysymys

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

    Vastaus

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

12. Kortti 12

    Kysymys

    How is the FE reference handbook available during the exam?

    Vastaus

    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. Kortti 13

    Kysymys

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

    Vastaus

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

14. Kortti 14

    Kysymys

    SI base unit of time?

    Vastaus

    The second, symbol $s$.

15. Kortti 15

    Kysymys

    Linear equation from one point and a slope?

    Vastaus

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

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

16. Kortti 16

    Kysymys

    Position after time $t$ under constant acceleration?

    Vastaus

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

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

17. Kortti 17

    Kysymys

    A project assignment lies outside your competence. Proper response?

    Vastaus

    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. Kortti 18

    Kysymys

    SI base unit of electric current?

    Vastaus

    The ampere, symbol $A$.

19. Kortti 19

    Kysymys

    Future amount under simple interest?

    Vastaus

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

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

20. Kortti 20

    Kysymys

    Distance between two Cartesian points?

    Vastaus

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

    The coordinate components must use the same length unit.

21. Kortti 21

    Kysymys

    Constant-acceleration relation without elapsed time?

    Vastaus

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

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

22. Kortti 22

    Kysymys

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

    Vastaus

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

23. Kortti 23

    Kysymys

    Conditional probability of $A$ given $B$?

    Vastaus

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

    This requires $P(B)>0$.

24. Kortti 24

    Kysymys

    SI base unit of thermodynamic temperature?

    Vastaus

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

25. Kortti 25

    Kysymys

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

    Vastaus

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

26. Kortti 26

    Kysymys

    Midpoint of two Cartesian points?

    Vastaus

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

    Average corresponding coordinates.

27. Kortti 27

    Kysymys

    Linear momentum of a particle?

    Vastaus

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

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

28. Kortti 28

    Kysymys

    Future amount under compound interest?

    Vastaus

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

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

29. Kortti 29

    Kysymys

    Should an FE search workflow depend on Ctrl+F?

    Vastaus

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

30. Kortti 30

    Kysymys

    SI base unit of amount of substance?

    Vastaus

    The mole, symbol $mol$.

31. Kortti 31

    Kysymys

    Joint probability expressed with a conditional probability?

    Vastaus

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

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

32. Kortti 32

    Kysymys

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

    Vastaus

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

    The coordinate units and radius unit must match.

33. Kortti 33

    Kysymys

    Impulse-momentum relationship?

    Vastaus

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

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

34. Kortti 34

    Kysymys

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

    Vastaus

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

35. Kortti 35

    Kysymys

    SI base unit of luminous intensity?

    Vastaus

    The candela, symbol $cd$.

36. Kortti 36

    Kysymys

    Present worth of one future amount?

    Vastaus

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

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

37. Kortti 37

    Kysymys

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

    Vastaus

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

    This form requires $a\ne0$.

38. Kortti 38

    Kysymys

    When is total linear momentum conserved?

    Vastaus

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

39. Kortti 39

    Kysymys

    Why avoid memorizing FE handbook page numbers?

    Vastaus

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

40. Kortti 40

    Kysymys

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

    Vastaus

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

    Independence is not the same as mutual exclusivity.

41. Kortti 41

    Kysymys

    SI unit of frequency in base-unit form?

    Vastaus

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

42. Kortti 42

    Kysymys

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

    Vastaus

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

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

43. Kortti 43

    Kysymys

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

    Vastaus

    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. Kortti 44

    Kysymys

    Work done by a constant force through displacement?

    Vastaus

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

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

45. Kortti 45

    Kysymys

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

    Vastaus

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

46. Kortti 46

    Kysymys

    SI unit of force in base-unit form?

    Vastaus

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

47. Kortti 47

    Kysymys

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

    Vastaus

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

48. Kortti 48

    Kysymys

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

    Vastaus

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

    This requires $P(B)>0$.

49. Kortti 49

    Kysymys

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

    Vastaus

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

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

50. Kortti 50

    Kysymys

    Translational kinetic energy of a particle?

    Vastaus

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

    Mass and speed must use a coherent unit system.

51. Kortti 51

    Kysymys

    SI unit of pressure in force-area form?

    Vastaus

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

52. Kortti 52

    Kysymys

    Equivalent annual amount for present worth $P$?

    Vastaus

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

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

53. Kortti 53

    Kysymys

    May confidential client information be shared for personal benefit?

    Vastaus

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

54. Kortti 54

    Kysymys

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

    Vastaus

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

55. Kortti 55

    Kysymys

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

    Vastaus

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

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

56. Kortti 56

    Kysymys

    Net work-energy relationship for a particle?

    Vastaus

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

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

57. Kortti 57

    Kysymys

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

    Vastaus

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

    Use this when order matters and selection is without replacement.

58. Kortti 58

    Kysymys

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

    Vastaus

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

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

59. Kortti 59

    Kysymys

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

    Vastaus

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

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

60. Kortti 60

    Kysymys

    SI unit of energy in force-distance form?

    Vastaus

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

61. Kortti 61

    Kysymys

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

    Vastaus

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

62. Kortti 62

    Kysymys

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

    Vastaus

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

63. Kortti 63

    Kysymys

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

    Vastaus

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

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

64. Kortti 64

    Kysymys

    Change in gravitational potential energy near Earth's surface?

    Vastaus

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

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

65. Kortti 65

    Kysymys

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

    Vastaus

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

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

66. Kortti 66

    Kysymys

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

    Vastaus

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

    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. Kortti 67

    Kysymys

    A handbook search for an acronym fails. Next move?

    Vastaus

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

68. Kortti 68

    Kysymys

    SI unit of power in energy-time form?

    Vastaus

    The watt, $W=J/s$.

69. Kortti 69

    Kysymys

    Logarithm of a product?

    Vastaus

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

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

70. Kortti 70

    Kysymys

    Conditions for a binomial model?

    Vastaus

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

71. Kortti 71

    Kysymys

    Using another engineer's analysis in your deliverable?

    Vastaus

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

72. Kortti 72

    Kysymys

    SI unit of electric charge in current-time form?

    Vastaus

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

73. Kortti 73

    Kysymys

    Average power from work over elapsed time?

    Vastaus

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

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

74. Kortti 74

    Kysymys

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

    Vastaus

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

    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. Kortti 75

    Kysymys

    Logarithm of a positive quantity raised to a power?

    Vastaus

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

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

76. Kortti 76

    Kysymys

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

    Vastaus

    The volt, $V=J/C$.

77. Kortti 77

    Kysymys

    Instantaneous mechanical power delivered by a force?

    Vastaus

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

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

78. Kortti 78

    Kysymys

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

    Vastaus

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

79. Kortti 79

    Kysymys

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

    Vastaus

    $$E[X]=np$$

80. Kortti 80

    Kysymys

    SI unit of electrical resistance in voltage-current form?

    Vastaus

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

81. Kortti 81

    Kysymys

    Moment of a force about a point?

    Vastaus

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

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

82. Kortti 82

    Kysymys

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

    Vastaus

    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. Kortti 83

    Kysymys

    Inverse relationship between the natural log and exponential?

    Vastaus

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

    The second identity requires $x>0$.

84. Kortti 84

    Kysymys

    SI prefix giga?

    Vastaus

    $G=10^9$.

85. Kortti 85

    Kysymys

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

    Vastaus

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

    Use consistent angular sign convention and radians for angular measures.

86. Kortti 86

    Kysymys

    When should an engineer sign or seal technical work?

    Vastaus

    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. Kortti 87

    Kysymys

    Best memory target for returning to a handbook formula?

    Vastaus

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

88. Kortti 88

    Kysymys

    SI prefix mega?

    Vastaus

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

89. Kortti 89

    Kysymys

    Convert degrees to radians?

    Vastaus

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

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

90. Kortti 90

    Kysymys

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

    Vastaus

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

91. Kortti 91

    Kysymys

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

    Vastaus

    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. Kortti 92

    Kysymys

    Where do licensure and sealing requirements come from?

    Vastaus

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

93. Kortti 93

    Kysymys

    Rotational kinetic energy about a fixed axis?

    Vastaus

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

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

94. Kortti 94

    Kysymys

    SI prefix kilo?

    Vastaus

    $k=10^3$.

95. Kortti 95

    Kysymys

    Right-triangle definition of sine?

    Vastaus

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

96. Kortti 96

    Kysymys

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

    Vastaus

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

97. Kortti 97

    Kysymys

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

    Vastaus

    $$\sum M=I\alpha$$

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

98. Kortti 98

    Kysymys

    SI prefix milli?

    Vastaus

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

99. Kortti 99

    Kysymys

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

    Vastaus

    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. Kortti 100

     Kysymys

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

     Vastaus

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

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

101. Kortti 101

     Kysymys

     Right-triangle definition of cosine?

     Vastaus

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

102. Kortti 102

     Kysymys

     Force-displacement relation for an ideal linear spring?

     Vastaus

     $$F=-kx$$

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

103. Kortti 103

     Kysymys

     May qualifications or project responsibility be exaggerated in a proposal?

     Vastaus

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

104. Kortti 104

     Kysymys

     SI prefix micro?

     Vastaus

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

105. Kortti 105

     Kysymys

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

     Vastaus

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

106. Kortti 106

     Kysymys

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

     Vastaus

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

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

107. Kortti 107

     Kysymys

     Right-triangle definition of tangent?

     Vastaus

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

108. Kortti 108

     Kysymys

     Average normal stress under centered axial load?

     Vastaus

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

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

109. Kortti 109

     Kysymys

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

     Vastaus

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

110. Kortti 110

     Kysymys

     SI prefix nano?

     Vastaus

     $n=10^{-9}$.

111. Kortti 111

     Kysymys

     Fundamental Pythagorean trigonometric identity?

     Vastaus

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

112. Kortti 112

     Kysymys

     Engineering normal strain?

     Vastaus

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

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

113. Kortti 113

     Kysymys

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

     Vastaus

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

114. Kortti 114

     Kysymys

     Technical duty to an employer or client?

     Vastaus

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

115. Kortti 115

     Kysymys

     How does a length conversion factor apply to area?

     Vastaus

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

116. Kortti 116

     Kysymys

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

     Vastaus

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

117. Kortti 117

     Kysymys

     When is a Poisson count model reasonable?

     Vastaus

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

118. Kortti 118

     Kysymys

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

     Vastaus

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

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

119. Kortti 119

     Kysymys

     Linear elastic stress-strain relation in uniaxial loading?

     Vastaus

     $$\sigma=E\varepsilon$$

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

120. Kortti 120

     Kysymys

     How does a length conversion factor apply to volume?

     Vastaus

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

121. Kortti 121

     Kysymys

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

     Vastaus

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

122. Kortti 122

     Kysymys

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

     Vastaus

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

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

123. Kortti 123

     Kysymys

     A supervisor directs an illegal engineering action. Proper response?

     Vastaus

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

124. Kortti 124

     Kysymys

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

     Vastaus

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

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

125. Kortti 125

     Kysymys

     Mass density?

     Vastaus

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

     Use mass, not weight, in the numerator.

126. Kortti 126

     Kysymys

     Inches in one foot?

     Vastaus

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

127. Kortti 127

     Kysymys

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

     Vastaus

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

128. Kortti 128

     Kysymys

     How should subscripts be handled in a retrieved relationship?

     Vastaus

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

129. Kortti 129

     Kysymys

     Dot product from vector magnitudes and included angle?

     Vastaus

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

     The result is a scalar.

130. Kortti 130

     Kysymys

     Average normal pressure on an area?

     Vastaus

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

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

131. Kortti 131

     Kysymys

     Feet in one yard?

     Vastaus

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

132. Kortti 132

     Kysymys

     When are two interest rates equivalent?

     Vastaus

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

133. Kortti 133

     Kysymys

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

     Vastaus

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

134. Kortti 134

     Kysymys

     Expected value of a discrete random variable?

     Vastaus

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

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

135. Kortti 135

     Kysymys

     Dot-product test for perpendicular nonzero vectors?

     Vastaus

     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. Kortti 136

     Kysymys

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

     Vastaus

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

     Depth increases downward and the fluid is at rest.

137. Kortti 137

     Kysymys

     Feet in one statute mile?

     Vastaus

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

138. Kortti 138

     Kysymys

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

     Vastaus

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

     

     276 korttia

     FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

     Opiskele tätä pakkaa ilmaiseksi

     Nibomo avautuu, jotta voit aloittaa opiskelun.

139. Kortti 139

     Kysymys

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

     Vastaus

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

140. Kortti 140

     Kysymys

     Magnitude of a vector cross product?

     Vastaus

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

     The direction follows the right-hand rule.

141. Kortti 141

     Kysymys

     Exact inch-to-metre relationship?

     Vastaus

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

142. Kortti 142

     Kysymys

     Volumetric flow rate through a cross-section?

     Vastaus

     $$Q=Av_{avg}$$

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

143. Kortti 143

     Kysymys

     Variance from the mean $\mu$?

     Vastaus

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

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

144. Kortti 144

     Kysymys

     Future-worth acceptance rule for an independent investment?

     Vastaus

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

145. Kortti 145

     Kysymys

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

     Vastaus

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

146. Kortti 146

     Kysymys

     Exact foot-to-metre relationship?

     Vastaus

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

147. Kortti 147

     Kysymys

     How should important uncertainty be communicated?

     Vastaus

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

148. Kortti 148

     Kysymys

     When is the matrix product $AB$ defined?

     Vastaus

     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. Kortti 149

     Kysymys

     Steady one-inlet, one-outlet mass continuity?

     Vastaus

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Kortti 150

     Kysymys

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

     Vastaus

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

151. Kortti 151

     Kysymys

     Square feet in one acre?

     Vastaus

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

152. Kortti 152

     Kysymys

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

     Vastaus

     $$ad-bc$$

153. Kortti 153

     Kysymys

     Bernoulli relationship along a streamline for ideal steady flow?

     Vastaus

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

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

154. Kortti 154

     Kysymys

     May inconvenient measurements be discarded without a technical basis?

     Vastaus

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

155. Kortti 155

     Kysymys

     Annual-worth acceptance rule for an independent investment?

     Vastaus

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

156. Kortti 156

     Kysymys

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

     Vastaus

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

157. Kortti 157

     Kysymys

     Cubic metres in one litre?

     Vastaus

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

158. Kortti 158

     Kysymys

     What does a zero determinant mean for a square matrix?

     Vastaus

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

159. Kortti 159

     Kysymys

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

     Vastaus

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

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

160. Kortti 160

     Kysymys

     Unbiased sample variance for an independent sample from one population?

     Vastaus

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

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

161. Kortti 161

     Kysymys

     How should mutually exclusive alternatives be compared?

     Vastaus

     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. Kortti 162

     Kysymys

     Cubic centimetres in one millilitre?

     Vastaus

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

163. Kortti 163

     Kysymys

     Why keep contemporaneous engineering records?

     Vastaus

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

164. Kortti 164

     Kysymys

     Fast check after substituting into a handbook formula?

     Vastaus

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

165. Kortti 165

     Kysymys

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

     Vastaus

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

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

166. Kortti 166

     Kysymys

     Sensible heat for constant specific heat and no phase change?

     Vastaus

     $$Q=mc\Delta T$$

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

167. Kortti 167

     Kysymys

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

     Vastaus

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

     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. Kortti 168

     Kysymys

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

     Vastaus

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

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

169. Kortti 169

     Kysymys

     Purpose of incremental analysis for mutually exclusive alternatives?

     Vastaus

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

170. Kortti 170

     Kysymys

     Physical meaning of $dy/dx$?

     Vastaus

     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. Kortti 171

     Kysymys

     One-dimensional steady conduction through a plane wall?

     Vastaus

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

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

172. Kortti 172

     Kysymys

     When should you algebraically rearrange a handbook relationship?

     Vastaus

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

173. Kortti 173

     Kysymys

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

     Vastaus

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

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

174. Kortti 174

     Kysymys

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

     Vastaus

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

175. Kortti 175

     Kysymys

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

     Vastaus

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

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

176. Kortti 176

     Kysymys

     Convective heat-transfer rate at a surface?

     Vastaus

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

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

177. Kortti 177

     Kysymys

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

     Vastaus

     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. Kortti 178

     Kysymys

     Internal rate of return?

     Vastaus

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

179. Kortti 179

     Kysymys

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

     Vastaus

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

180. Kortti 180

     Kysymys

     Convert Celsius temperature to kelvin?

     Vastaus

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

     This conversion is for absolute temperature, not temperature difference.

181. Kortti 181

     Kysymys

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

     Vastaus

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

182. Kortti 182

     Kysymys

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

     Vastaus

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

183. Kortti 183

     Kysymys

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

     Vastaus

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

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

184. Kortti 184

     Kysymys

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

     Vastaus

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

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

185. Kortti 185

     Kysymys

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

     Vastaus

     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. Kortti 186

     Kysymys

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

     Vastaus

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

187. Kortti 187

     Kysymys

     Why inspect boundary or initial conditions near a formula?

     Vastaus

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

188. Kortti 188

     Kysymys

     Convert Celsius temperature to Fahrenheit?

     Vastaus

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

189. Kortti 189

     Kysymys

     Ethical basis for expert technical testimony?

     Vastaus

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

190. Kortti 190

     Kysymys

     Meaning of a breakeven value?

     Vastaus

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

191. Kortti 191

     Kysymys

     Derivative of $e^x$?

     Vastaus

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

192. Kortti 192

     Kysymys

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

     Vastaus

     $$\Delta E=Q-W$$

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

193. Kortti 193

     Kysymys

     Frequentist meaning of a 95% confidence procedure?

     Vastaus

     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. Kortti 194

     Kysymys

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

     Vastaus

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

195. Kortti 195

     Kysymys

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

     Vastaus

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

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

196. Kortti 196

     Kysymys

     Relationship between Celsius and kelvin temperature differences?

     Vastaus

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

197. Kortti 197

     Kysymys

     How should credit for team engineering work be assigned?

     Vastaus

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

198. Kortti 198

     Kysymys

     Main limitation of simple payback period?

     Vastaus

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

199. Kortti 199

     Kysymys

     Roles of the null and alternative hypotheses?

     Vastaus

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

200. Kortti 200

     Kysymys

     Ideal-gas equation of state?

     Vastaus

     $$pV=nRT$$

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

201. Kortti 201

     Kysymys

     First check when reading a handbook table?

     Vastaus

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

202. Kortti 202

     Kysymys

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

     Vastaus

     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. Kortti 203

     Kysymys

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

     Vastaus

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

204. Kortti 204

     Kysymys

     Convert Fahrenheit absolute temperature to Rankine?

     Vastaus

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Kortti 205

     Kysymys

     Ohm's law for an ideal resistor?

     Vastaus

     $$V=IR$$

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

206. Kortti 206

     Kysymys

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

     Vastaus

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

207. Kortti 207

     Kysymys

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

     Vastaus

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

208. Kortti 208

     Kysymys

     Is a radian dimensionless?

     Vastaus

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

209. Kortti 209

     Kysymys

     Electrical power absorbed under the passive sign convention?

     Vastaus

     $$P=VI$$

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

210. Kortti 210

     Kysymys

     Meaning of a p-value?

     Vastaus

     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. Kortti 211

     Kysymys

     First check when reading a handbook graph?

     Vastaus

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

212. Kortti 212

     Kysymys

     Difference between mass and weight?

     Vastaus

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

213. Kortti 213

     Kysymys

     Kirchhoff's current law at a node?

     Vastaus

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

214. Kortti 214

     Kysymys

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

     Vastaus

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

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

215. Kortti 215

     Kysymys

     Opportunity cost?

     Vastaus

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

216. Kortti 216

     Kysymys

     Weight from mass in a gravitational field?

     Vastaus

     $$W=mg$$

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

217. Kortti 217

     Kysymys

     Kirchhoff's voltage law around a closed loop?

     Vastaus

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

218. Kortti 218

     Kysymys

     Decision rule using significance level $\alpha$?

     Vastaus

     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. Kortti 219

     Kysymys

     Can proprietary methods from a former employer be reused freely?

     Vastaus

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

220. Kortti 220

     Kysymys

     Pound-force to newtons?

     Vastaus

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

221. Kortti 221

     Kysymys

     When does a definite integral equal ordinary geometric area?

     Vastaus

     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. Kortti 222

     Kysymys

     Charge-voltage relation for an ideal capacitor?

     Vastaus

     $$q=CV$$

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

223. Kortti 223

     Kysymys

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

     Vastaus

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

224. Kortti 224

     Kysymys

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

     Vastaus

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

225. Kortti 225

     Kysymys

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

     Vastaus

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

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

226. Kortti 226

     Kysymys

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

     Vastaus

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

227. Kortti 227

     Kysymys

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

     Vastaus

     $$\tau=RC$$

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

228. Kortti 228

     Kysymys

     What physical quantity does pound-mass measure?

     Vastaus

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

229. Kortti 229

     Kysymys

     Type I error?

     Vastaus

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

230. Kortti 230

     Kysymys

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

     Vastaus

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

231. Kortti 231

     Kysymys

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

     Vastaus

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

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

232. Kortti 232

     Kysymys

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

     Vastaus

     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. Kortti 233

     Kysymys

     Recognizable form of a separable first-order differential equation?

     Vastaus

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

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

234. Kortti 234

     Kysymys

     Using unverified software output in a safety-relevant calculation?

     Vastaus

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

235. Kortti 235

     Kysymys

     Equivalent resistance of ideal resistors in series?

     Vastaus

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

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

236. Kortti 236

     Kysymys

     Pressure conversion from psi to pascals?

     Vastaus

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

237. Kortti 237

     Kysymys

     Type II error?

     Vastaus

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

238. Kortti 238

     Kysymys

     Useful flashcard to make after a failed handbook search?

     Vastaus

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

239. Kortti 239

     Kysymys

     Standard form of a first-order linear differential equation?

     Vastaus

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

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

240. Kortti 240

     Kysymys

     Equivalent resistance of ideal resistors in parallel?

     Vastaus

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

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

241. Kortti 241

     Kysymys

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

     Vastaus

     $$BV_k=B-kD$$

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

242. Kortti 242

     Kysymys

     Standard atmosphere in pascals?

     Vastaus

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

243. Kortti 243

     Kysymys

     Using AI-generated technical content in engineering work?

     Vastaus

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

244. Kortti 244

     Kysymys

     Trial form for a homogeneous linear ODE with constant coefficients?

     Vastaus

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

245. Kortti 245

     Kysymys

     Wave-speed relationship among frequency and wavelength?

     Vastaus

     $$v=f\lambda$$

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

246. Kortti 246

     Kysymys

     Statistical power?

     Vastaus

     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. Kortti 247

     Kysymys

     Bar in pascals?

     Vastaus

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

248. Kortti 248

     Kysymys

     How should handbook navigation be practiced before the FE exam?

     Vastaus

     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. Kortti 249

     Kysymys

     Is depreciation itself a cash outflow?

     Vastaus

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

250. Kortti 250

     Kysymys

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

     Vastaus

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

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

251. Kortti 251

     Kysymys

     Relationship between period and frequency?

     Vastaus

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

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

252. Kortti 252

     Kysymys

     How should foreseeable environmental harm enter an engineering decision?

     Vastaus

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

253. Kortti 253

     Kysymys

     Foot-pound-force in joules?

     Vastaus

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

254. Kortti 254

     Kysymys

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

     Vastaus

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

255. Kortti 255

     Kysymys

     When should you search instead of relying on recall?

     Vastaus

     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. Kortti 256

     Kysymys

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

     Vastaus

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

257. Kortti 257

     Kysymys

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

     Vastaus

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

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

258. Kortti 258

     Kysymys

     General material balance on a control volume?

     Vastaus

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

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

259. Kortti 259

     Kysymys

     Approximate joules in one International Table Btu?

     Vastaus

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

260. Kortti 260

     Kysymys

     What is the defender in a replacement analysis?

     Vastaus

     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. Kortti 261

     Kysymys

     Speaking publicly about a technical matter outside your expertise?

     Vastaus

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

262. Kortti 262

     Kysymys

     Single-interval trapezoidal approximation to an integral?

     Vastaus

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

     It is exact for linear $f$.

263. Kortti 263

     Kysymys

     Energy efficiency of a device?

     Vastaus

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

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

264. Kortti 264

     Kysymys

     Purpose of sensitivity analysis in engineering economics?

     Vastaus

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

265. Kortti 265

     Kysymys

     Can a strong correlation establish causation?

     Vastaus

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

266. Kortti 266

     Kysymys

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

     Vastaus

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

267. Kortti 267

     Kysymys

     Mechanical horsepower in watts?

     Vastaus

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

268. Kortti 268

     Kysymys

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

     Vastaus

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

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

269. Kortti 269

     Kysymys

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

     Vastaus

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

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

270. Kortti 270

     Kysymys

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

     Vastaus

     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. Kortti 271

     Kysymys

     Kilowatt-hour in joules?

     Vastaus

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

272. Kortti 272

     Kysymys

     Percent error relative to an accepted nonzero value?

     Vastaus

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

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

273. Kortti 273

     Kysymys

     Does legal compliance alone settle an engineering ethics question?

     Vastaus

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

274. Kortti 274

     Kysymys

     Does handbook fluency replace discipline-specific problem practice?

     Vastaus

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

275. Kortti 275

     Kysymys

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

     Vastaus

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

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

276. Kortti 276

     Kysymys

     Dimensional requirement for a physically meaningful additive equation?

     Vastaus

     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.