AP Calculus AB Flashcards: 8-Unit Formulas, Theorems & Concepts

这套卡组中的卡片

1. 卡片 1

   问题

   What does $\lim_{x \to a} f(x)=L$ say?

   答案

   The values of $f(x)$ approach $L$ as $x$ approaches $a$ from both sides. The statement doesn't require $f(a)=L$ or even require $f(a)$ to exist.

2. 卡片 2

   问题

   How can a table estimate $\lim_{x \to a} f(x)$?

   答案

   Use inputs approaching $a$ from below and above, then look for a common output value. Values exactly at $x=a$ don't determine the limit.

3. 卡片 3

   问题

   When does direct substitution evaluate a limit?

   答案

   When the function is continuous at the target input. Then

   $$\lim_{x \to a} f(x)=f(a).$$

4. 卡片 4

   问题

   Three conditions for continuity at $x=a$?

   答案

   $f(a)$ exists, $\lim_{x \to a}f(x)$ exists, and

   $$\lim_{x \to a}f(x)=f(a).$$

5. 卡片 5

   问题

   Intermediate Value Theorem: hypotheses and conclusion?

   答案

   If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then some $c$ in $[a,b]$ satisfies $f(c)=N$. If $N$ is strictly between the endpoint values, $c$ lies in $(a,b)$.

6. 卡片 6

   问题

   When does a two-sided limit equal $L$?

   答案

   Exactly when both one-sided limits equal $L$:

   $$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.$$

   If the one-sided limits differ, the two-sided limit doesn't exist.

7. 卡片 7

   问题

   How do you read a finite limit from a graph?

   答案

   Follow the graph toward the target $x$-value from both sides. The common approached $y$-value is the limit, regardless of a hole or a differently placed filled point.

8. 卡片 8

   问题

   Limit law for a sum or difference?

   答案

   If both component limits exist,

   $$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$$

9. 卡片 9

   问题

   What makes a discontinuity removable?

   答案

   The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.

10. 卡片 10

    问题

    Squeeze Theorem: usable form?

    答案

    If $g(x)\le f(x)\le h(x)$ near $a$ and

    $$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$$

    then $\lim_{x\to a}f(x)=L$.

11. 卡片 11

    问题

    What does $\lim_{x\to a}f(x)=+\infty$ mean?

    答案

    $f(x)$ grows without bound above as $x$ approaches $a$. It describes unbounded behavior, not a finite limit value.

12. 卡片 12

    问题

    What must a table show for a left-hand limit?

    答案

    Inputs less than the target and moving toward it. For $\lim_{x\to a^-}f(x)$, use $x<a$ with $x$ getting closer to $a$.

13. 卡片 13

    问题

    Limit law for a product?

    答案

    If both limits exist,

    $$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$$

14. 卡片 14

    问题

    Graph signature of a jump discontinuity?

    答案

    The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.

15. 卡片 15

    问题

    Which theorem can guarantee a root on $[a,b]$?

    答案

    The Intermediate Value Theorem. If $f$ is continuous on $[a,b]$ and $0$ lies between $f(a)$ and $f(b)$, then $f(c)=0$ for some $c$ in the interval.

16. 卡片 16

    问题

    Horizontal asymptote from a limit at infinity?

    答案

    If $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$, then $y=L$ is a horizontal asymptote in that direction.

17. 卡片 17

    问题

    What does an open circle say about a graph's limit?

    答案

    Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.

18. 卡片 18

    问题

    Limit law for a quotient—and its condition?

    答案

    If both limits exist and the denominator limit is nonzero,

    $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}.$$

    The law doesn't apply when the denominator limit is $0$.

19. 卡片 19

    问题

    What does continuity on $[a,b]$ require at the endpoints?

    答案

    Continuity on $(a,b)$, right-continuity at $a$, and left-continuity at $b$:

    $$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$$

20. 卡片 20

    问题

    When is the Squeeze Theorem a natural choice?

    答案

    When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.

21. 卡片 21

    问题

    Vertical asymptote from one-sided behavior?

    答案

    If at least one one-sided limit at $x=a$ is $+\infty$ or $-\infty$, then $x=a$ is a vertical asymptote.

22. 卡片 22

    问题

    Limit at infinity of equal-degree rational functions?

    答案

    The ratio of the leading coefficients:

    $$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}.$$

    This assumes $b_n\ne0$.

23. 卡片 23

    问题

    When can a limit pass through a continuous outer function?

    答案

    If $\lim_{x\to a}g(x)=L$ and $f$ is continuous at $L$, then

    $$\lim_{x\to a}f(g(x))=f(L).$$

24. 卡片 24

    问题

    What makes a discontinuity infinite?

    答案

    The function becomes unbounded on at least one side of the input, usually producing a vertical asymptote.

25. 卡片 25

    问题

    Left limit $=2$ and right limit $=5$: two-sided limit?

    答案

    It doesn't exist. A finite two-sided limit requires the two one-sided limits to agree.

26. 卡片 26

    问题

    Standard trigonometric limit behind $\frac{\sin x}{x}$?

    答案

    With angles in radians,

    $$\lim_{x\to0}\frac{\sin x}{x}=1.$$

    Equivalent scaled forms follow by substitution.

27. 卡片 27

    问题

    Continuity of a composition?

    答案

    If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f\circ g$ is continuous at $a$.

28. 卡片 28

    问题

    Limit at infinity when a rational numerator has lower degree?

    答案

    $0$. If the numerator's degree is less than the denominator's, the denominator dominates as $x\to\pm\infty$.

29. 卡片 29

    问题

    What does the indeterminate form $0/0$ tell you?

    答案

    Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation; $0/0$ isn't the limit value.

30. 卡片 30

    问题

    When do opposite infinite one-sided limits give a two-sided limit?

    答案

    They don't. For example, $+\infty$ from the left and $-\infty$ from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.

31. 卡片 31

    问题

    How do you choose a parameter to make a piecewise function continuous?

    答案

    Set the relevant one-sided formulas and the function value equal at the join, then solve the resulting equation for the parameter.

32. 卡片 32

    问题

    Value of $\lim_{x\to0}\frac{1-\cos x}{x}$?

    答案

    $0$. Rationalizing gives a product involving $\sin x/x$ and a factor that approaches $0$.

33. 卡片 33

    问题

    Can $\lim_{x\to a}f(x)$ exist when $f(a)$ doesn't?

    答案

    Yes. A limit uses nearby values, so a hole at $x=a$ can coexist with a finite two-sided limit.

34. 卡片 34

    问题

    What graph behavior makes a finite limit fail even without a jump?

    答案

    Unbounded or persistent oscillating behavior near the input can prevent a finite limit. Check both sides rather than relying on the plotted point.

35. 卡片 35

    问题

    Average rate of change of $f$ on $[a,b]$?

    答案

    $$\frac{f(b)-f(a)}{b-a}$$

    It is the slope of the secant line through $(a,f(a))$ and $(b,f(b))$.

36. 卡片 36

    问题

    Derivative at $x=a$ using an increment $h$?

    答案

    $$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$

    The derivative exists only if this finite limit exists.

37. 卡片 37

    问题

    Tangent-line equation to $y=f(x)$ at $x=a$?

    答案

    $$y-f(a)=f'(a)(x-a)$$

    This requires $f'(a)$ to exist.

38. 卡片 38

    问题

    What does differentiability imply about continuity?

    答案

    If $f$ is differentiable at $a$, then $f$ is continuous at $a$. The converse is false: continuity alone doesn't guarantee differentiability.

39. 卡片 39

    问题

    Power rule for derivatives?

    答案

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

    Apply it where the original real-valued power function and its derivative are defined.

40. 卡片 40

    问题

    Units of $f'(x)$?

    答案

    Output units of $f$ per input unit of $x$. A derivative is a rate of change, so its units are a quotient.

41. 卡片 41

    问题

    Derivative at $x=a$ using $x\to a$?

    答案

    $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$

    This is equivalent to the $h$-form after setting $h=x-a$.

42. 卡片 42

    问题

    How does a graph of $f$ show the sign of $f'$?

    答案

    $f'(x)>0$ where $f$ rises as $x$ increases, and $f'(x)<0$ where $f$ falls. A horizontal tangent gives $f'(x)=0$ when the derivative exists.

43. 卡片 43

    问题

    Derivative of a constant?

    答案

    $$\frac{d}{dx}C=0$$

    A constant function has zero rate of change.

44. 卡片 44

    问题

    Derivative of $\sin x$?

    答案

    $$\frac{d}{dx}(\sin x)=\cos x$$

    The angle must be measured in radians for the standard formula.

45. 卡片 45

    问题

    Product rule?

    答案

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

    Differentiating each factor and multiplying the results is not the product rule.

46. 卡片 46

    问题

    How can nearby table values estimate $f'(a)$?

    答案

    Use a difference quotient with inputs close to $a$. A symmetric estimate is

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

    Smaller $h$ often helps, subject to the data's precision.

47. 卡片 47

    问题

    What does $f''(x)$ measure?

    答案

    The rate of change of $f'(x)$ with respect to $x$. Its units are the units of $f$ per square input unit.

48. 卡片 48

    问题

    Instantaneous rate of change of $f$ at $a$?

    答案

    $f'(a)$. It is the limit of average rates over intervals shrinking to $a$, and geometrically it is the tangent-line slope.

49. 卡片 49

    问题

    Derivative of a sum or difference?

    答案

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

50. 卡片 50

    问题

    Derivative of $\cos x$?

    答案

    $$\frac{d}{dx}(\cos x)=-\sin x$$

    The standard formula assumes radians.

51. 卡片 51

    问题

    Quotient rule?

    答案

    For $g(x)\ne0$,

    $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.$$

    The order in the numerator matters.

52. 卡片 52

    问题

    Common notations for the first derivative?

    答案

    $f'(x)$, $y'$, $\dfrac{dy}{dx}$, and $\dfrac{d}{dx}f(x)$. They describe the same derivative in different contexts.

53. 卡片 53

    问题

    Derivative of $e^x$?

    答案

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

54. 卡片 54

    问题

    What graph features can make $f$ nondifferentiable?

    答案

    A discontinuity, corner, cusp, vertical tangent, or other failure of the difference-quotient limit. Continuity is necessary but not sufficient.

55. 卡片 55

    问题

    Derivative of $\tan x$?

    答案

    Where $\tan x$ is defined,

    $$\frac{d}{dx}(\tan x)=\sec^2x.$$

    Angles are in radians.

56. 卡片 56

    问题

    What does the derivative function $f'$ assign to each input?

    答案

    The instantaneous rate of change—or tangent slope—of $f$ at that input, wherever the derivative exists.

57. 卡片 57

    问题

    Derivative of $\ln x$?

    答案

    For $x>0$,

    $$\frac{d}{dx}\ln x=\frac1x.$$

    More generally, $d(\ln|x|)/dx=1/x$ for $x\ne0$.

58. 卡片 58

    问题

    How does the power rule handle roots or negative powers?

    答案

    Rewrite them as $x^n$ and apply $nx^{n-1}$ on intervals where the real-valued expression is defined. Domain restrictions still matter.

59. 卡片 59

    问题

    Derivative of $\csc x$?

    答案

    Where $\csc x$ is defined,

    $$\frac{d}{dx}(\csc x)=-\csc x\cot x.$$

    Angles are in radians.

60. 卡片 60

    问题

    If $f'(x)>0$ throughout an interval, what does $f$ do there?

    答案

    $f$ is increasing on that interval.

61. 卡片 61

    问题

    Derivative of $a^x$ for a constant base?

    答案

    For $a>0$,

    $$\frac{d}{dx}a^x=a^x\ln a.$$

    When $a=1$, the derivative is $0$.

62. 卡片 62

    问题

    How can a graph estimate $f'(a)$?

    答案

    Estimate the slope of the tangent line at $x=a$, using two convenient points on a drawn tangent when available. A steeper tangent has a derivative with larger magnitude.

63. 卡片 63

    问题

    Derivative of $\sec x$?

    答案

    Where $\sec x$ is defined,

    $$\frac{d}{dx}(\sec x)=\sec x\tan x.$$

    Angles are in radians.

64. 卡片 64

    问题

    If $f''(x)>0$, how is $f'$ changing?

    答案

    $f'$ is increasing. This is also the derivative condition associated with $f$ being concave up.

65. 卡片 65

    问题

    Derivative of $\log_a x$?

    答案

    For $x>0$, $a>0$, and $a\ne1$,

    $$\frac{d}{dx}\log_a x=\frac{1}{x\ln a}.$$

66. 卡片 66

    问题

    Product rule from a table at $x=a$?

    答案

    For $h=fg$,

    $$h'(a)=f'(a)g(a)+f(a)g'(a).$$

    Use the four table entries at the same input.

67. 卡片 67

    问题

    Derivative of $\cot x$?

    答案

    Where $\cot x$ is defined,

    $$\frac{d}{dx}(\cot x)=-\csc^2x.$$

    Angles are in radians.

68. 卡片 68

    问题

    Why isn't $|x|$ differentiable at $x=0$?

    答案

    Its left-hand slope is $-1$ and right-hand slope is $1$. The one-sided derivative limits disagree, creating a corner.

69. 卡片 69

    问题

    Constant-multiple rule?

    答案

    For a constant $c$,

    $$\frac{d}{dx}[c f(x)]=c f'(x).$$

70. 卡片 70

    问题

    Quotient rule from a table at $x=a$?

    答案

    For $h=f/g$ with $g(a)\ne0$,

    $$h'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^2}.$$

71. 卡片 71

    问题

    Chain rule for $f(g(x))$?

    答案

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

    Differentiate the outer function at the unchanged inner input, then multiply by the inner derivative.

72. 卡片 72

    问题

    How do you identify inner and outer functions in a composite?

    答案

    Find the expression evaluated first—that is the inner function. The operation applied to its result is the outer function.

73. 卡片 73

    问题

    Core rule when differentiating an implicit equation in $x$ and $y$?

    答案

    Treat $y$ as a differentiable function of $x$. Every derivative of an expression involving $y$ gains a factor of $dy/dx$ by the chain rule.

74. 卡片 74

    问题

    Derivative of an inverse function at $x$?

    答案

    If $f$ is differentiable and one-to-one near $f^{-1}(x)$, with $f'(f^{-1}(x))\ne0$,

    $$[f^{-1}]'(x)=\frac{1}{f'(f^{-1}(x))}.$$

75. 卡片 75

    问题

    Derivative of $\arcsin x$?

    答案

    For $|x|<1$,

    $$\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}.$$

76. 卡片 76

    问题

    Notation for the third derivative of $f$?

    答案

    $f'''(x)$ or $\dfrac{d^3f}{dx^3}$. The exponent on $d$ indicates derivative order; it is not an ordinary power.

77. 卡片 77

    问题

    If $h(x)=f(g(x))$, what table entries give $h'(a)$?

    答案

    $$h'(a)=f'(g(a))g'(a)$$

    Use $g(a)$ to find the input needed for the table entry of $f'$.

78. 卡片 78

    问题

    For $x^2+y^2=r^2$, what is $dy/dx$?

    答案

    Where $y\ne0$,

    $$\frac{dy}{dx}=-\frac{x}{y}.$$

    Differentiate to get $2x+2y(dy/dx)=0$.

79. 卡片 79

    问题

    If $f(a)=b$, how do you find $[f^{-1}]'(b)$?

    答案

    Provided $f'(a)\ne0$,

    $$[f^{-1}]'(b)=\frac{1}{f'(a)}.$$

    The inverse swaps the input-output pair $(a,b)$.

80. 卡片 80

    问题

    Derivative of $\arctan x$?

    答案

    For every real $x$,

    $$\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}.$$

81. 卡片 81

    问题

    Derivative of $e^{g(x)}$?

    答案

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

    The extra factor is the chain rule.

82. 卡片 82

    问题

    Slope of a tangent to an implicit curve $F(x,y)=0$?

    答案

    Differentiate the relation with respect to $x$, solve for $dy/dx$, then substitute the point. Confirm the point lies on the curve and the resulting slope is defined.

83. 卡片 83

    问题

    Why must $f'(a)\ne0$ to use $[f^{-1}]'(f(a))=1/f'(a)$?

    答案

    Because the reciprocal slope would be undefined when $f'(a)=0$. The inverse may have a vertical tangent or fail to be differentiable there.

84. 卡片 84

    问题

    Derivative of $\arccos x$?

    答案

    For $|x|<1$,

    $$\frac{d}{dx}(\arccos x)=-\frac{1}{\sqrt{1-x^2}}.$$

85. 卡片 85

    问题

    How do you find $d^2y/dx^2$ for an implicit relation?

    答案

    Differentiate the first-derivative equation again with respect to $x$, include $dy/dx$ factors, then substitute the known expression for $dy/dx$ if needed.

86. 卡片 86

    问题

    Derivative of $\ln(g(x))$?

    答案

    Where $g(x)>0$,

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

    For $\ln|g(x)|$, the same derivative holds where $g(x)\ne0$.

87. 卡片 87

    问题

    Derivative of $y^n$ when $y=y(x)$?

    答案

    $$\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}$$

    The $dy/dx$ factor comes from the chain rule.

88. 卡片 88

    问题

    How are tangent slopes of inverse graphs related?

    答案

    At reflected points $(a,b)$ and $(b,a)$, the slopes are reciprocals when both are defined and nonzero.

89. 卡片 89

    问题

    Derivative of $\arcsin(g(x))$?

    答案

    $$\frac{d}{dx}\arcsin(g(x))=\frac{g'(x)}{\sqrt{1-[g(x)]^2}},\qquad |g(x)|<1.$$

90. 卡片 90

    问题

    Derivative of $\sin(g(x))$?

    答案

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

91. 卡片 91

    问题

    Horizontal tangent on an implicit curve: derivative condition?

    答案

    $dy/dx=0$ at a valid point, with the derivative defined there. In a fraction for $dy/dx$, the numerator is typically zero while the denominator is nonzero.

92. 卡片 92

    问题

    How do you differentiate $f^{-1}(x)$ without solving for the inverse?

    答案

    Use the reciprocal derivative formula and the matching original input: find $a$ with $f(a)=x$, then compute $1/f'(a)$.

93. 卡片 93

    问题

    Difference between $f''(x)$ and $[f'(x)]^2$?

    答案

    $f''(x)$ is the derivative of $f'(x)$. The expression $[f'(x)]^2$ is the square of the first derivative; they are generally unrelated.

94. 卡片 94

    问题

    Derivative of $[g(x)]^n$?

    答案

    $$\frac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x)$$

    This combines the power rule with the chain rule.

95. 卡片 95

    问题

    Vertical tangent on an implicit curve: derivative clue?

    答案

    $dy/dx$ becomes unbounded or undefined while the curve still has a vertical tangent. In a simplified derivative fraction, the denominator is often zero and the numerator nonzero.

96. 卡片 96

    问题

    Table formula for an inverse derivative at $x=b$?

    答案

    Find $a$ in the table with $f(a)=b$. If $f'(a)\ne0$, then

    $$[f^{-1}]'(b)=\frac{1}{f'(a)}.$$

97. 卡片 97

    问题

    Derivative of $\arctan(g(x))$?

    答案

    $$\frac{d}{dx}\arctan(g(x))=\frac{g'(x)}{1+[g(x)]^2}$$

98. 卡片 98

    问题

    How do product and chain rules combine in $f(x)g(h(x))$?

    答案

    $$\frac{d}{dx}[f(x)g(h(x))]=f'(x)g(h(x))+f(x)g'(h(x))h'(x).$$

    Use the product rule outside and the chain rule on the composite factor.

99. 卡片 99

    问题

    Why can $d^2y/dx^2$ depend on both $x$ and $y$?

    答案

    An implicit relation may never solve explicitly for $y$. After differentiating twice and replacing $dy/dx$, the result can naturally remain a function of both coordinates.

100. 卡片 100

     问题

     A quantity $y$ changes through $u$, which changes with $x$. How are the rates connected?

     答案

     When the functions are differentiable, the chain rule gives

     $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$

101. 卡片 101

     问题

     What local property lets a function have an inverse derivative?

     答案

     The function must be one-to-one near the point, differentiable there, and have a nonzero derivative. Then the inverse is locally differentiable with reciprocal slope.

102. 卡片 102

     问题

     Derivative of $a^{g(x)}$?

     答案

     For $a>0$,

     $$\frac{d}{dx}a^{g(x)}=a^{g(x)}\ln(a)\,g'(x).$$

103. 卡片 103

     问题

     How should $Q'(t)$ be interpreted in context?

     答案

     At time $t$, the quantity $Q$ changes at an instantaneous rate of $Q'(t)$ output units per unit of time. Include the quantity, time, direction or sign, and units.

104. 卡片 104

     问题

     Position, velocity, and acceleration relationships?

     答案

     For position $s(t)$,

     $$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$

105. 卡片 105

     问题

     Central idea of a related-rates problem?

     答案

     Write an equation connecting the changing quantities, differentiate it with respect to time, then use the data for the specified instant.

106. 卡片 106

     问题

     Linearization of $f$ near $x=a$?

     答案

     $$L(x)=f(a)+f'(a)(x-a)$$

     For $x$ close to $a$, $f(x)\approx L(x)$.

107. 卡片 107

     问题

     L’Hospital’s Rule: basic conditions?

     答案

     For a quotient with differentiable numerator and denominator near the target, denominator derivative nonzero nearby, and indeterminate form $0/0$ or $\infty/\infty$, the original limit equals the limit of the derivative quotient when that latter limit exists or is infinite.

108. 卡片 108

     问题

     If distance is in meters and time in seconds, units of acceleration?

     答案

     Meters per second squared, $\text{m}/\text{s}^2$. Acceleration is the rate of change of velocity with respect to time.

109. 卡片 109

     问题

     Speed in terms of velocity?

     答案

     $$\text{speed}=|v(t)|$$

     Velocity includes direction; speed is nonnegative magnitude.

110. 卡片 110

     问题

     Why do $x$ and $y$ gain $dx/dt$ and $dy/dt$ in related rates?

     答案

     They are functions of time. Differentiating an expression such as $x^2$ with respect to $t$ gives $2x(dx/dt)$ by the chain rule.

111. 卡片 111

     问题

     Differential approximation connecting $dx$ and $dy$?

     答案

     $$dy=f'(x)\,dx$$

     For a small change $\Delta x$, the actual change satisfies $\Delta y\approx f'(x)\Delta x$.

112. 卡片 112

     问题

     Which indeterminate forms directly allow L’Hospital’s Rule?

     答案

     $0/0$ and $\infty/\infty$. Other indeterminate forms must first be rewritten as an appropriate quotient.

113. 卡片 113

     问题

     How do you estimate an instantaneous contextual rate from a table?

     答案

     Use a nearby difference quotient, preferably with times on both sides of the target. Report the result with output units per time unit.

114. 卡片 114

     问题

     What does positive acceleration say about velocity?

     答案

     Velocity is increasing. It does not by itself say the object moves forward or speeds up; the sign of velocity also matters.

115. 卡片 115

     问题

     Related rates: when should numerical values be substituted?

     答案

     After differentiating the general relationship with respect to time. Substituting fixed values too early can erase the rates being sought.

116. 卡片 116

     问题

     How does concavity predict linearization error?

     答案

     Near the tangency point, a concave-up graph generally lies above its tangent line, so the linearization underestimates. A concave-down graph generally lies below, so it overestimates.

117. 卡片 117

     问题

     Why can't L’Hospital’s Rule be applied directly to a product?

     答案

     The rule applies to quotients with $0/0$ or $\infty/\infty$ form. Rewrite an indeterminate product such as $0\cdot\infty$ as a quotient first.

118. 卡片 118

     问题

     When is a particle moving in the positive direction?

     答案

     When $v(t)>0$. Position then increases as time increases.

119. 卡片 119

     问题

     How can velocity show a change of direction?

     答案

     Velocity changes sign. A time with $v(t)=0$ is only a candidate; confirm the sign differs on the two sides.

120. 卡片 120

     问题

     First equation to seek in a geometric related-rates problem?

     答案

     A geometric constraint involving the changing quantities, such as a Pythagorean, area, volume, or similar-triangle relation.

121. 卡片 121

     问题

     Tangent-line approximation of $f(a+\Delta x)$?

     答案

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

     It is most reliable for small $|\Delta x|$ where the function is well approximated by its tangent.

122. 卡片 122

     问题

     When may L’Hospital’s Rule be applied more than once?

     答案

     When the derivative quotient still has $0/0$ or $\infty/\infty$ form and the rule's conditions continue to hold.

123. 卡片 123

     问题

     What must a contextual derivative sentence include?

     答案

     The changing quantity, the instant or input, the rate's sign or direction when relevant, and correct output-per-input units.

124. 卡片 124

     问题

     Velocity negative and acceleration positive: what happens?

     答案

     The particle moves in the negative direction while its velocity increases toward zero. Its speed decreases as long as velocity and acceleration have opposite signs.

125. 卡片 125

     问题

     How should a negative related rate be interpreted?

     答案

     The measured quantity is decreasing at that instant. Keep the sign and state the units; don't silently report only its magnitude.

126. 卡片 126

     问题

     When is local linearity a sound approximation tool?

     答案

     When $f$ is differentiable near the base point and the target input is close enough that curvature has limited effect.

127. 卡片 127

     问题

     Can L’Hospital’s Rule handle a one-sided limit?

     答案

     Yes, if the required indeterminate form and differentiability conditions hold on that side and the derivative-quotient limit exists or is infinite.

128. 卡片 128

     问题

     When is speed increasing?

     答案

     When velocity and acceleration have the same sign, so $v(t)a(t)>0$.

129. 卡片 129

     问题

     Volume changes with time: notation for its rate?

     答案

     $dV/dt$. Its units are cubic length units per time unit.

130. 卡片 130

     问题

     Why are similar triangles useful in related rates?

     答案

     They can reduce several changing lengths to one constraint before differentiation, keeping the rate equation tied to the geometry.

131. 卡片 131

     问题

     Meaning of $dy=f'(x)dx$ in approximation?

     答案

     $dy$ is the tangent-line estimate of the actual output change $\Delta y$ caused by an input change $dx$.

132. 卡片 132

     问题

     What conclusion does L’Hospital’s Rule permit?

     答案

     Under its conditions,

     $$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}.$$

     It does not say the two quotients are equal as functions.

133. 卡片 133

     问题

     When is speed decreasing?

     答案

     When velocity and acceleration have opposite signs, so $v(t)a(t)<0$.

134. 卡片 134

     问题

     What does a tangent slope read from a contextual graph represent?

     答案

     The instantaneous rate of the vertical quantity with respect to the horizontal quantity, with vertical units per horizontal unit.

135. 卡片 135

     问题

     Does $v(t)=0$ guarantee a particle changes direction?

     答案

     No. It is momentarily at rest, but direction changes only if velocity changes sign across that time.

136. 卡片 136

     问题

     How do you translate “$Q$ increases by 3 units per minute” into derivative notation?

     答案

     $dQ/dt=3$ in the stated time interval or at the stated instant. “Decreases by 3” would give $dQ/dt=-3$.

137. 卡片 137

     问题

     Extreme Value Theorem: hypothesis and conclusion?

     答案

     If $f$ is continuous on the closed interval $[a,b]$, then $f$ has at least one absolute minimum value and at least one absolute maximum value on $[a,b]$.

138. 卡片 138

     问题

     What is a critical number of $f$?

     答案

     A number $c$ in the domain of $f$ where $f'(c)=0$ or $f'(c)$ doesn't exist.

139. 卡片 139

     问题

     First derivative test for a local maximum?

     答案

     $f'$ changes from positive to negative at the critical point, so $f$ changes from increasing to decreasing.

140. 卡片 140

     问题

     Second-derivative sign for concave up?

     答案

     If $f''(x)>0$ on an interval, then $f$ is concave up there and $f'$ is increasing.

141. 卡片 141

     问题

     If the graph of $f'$ is above the $x$-axis, what does $f$ do?

     答案

     $f$ is increasing because $f'(x)>0$.

142. 卡片 142

     问题

     First step in an optimization model?

     答案

     Define the objective quantity and the constraint, including the feasible domain. Rewrite the objective as a function of one variable before differentiating.

143. 卡片 143

     问题

     Mean Value Theorem: hypotheses and conclusion?

     答案

     If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c$ in $(a,b)$ satisfies

     $$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

144. 卡片 144

     问题

     Candidates test for absolute extrema on $[a,b]$?

     答案

     Assuming $f$ is continuous on $[a,b]$, evaluate $f$ at every critical number in $(a,b)$ and at both endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.

     

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145. 卡片 145

     问题

     First derivative test for a local minimum?

     答案

     $f'$ changes from negative to positive at the critical point, so $f$ changes from decreasing to increasing.

146. 卡片 146

     问题

     What must happen at an inflection point?

     答案

     The graph's concavity changes. A zero or undefined value of $f''$ is only a candidate; verify a concavity change.

147. 卡片 147

     问题

     If $f'$ has a local maximum, what can that say about $f$?

     答案

     $f''$ may change from positive to negative there, so $f$ may change from concave up to concave down. Confirm the sign change rather than relying only on the point.

148. 卡片 148

     问题

     How do you confirm an optimization answer is absolute?

     答案

     Compare objective values at all feasible critical points and relevant endpoints, or use a justified monotonicity argument over the feasible domain.

149. 卡片 149

     问题

     Rolle’s Theorem: hypotheses and conclusion?

     答案

     If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then some $c$ in $(a,b)$ satisfies $f'(c)=0$.

150. 卡片 150

     问题

     Difference between absolute and relative extrema?

     答案

     An absolute maximum is at least every other value on the stated domain, and an absolute minimum is at most every other value. A relative extremum makes the corresponding comparison only with nearby values.

151. 卡片 151

     问题

     If $f$ is continuous at a critical number $c$ and $f'$ is positive on both sides, is there a local extremum?

     答案

     No. The function is increasing through $c$, so it has no local extremum there.

152. 卡片 152

     问题

     Second derivative test for a local minimum?

     答案

     If $f'(c)=0$ and $f''(c)>0$, then $f$ has a local minimum at $c$.

153. 卡片 153

     问题

     Zeros of $f'$ correspond to what features of $f$?

     答案

     Horizontal tangents where $f'$ exists. They are critical numbers and possible extrema, but each needs further sign or value analysis.

154. 卡片 154

     问题

     Implicit relation: how can $dy/dx$ reveal local behavior?

     答案

     Its sign shows whether the relation's local branch rises or falls as $x$ increases; zeros and undefined values mark possible horizontal or vertical tangents.

155. 卡片 155

     问题

     Which theorem links an average slope to an instantaneous slope?

     答案

     The Mean Value Theorem, provided the function is continuous on the closed interval and differentiable on its interior.

156. 卡片 156

     问题

     How can an implicit derivative locate a horizontal tangent?

     答案

     At a valid point on the relation, find where the simplified numerator of $dy/dx$ is zero while its denominator is nonzero, then confirm the local branch has the expected behavior.

157. 卡片 157

     问题

     Derivative-sign chart: where is $f$ decreasing?

     答案

     On intervals where $f'(x)<0$.

158. 卡片 158

     问题

     Second derivative test for a local maximum?

     答案

     If $f'(c)=0$ and $f''(c)<0$, then $f$ has a local maximum at $c$.

159. 卡片 159

     问题

     If $f'$ is increasing, what is the concavity of $f$?

     答案

     $f$ is concave up on that interval, assuming the relevant derivatives exist.

160. 卡片 160

     问题

     Why must an optimization domain be stated?

     答案

     The context can exclude algebraic candidates and add boundary cases. Absolute extrema depend on the feasible inputs, not just on where the derivative is zero.

161. 卡片 161

     问题

     Which theorem guarantees absolute extrema, not where they occur?

     答案

     The Extreme Value Theorem. Continuity on a closed interval guarantees at least one maximum value and one minimum value, but finding them requires analysis.

162. 卡片 162

     问题

     Can $f'(c)$ fail to exist at a local extremum?

     答案

     Yes. Corners, cusps, or other nondifferentiable domain points can be local extrema, which is why critical numbers include points where $f'$ is undefined.

163. 卡片 163

     问题

     For a function continuous at $c$, what same-sign pattern in $f'$ rules out a local extremum there?

     答案

     If $f'$ is positive on both sides of $c$, or negative on both sides, then $f$ keeps the same monotonic direction through $c$ and has no local extremum there.

164. 卡片 164

     问题

     If $f'(c)=0$ and $f''(c)=0$, what does the second derivative test conclude?

     答案

     Nothing. The test is inconclusive; use a first-derivative sign change, higher analysis, or direct comparison.

165. 卡片 165

     问题

     If the graph of $f'$ crosses from negative to positive, what feature does $f$ have?

     答案

     A local minimum at the crossing input, provided the input is in the domain of $f$.

166. 卡片 166

     问题

     How can an implicit derivative locate a vertical tangent?

     答案

     Find valid points where the simplified $dy/dx$ denominator is zero and numerator is nonzero, then confirm the curve has the expected local behavior.

167. 卡片 167

     问题

     Which extra condition turns the Mean Value Theorem into Rolle’s Theorem?

     答案

     $f(a)=f(b)$. The average slope becomes zero, so the guaranteed instantaneous slope is also zero.

168. 卡片 168

     问题

     Why are endpoints included in the candidates test?

     答案

     An absolute maximum or minimum on a closed interval can occur at an endpoint even though no two-sided derivative test applies there.

169. 卡片 169

     问题

     If $f'(x)=0$ throughout an interval, what is $f$ there?

     答案

     $f$ is constant on that interval, assuming the usual continuity and differentiability conditions that let the Mean Value Theorem apply.

170. 卡片 170

     问题

     Second-derivative sign for concave down?

     答案

     If $f''(x)<0$ on an interval, then $f$ is concave down there and $f'$ is decreasing.

171. 卡片 171

     问题

     Graph of $f'$ has a local minimum: possible effect on $f$?

     答案

     $f''$ may change from negative to positive, so $f$ may change from concave down to concave up. Verify the sign change.

172. 卡片 172

     问题

     What should the final line of an optimization solution state?

     答案

     The requested maximum or minimum quantity in context, with units and the feasible input that produces it when relevant.

173. 卡片 173

     问题

     Can Rolle’s Theorem be used if $f$ has a corner inside $(a,b)$?

     答案

     No. A corner breaks differentiability on the open interval, so the theorem's hypotheses are not satisfied.

174. 卡片 174

     问题

     How do $f''$ zeros help analyze a graph?

     答案

     They are candidates for changes in concavity. Test the sign of $f''$ on both sides; a zero alone doesn't guarantee an inflection point.

175. 卡片 175

     问题

     What does the accumulation function $F(x)=\int_a^x f(t)\,dt$ measure?

     答案

     The signed net accumulation of $f$ from $a$ to $x$. Contributions above the axis are positive; contributions below it are negative.

176. 卡片 176

     问题

     Left Riemann sum on equal subintervals?

     答案

     If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$, then

     $$L_n=\sum_{i=1}^{n} f(x_{i-1})\Delta x.$$

177. 卡片 177

     问题

     What does $\int_a^b f(x)\,dx$ represent geometrically?

     答案

     Signed area between the graph and the $x$-axis from $a$ to $b$, when $f$ is integrable. Regions below the axis subtract from regions above it.

178. 卡片 178

     问题

     Fundamental Theorem of Calculus: evaluate a definite integral?

     答案

     If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then

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

179. 卡片 179

     问题

     Derivative of $F(x)=\int_a^x f(t)\,dt$?

     答案

     If $f$ is continuous, then

     $$F'(x)=f(x).$$

     This connects accumulation with instantaneous rate.

180. 卡片 180

     问题

     Why do all antiderivatives of the same function differ by a constant?

     答案

     If $F'=f$ and $G'=f$ on an interval, then $(F-G)'=0$, so $F-G=C$ on that interval.

181. 卡片 181

     问题

     Right Riemann sum on equal subintervals?

     答案

     If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$, then

     $$R_n=\sum_{i=1}^{n} f(x_i)\Delta x.$$

182. 卡片 182

     问题

     How does reversing integral bounds change the value?

     答案

     It changes the sign:

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

183. 卡片 183

     问题

     Net Change Theorem?

     答案

     If $Q'(t)$ is the rate of change of a quantity, then

     $$Q(b)-Q(a)=\int_a^b Q'(t)\,dt.$$

184. 卡片 184

     问题

     Derivative of $\int_a^{g(x)} f(t)\,dt$?

     答案

     If $f$ is continuous on an interval containing $a$ and the range of $g$, and $g$ is differentiable, then

     $$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x).$$

185. 卡片 185

     问题

     Power rule for antiderivatives?

     答案

     For $n\ne-1$,

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

186. 卡片 186

     问题

     Midpoint Riemann sum on equal subintervals?

     答案

     With midpoint $m_i=(x_{i-1}+x_i)/2$,

     $$M_n=\sum_{i=1}^{n} f(m_i)\Delta x.$$

187. 卡片 187

     问题

     How can an integral be split at an interior point $c$?

     答案

     For $a\le c\le b$,

     $$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.$$

188. 卡片 188

     问题

     Derivative of $G(x)=\int_x^b f(t)\,dt$?

     答案

     If $f$ is continuous, then

     $$G'(x)=-f(x).$$

     The variable lower bound produces the negative sign.

189. 卡片 189

     问题

     Antiderivative of $1/x$?

     答案

     On any interval not crossing zero,

     $$\int \frac1x\,dx=\ln|x|+C.$$

190. 卡片 190

     问题

     Trapezoidal approximation on equal subintervals?

     答案

     $$T_n=\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right].$$

191. 卡片 191

     问题

     How do geometric regions help evaluate a definite integral?

     答案

     Replace each familiar region with its geometric area, then attach a positive sign above the axis and a negative sign below it.

192. 卡片 192

     问题

     Basic antiderivatives of sine and cosine?

     答案

     $$\int \sin x\,dx=-\cos x+C,\qquad \int \cos x\,dx=\sin x+C.$$

193. 卡片 193

     问题

     Definite integral as a limit of Riemann sums?

     答案

     For an integrable function and sample points $x_i^*$,

     $$\int_a^b f(x)\,dx=\lim_{\max\Delta x_i\to0}\sum_i f(x_i^*)\Delta x_i.$$

194. 卡片 194

     问题

     Constant-multiple rule for integrals?

     答案

     For a constant $k$,

     $$\int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx.$$

     The analogous rule holds for indefinite integrals.

195. 卡片 195

     问题

     What pattern suggests $u$-substitution?

     答案

     A composite expression paired with its derivative, such as $f(g(x))g'(x)$. Set $u=g(x)$ so $du=g'(x)\,dx$.

196. 卡片 196

     问题

     How should bounds change in a definite $u$-substitution?

     答案

     If $u=g(x)$, replace the $x$-bounds $a,b$ with $u$-bounds $g(a),g(b)$. Then finish entirely in $u$, or return to $x$ before applying the original bounds.

197. 卡片 197

     问题

     What condition makes $F(x)=\int_a^x f(t)\,dt$ differentiable with $F'(x)=f(x)$?

     答案

     Continuity of $f$ on an interval containing $a$ and $x$ is the standard AP Calculus condition.

198. 卡片 198

     问题

     Sum-and-difference rule for definite integrals?

     答案

     For integrable $f$ and $g$,

     $$\int_a^b [f(x)\pm g(x)]\,dx=\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx.$$

199. 卡片 199

     问题

     Basic antiderivative of $e^x$?

     答案

     $$\int e^x\,dx=e^x+C.$$

200. 卡片 200

     问题

     Basic antiderivatives of $\sec^2x$ and $\csc^2x$?

     答案

     $$\int\sec^2x\,dx=\tan x+C,\qquad \int\csc^2x\,dx=-\cot x+C.$$

201. 卡片 201

     问题

     For an increasing integrable function, how do left and right sums compare with the integral?

     答案

     On the same partition, the left sum is an underestimate and the right sum is an overestimate. Reverse those conclusions when the function is decreasing.

202. 卡片 202

     问题

     How does concavity predict trapezoidal and midpoint error?

     答案

     For a concave-up function, trapezoidal sums overestimate and midpoint sums underestimate. For a concave-down function, the directions reverse.

203. 卡片 203

     问题

     Why might polynomial long division help before integrating a rational function?

     答案

     When the numerator's degree is at least the denominator's, division rewrites the expression as a polynomial plus a simpler proper rational function.

204. 卡片 204

     问题

     What denominator pattern suggests an arctangent antiderivative?

     答案

     After completing the square and scaling, a form like

     $$\int\frac{1}{1+u^2}\,du=\arctan u+C.$$

205. 卡片 205

     问题

     Basic antiderivatives of $\sec x\tan x$ and $\csc x\cot x$?

     答案

     $$\int\sec x\tan x\,dx=\sec x+C,\qquad \int\csc x\cot x\,dx=-\csc x+C.$$

206. 卡片 206

     问题

     How does an initial condition determine an antiderivative?

     答案

     First find the family $F(x)+C$. Substitute the given point, such as $y(a)=b$, and solve for $C$.

207. 卡片 207

     问题

     Should a definite-integral answer include $+C$?

     答案

     No. A definite integral is one number or quantity. The constant of integration belongs to an indefinite integral or general antiderivative.

208. 卡片 208

     问题

     Why does an indefinite integral include $+C$?

     答案

     Differentiation loses additive constants. The $+C$ represents every function with the stated derivative.

209. 卡片 209

     问题

     When is $F(x)=\int_a^x f(t)\,dt$ increasing?

     答案

     Where $F'(x)=f(x)>0$. It is decreasing where $f(x)<0$.

210. 卡片 210

     问题

     How is the concavity of $F(x)=\int_a^x f(t)\,dt$ determined?

     答案

     Since $F'(x)=f(x)$, $F$ is concave up where $f$ is increasing and concave down where $f$ is decreasing, assuming the needed derivatives exist.

211. 卡片 211

     问题

     How is $\int_a^b |f(x)|\,dx$ interpreted?

     答案

     As total geometric area between $f$ and the $x$-axis. Split at zeros of $f$ and make every regional contribution nonnegative.

212. 卡片 212

     问题

     What constant-factor check completes many $u$-substitutions?

     答案

     Compare $du=g'(x)\,dx$ with the remaining factor. Multiply or divide by a constant so the integrand contains exactly the needed differential.

213. 卡片 213

     问题

     How do you recover $\Delta x$ from a sigma-form Riemann sum on $[a,b]$?

     答案

     Identify the factor multiplying each function value. For $n$ equal subintervals, it should be

     $$\Delta x=\frac{b-a}{n}.$$

214. 卡片 214

     问题

     Riemann sum for unequal subinterval widths?

     答案

     If $x_{i-1}$ to $x_i$ has width $\Delta x_i$ and sample point $x_i^*$, use

     $$\sum_i f(x_i^*)\Delta x_i.$$

215. 卡片 215

     问题

     Does continuity guarantee integrability on a closed interval?

     答案

     Yes. A function continuous on $[a,b]$ is integrable there. Some discontinuous functions are also integrable, so continuity is sufficient, not necessary.

216. 卡片 216

     问题

     Antiderivative pattern for $g'(x)/g(x)$?

     答案

     Where $g(x)\ne0$,

     $$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C.$$

217. 卡片 217

     问题

     What algebraic rewrites often reveal a basic antiderivative?

     答案

     Split sums, factor out constants, simplify rational powers, and rewrite radicals as exponents. Then apply known antiderivative rules term by term.

218. 卡片 218

     问题

     What units does $\int_a^b r(t)\,dt$ have?

     答案

     Rate units multiplied by input units. For example, liters per minute integrated over minutes gives liters.

219. 卡片 219

     问题

     What is a differential equation?

     答案

     An equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true on an interval.

220. 卡片 220

     问题

     How does a verbal rate statement become a differential equation?

     答案

     Name the changing quantity and its independent variable, translate its rate as a derivative, and express that derivative using the stated relationship. For example, “rate proportional to $y$” becomes $dy/dt=ky$.

221. 卡片 221

     问题

     How do you verify that $y=f(x)$ solves a differential equation?

     答案

     Differentiate $f$ as needed, substitute $y$ and its derivatives into the equation, and confirm both sides agree on the claimed interval.

222. 卡片 222

     问题

     General solution versus particular solution?

     答案

     A general solution contains an arbitrary constant and represents a family of solution curves. A particular solution uses an initial condition to determine that constant.

223. 卡片 223

     问题

     What does one segment in a slope field show?

     答案

     At $(x,y)$, its slope equals the value of $dy/dx$ given by the differential equation at that point.

224. 卡片 224

     问题

     What units does the constant $k$ have in $dy/dt=ky$?

     答案

     Inverse time units, such as per hour. That makes the exponent $kt$ dimensionless.

225. 卡片 225

     问题

     How do you verify a proposed solution to an initial value problem?

     答案

     Check both requirements: the function must satisfy the differential equation throughout the stated interval, and it must satisfy the given initial condition.

226. 卡片 226

     问题

     What makes a first-order differential equation separable?

     答案

     It can be rearranged so all $y$ factors accompany $dy$ and all $x$ factors accompany $dx$, such as

     $$g(y)\,dy=f(x)\,dx.$$

227. 卡片 227

     问题

     What is an initial value problem?

     答案

     A differential equation paired with a value such as $y(x_0)=y_0$. It asks for a solution through $(x_0,y_0)$; depending on the equation, there may be zero, one, or multiple such solutions.

228. 卡片 228

     问题

     What is an isocline in a slope field?

     答案

     A curve along which the differential equation gives the same slope. For $dy/dx=F(x,y)$, an isocline satisfies $F(x,y)=k$ for a constant $k$.

229. 卡片 229

     问题

     How do you draw a slope-field segment at $(x_0,y_0)$?

     答案

     Evaluate the differential equation at $(x_0,y_0)$ to find the slope, then draw a short segment through that point with the resulting slope.

230. 卡片 230

     问题

     General solution of $dy/dt=ky$?

     答案

     $$y=Ce^{kt}$$

     for a constant $C$. The zero solution is included by $C=0$.

231. 卡片 231

     问题

     Core method for solving a separable differential equation?

     答案

     Separate the variables, integrate both sides, include a constant of integration, and solve for $y$ when practical. Then apply any initial condition.

232. 卡片 232

     问题

     How should a solution curve follow a slope field?

     答案

     It must pass through its initial point and remain tangent to the short field segments it crosses. It should not be drawn by connecting segment endpoints.

233. 卡片 233

     问题

     If $dy/dx=f(x)$, what pattern appears in its slope field?

     答案

     Every point with the same $x$-coordinate has the same slope, so the segments repeat vertically in columns.

234. 卡片 234

     问题

     Why is one integration constant enough after integrating both sides?

     答案

     Two constants can be combined: $C_2-C_1$ is still an arbitrary constant. Write a single $C$.

235. 卡片 235

     问题

     Solution of $dy/dt=ky$ with $y(0)=y_0$?

     答案

     $$y(t)=y_0e^{kt}.$$

236. 卡片 236

     问题

     Can one differential equation have infinitely many solutions?

     答案

     Yes. A differential equation commonly describes a family of solution functions. An initial condition may select one particular solution when the relevant conditions support uniqueness.

237. 卡片 237

     问题

     What is an equilibrium solution of $dy/dx=F(y)$?

     答案

     A constant solution $y=c$ where $F(c)=0$. In the slope field, the segments along that horizontal line have zero slope.

238. 卡片 238

     问题

     For continuous $f$, particular solution of $dy/dx=f(x)$ with $y(a)=b$?

     答案

     $$y(x)=b+\int_a^x f(t)\,dt.$$

     The Fundamental Theorem of Calculus gives $y'=f(x)$, and $y(a)=b$.

239. 卡片 239

     问题

     What can be lost when dividing to separate variables?

     答案

     Equilibrium solutions that make the divided factor zero. Check those constant solutions directly in the original differential equation.

240. 卡片 240

     问题

     In $dy/dt=ky$, what do the signs of $k$ mean?

     答案

     For a positive quantity, $k>0$ produces exponential growth, $k<0$ produces exponential decay, and $k=0$ keeps the quantity constant.

241. 卡片 241

     问题

     How can a table of slopes identify the matching differential equation?

     答案

     Test representative $(x,y)$ entries in each candidate equation. Match zeros, signs, and repeated slope patterns before checking exact numerical values.

242. 卡片 242

     问题

     How does the sign of $dy/dx$ describe a solution?

     答案

     The solution is increasing where $dy/dx>0$ and decreasing where $dy/dx<0$. At one point, $dy/dx=0$ gives a horizontal tangent; a constant $y=c$ is an equilibrium only when the derivative equation gives zero all along that level.

243. 卡片 243

     问题

     How can a differential equation determine a solution's concavity?

     答案

     Differentiate the equation with respect to the independent variable to obtain $y''$, using the chain rule for any $y$-dependence. Then use the sign of $y''$ along the solution.

244. 卡片 244

     问题

     Why must a differential-equation solution include an interval or domain?

     答案

     Algebraic steps may introduce restrictions, and the formula must remain differentiable and satisfy the original equation throughout the interval containing the initial point.

245. 卡片 245

     问题

     Doubling time for exponential growth $y=y_0e^{kt}$?

     答案

     For $k>0$,

     $$T_d=\frac{\ln2}{k}.$$

     It is independent of the initial amount.

246. 卡片 246

     问题

     How can a slope field reveal whether $dy/dx$ depends only on $y$?

     答案

     Slopes repeat horizontally: every point at the same height has the same segment slope.

247. 卡片 247

     问题

     How is an initial condition used after separation?

     答案

     Substitute the given $x$ and $y$ values into the integrated relationship to solve for the arbitrary constant, then state the resulting particular solution.

248. 卡片 248

     问题

     How do units check a model $dy/dt=F(t,y)$?

     答案

     The right side must have the same units as $dy/dt$: units of $y$ per unit of $t$. A mismatch signals an incorrect translation or parameter unit.

249. 卡片 249

     问题

     Why should a separated solution be checked in the original equation?

     答案

     Division, logarithms, square roots, or algebraic rearrangement can lose or introduce branches. Direct substitution confirms the formula and its stated domain.

250. 卡片 250

     问题

     Half-life for exponential decay $y=y_0e^{kt}$?

     答案

     For $k<0$,

     $$T_{1/2}=\frac{\ln(1/2)}{k}=\frac{\ln2}{|k|}.$$

251. 卡片 251

     问题

     Average value of $f$ on $[a,b]$?

     答案

     For integrable $f$ and $a<b$,

     $$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

252. 卡片 252

     问题

     Displacement from velocity $v(t)$ on $[a,b]$?

     答案

     $$s(b)-s(a)=\int_a^b v(t)\,dt.$$

     Velocity below zero contributes negative displacement.

253. 卡片 253

     问题

     Area between vertical curves $y=f(x)$ and $y=g(x)$?

     答案

     On intervals where $f(x)\ge g(x)$,

     $$A=\int_a^b [f(x)-g(x)]\,dx.$$

     Think top minus bottom.

254. 卡片 254

     问题

     Volume from known cross-sectional area $A(x)$?

     答案

     If slices are perpendicular to the $x$-axis,

     $$V=\int_a^b A(x)\,dx.$$

255. 卡片 255

     问题

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     答案

     If $f$ is continuous on $[a,b]$, then some $c\in[a,b]$ satisfies

     $$f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

     If $a<b$, a point can also be chosen in $(a,b)$.

256. 卡片 256

     问题

     Velocity and acceleration from position $s(t)$?

     答案

     $$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$

257. 卡片 257

     问题

     Cross-sectional area when each slice is a square?

     答案

     If the base segment has length $s(x)$, then

     $$A(x)=[s(x)]^2.$$

258. 卡片 258

     问题

     How do you find accumulation from an inflow rate and an outflow rate?

     答案

     Integrate the net rate:

     $$Q(b)-Q(a)=\int_a^b [r_{\text{in}}(t)-r_{\text{out}}(t)]\,dt.$$

259. 卡片 259

     问题

     Area between horizontal curves written as $x=R(y)$ and $x=L(y)$?

     答案

     On intervals where $R(y)\ge L(y)$,

     $$A=\int_c^d [R(y)-L(y)]\,dy.$$

     Think right minus left.

260. 卡片 260

     问题

     Disc-method volume formula?

     答案

     For radius $R(x)$ and slices perpendicular to the $x$-axis,

     $$V=\pi\int_a^b [R(x)]^2\,dx.$$

261. 卡片 261

     问题

     What units does average value have?

     答案

     The same units as the original function. Integration adds an input unit, and division by interval length removes it.

262. 卡片 262

     问题

     Total distance traveled from velocity $v(t)$?

     答案

     $$\text{distance}=\int_a^b |v(t)|\,dt.$$

     Split the interval wherever $v(t)=0$ and its sign changes.

263. 卡片 263

     问题

     Cross-sectional area when each slice is a rectangle?

     答案

     If the slice has base $b(x)$ and height $h(x)$, then

     $$A(x)=b(x)h(x).$$

     Use the stated relationship to express both in the integration variable.

264. 卡片 264

     问题

     How do you determine bounds for area between curves?

     答案

     Use the stated region boundaries or solve the curve-intersection equations. Check whether additional intersections inside the interval require splitting.

265. 卡片 265

     问题

     How is a rotation radius measured from a horizontal axis $y=k$?

     答案

     As vertical distance: $|y-k|$. For washers, identify which boundary stays farther from the axis over the interval.

266. 卡片 266

     问题

     When is a particle moving to the right or left?

     答案

     It moves right where $v(t)>0$ and left where $v(t)<0$. Position alone does not determine direction.

267. 卡片 267

     问题

     Cross-sectional area when the diameter of a semicircle is $d(x)$?

     答案

     The radius is $d(x)/2$, so

     $$A(x)=\frac12\pi\left(\frac{d(x)}2\right)^2=\frac{\pi}{8}[d(x)]^2.$$

268. 卡片 268

     问题

     Why must an area integral be split where curves intersect?

     答案

     The top/bottom or right/left relationship may switch there. Splitting keeps each integrand nonnegative and prevents signed cancellation.

269. 卡片 269

     问题

     How can a velocity table approximate displacement?

     答案

     Use a left, right, midpoint, or trapezoidal sum for $\int v(t)\,dt$. Each term is velocity times a time width.

270. 卡片 270

     问题

     Washer-method volume formula?

     答案

     For outer radius $R(x)$ and inner radius $r(x)$,

     $$V=\pi\int_a^b ([R(x)]^2-[r(x)]^2)\,dx.$$

271. 卡片 271

     问题

     How do you recover position from velocity and an initial position?

     答案

     If $s(a)$ is known,

     $$s(t)=s(a)+\int_a^t v(u)\,du.$$

272. 卡片 272

     问题

     How do you choose between vertical and horizontal area slices?

     答案

     Choose the direction that describes the region with fewer pieces. Vertical slices use top minus bottom; horizontal slices use right minus left.

273. 卡片 273

     问题

     Cross-sectional area of an equilateral triangle with side $s(x)$?

     答案

     $$A(x)=\frac{\sqrt3}{4}[s(x)]^2.$$

274. 卡片 274

     问题

     How can a table approximate the average value of $f$ on $[a,b]$?

     答案

     First approximate $\int_a^b f(x)\,dx$ with an appropriate Riemann or trapezoidal sum, then divide by $b-a$.

275. 卡片 275

     问题

     Single expression for area between two curves?

     答案

     When the functions are integrable,

     $$A=\int_a^b |f(x)-g(x)|\,dx.$$

     For hand evaluation, split where their order changes.

276. 卡片 276

     问题

     How is a rotation radius measured from a vertical axis $x=k$?

     答案

     As horizontal distance: $|x-k|$. With $dy$ slices, write the relevant boundaries as $x$-functions of $y$.

277. 卡片 277

     问题

     How can a rate table approximate total change with unequal time gaps?

     答案

     Multiply a representative rate on each interval by that interval's actual width, then sum. Do not assume a common $\Delta t$ when the table is uneven.

278. 卡片 278

     问题

     When should a volume integral use $dy$?

     答案

     When slices perpendicular to the $y$-axis make the cross-sectional area easiest to express as $A(y)$. Then use $V=\int_c^d A(y)\,dy$.

279. 卡片 279

     问题

     What signals that a washer, not a disc, is needed?

     答案

     The rotated region leaves a hole around the axis. Subtract the inner circular area from the outer circular area.

280. 卡片 280

     问题

     What base length is used for cross sections over a planar region?

     答案

     Use the distance across the base region within each slice: top minus bottom for vertical slices, or right minus left for horizontal slices.

281. 卡片 281

     问题

     Why must total distance split at velocity sign changes?

     答案

     Distance accumulates speed $|v|$, not signed velocity. A single integral of $v$ would cancel motion in opposite directions.

282. 卡片 282

     问题

     When does an accumulated quantity reach a local maximum?

     答案

     When its net rate changes from positive to negative. A zero rate alone is only a candidate; check the sign change.

283. 卡片 283

     问题

     What distinguishes area from a definite integral?

     答案

     Area is nonnegative. A definite integral is signed and may be zero or negative because regions below the axis subtract.

284. 卡片 284

     问题

     How do position, velocity, and acceleration graphs correspond?

     答案

     Velocity is the slope of position, and acceleration is the slope of velocity. Conversely, signed area under velocity gives position change, and signed area under acceleration gives velocity change.

285. 卡片 285

     问题

     How do you interpret $\int_a^b [r(t)-c(t)]\,dt$ in context?

     答案

     As net change: total amount added by $r$ minus total amount removed by $c$ over the interval. State the resulting quantity and units.

286. 卡片 286

     问题

     What units does a volume integral $\int A(x)\,dx$ have?

     答案

     Cubic units. Cross-sectional area contributes square units and slice thickness contributes one more length unit.

287. 卡片 287

     问题

     How can a graph of a rate reveal the largest accumulated value?

     答案

     Compare accumulation at endpoints and at times where the rate is zero or undefined. Compute signed areas to track the quantity's changes from its initial value.

288. 卡片 288

     问题

     Why should a contextual integral answer include a sentence?

     答案

     The number alone does not say whether it represents displacement, distance, area, volume, or net change. Name the quantity, interval, and units.