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

Карточки в этой колоде

1. Карточка 1

   Вопрос

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

   Ответ

   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. Карточка 2

   Вопрос

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

   Ответ

   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. Карточка 3

   Вопрос

   When does direct substitution evaluate a limit?

   Ответ

   When the function is continuous at the target input. Then

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

4. Карточка 4

   Вопрос

   Three conditions for continuity at $x=a$?

   Ответ

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

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

5. Карточка 5

   Вопрос

   Intermediate Value Theorem: hypotheses and conclusion?

   Ответ

   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. Карточка 6

   Вопрос

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

   Ответ

   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. Карточка 7

   Вопрос

   How do you read a finite limit from a graph?

   Ответ

   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. Карточка 8

   Вопрос

   Limit law for a sum or difference?

   Ответ

   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. Карточка 9

   Вопрос

   What makes a discontinuity removable?

   Ответ

   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. Карточка 10

    Вопрос

    Squeeze Theorem: usable form?

    Ответ

    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. Карточка 11

    Вопрос

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

    Ответ

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

12. Карточка 12

    Вопрос

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

    Ответ

    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. Карточка 13

    Вопрос

    Limit law for a product?

    Ответ

    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. Карточка 14

    Вопрос

    Graph signature of a jump discontinuity?

    Ответ

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

15. Карточка 15

    Вопрос

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

    Ответ

    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. Карточка 16

    Вопрос

    Horizontal asymptote from a limit at infinity?

    Ответ

    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. Карточка 17

    Вопрос

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

    Ответ

    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. Карточка 18

    Вопрос

    Limit law for a quotient—and its condition?

    Ответ

    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. Карточка 19

    Вопрос

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

    Ответ

    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. Карточка 20

    Вопрос

    When is the Squeeze Theorem a natural choice?

    Ответ

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

21. Карточка 21

    Вопрос

    Vertical asymptote from one-sided behavior?

    Ответ

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

22. Карточка 22

    Вопрос

    Limit at infinity of equal-degree rational functions?

    Ответ

    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. Карточка 23

    Вопрос

    When can a limit pass through a continuous outer function?

    Ответ

    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. Карточка 24

    Вопрос

    What makes a discontinuity infinite?

    Ответ

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

25. Карточка 25

    Вопрос

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

    Ответ

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

26. Карточка 26

    Вопрос

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

    Ответ

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Карточка 27

    Вопрос

    Continuity of a composition?

    Ответ

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

28. Карточка 28

    Вопрос

    Limit at infinity when a rational numerator has lower degree?

    Ответ

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

29. Карточка 29

    Вопрос

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

    Ответ

    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. Карточка 30

    Вопрос

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

    Ответ

    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. Карточка 31

    Вопрос

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

    Ответ

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

32. Карточка 32

    Вопрос

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

    Ответ

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

33. Карточка 33

    Вопрос

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

    Ответ

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

34. Карточка 34

    Вопрос

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

    Ответ

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

35. Карточка 35

    Вопрос

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

    Ответ

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

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

36. Карточка 36

    Вопрос

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

    Ответ

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

    The derivative exists only if this finite limit exists.

37. Карточка 37

    Вопрос

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

    Ответ

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

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

38. Карточка 38

    Вопрос

    What does differentiability imply about continuity?

    Ответ

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

39. Карточка 39

    Вопрос

    Power rule for derivatives?

    Ответ

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

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

40. Карточка 40

    Вопрос

    Units of $f'(x)$?

    Ответ

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

41. Карточка 41

    Вопрос

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

    Ответ

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

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

42. Карточка 42

    Вопрос

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

    Ответ

    $f'(x)>0$ 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. Карточка 43

    Вопрос

    Derivative of a constant?

    Ответ

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

    A constant function has zero rate of change.

44. Карточка 44

    Вопрос

    Derivative of $\sin x$?

    Ответ

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

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

45. Карточка 45

    Вопрос

    Product rule?

    Ответ

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

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

46. Карточка 46

    Вопрос

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

    Ответ

    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. Карточка 47

    Вопрос

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

    Ответ

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

48. Карточка 48

    Вопрос

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

    Ответ

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

49. Карточка 49

    Вопрос

    Derivative of a sum or difference?

    Ответ

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

50. Карточка 50

    Вопрос

    Derivative of $\cos x$?

    Ответ

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

    The standard formula assumes radians.

51. Карточка 51

    Вопрос

    Quotient rule?

    Ответ

    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. Карточка 52

    Вопрос

    Common notations for the first derivative?

    Ответ

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

53. Карточка 53

    Вопрос

    Derivative of $e^x$?

    Ответ

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

54. Карточка 54

    Вопрос

    What graph features can make $f$ nondifferentiable?

    Ответ

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

55. Карточка 55

    Вопрос

    Derivative of $\tan x$?

    Ответ

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Карточка 56

    Вопрос

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

    Ответ

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

57. Карточка 57

    Вопрос

    Derivative of $\ln x$?

    Ответ

    For $x>0$,

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

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

58. Карточка 58

    Вопрос

    How does the power rule handle roots or negative powers?

    Ответ

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

59. Карточка 59

    Вопрос

    Derivative of $\csc x$?

    Ответ

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Карточка 60

    Вопрос

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

    Ответ

    $f$ is increasing on that interval.

61. Карточка 61

    Вопрос

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

    Ответ

    For $a>0$,

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

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

62. Карточка 62

    Вопрос

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

    Ответ

    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. Карточка 63

    Вопрос

    Derivative of $\sec x$?

    Ответ

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Карточка 64

    Вопрос

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

    Ответ

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

65. Карточка 65

    Вопрос

    Derivative of $\log_a x$?

    Ответ

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

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

66. Карточка 66

    Вопрос

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

    Ответ

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Карточка 67

    Вопрос

    Derivative of $\cot x$?

    Ответ

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Карточка 68

    Вопрос

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

    Ответ

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

69. Карточка 69

    Вопрос

    Constant-multiple rule?

    Ответ

    For a constant $c$,

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

70. Карточка 70

    Вопрос

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

    Ответ

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

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

71. Карточка 71

    Вопрос

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

    Ответ

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

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

72. Карточка 72

    Вопрос

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

    Ответ

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

73. Карточка 73

    Вопрос

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

    Ответ

    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. Карточка 74

    Вопрос

    Derivative of an inverse function at $x$?

    Ответ

    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. Карточка 75

    Вопрос

    Derivative of $\arcsin x$?

    Ответ

    For $|x|<1$,

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

76. Карточка 76

    Вопрос

    Notation for the third derivative of $f$?

    Ответ

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

77. Карточка 77

    Вопрос

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

    Ответ

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

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

78. Карточка 78

    Вопрос

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

    Ответ

    Where $y\ne0$,

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

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

79. Карточка 79

    Вопрос

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

    Ответ

    Provided $f'(a)\ne0$,

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

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

80. Карточка 80

    Вопрос

    Derivative of $\arctan x$?

    Ответ

    For every real $x$,

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

81. Карточка 81

    Вопрос

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

    Ответ

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

    The extra factor is the chain rule.

82. Карточка 82

    Вопрос

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

    Ответ

    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. Карточка 83

    Вопрос

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

    Ответ

    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. Карточка 84

    Вопрос

    Derivative of $\arccos x$?

    Ответ

    For $|x|<1$,

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

85. Карточка 85

    Вопрос

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

    Ответ

    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. Карточка 86

    Вопрос

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

    Ответ

    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. Карточка 87

    Вопрос

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

    Ответ

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

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

88. Карточка 88

    Вопрос

    How are tangent slopes of inverse graphs related?

    Ответ

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

89. Карточка 89

    Вопрос

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

    Ответ

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

90. Карточка 90

    Вопрос

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

    Ответ

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

91. Карточка 91

    Вопрос

    Horizontal tangent on an implicit curve: derivative condition?

    Ответ

    $dy/dx=0$ 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. Карточка 92

    Вопрос

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

    Ответ

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

93. Карточка 93

    Вопрос

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

    Ответ

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

94. Карточка 94

    Вопрос

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

    Ответ

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

    This combines the power rule with the chain rule.

95. Карточка 95

    Вопрос

    Vertical tangent on an implicit curve: derivative clue?

    Ответ

    $dy/dx$ 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. Карточка 96

    Вопрос

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

    Ответ

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

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

97. Карточка 97

    Вопрос

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

    Ответ

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

98. Карточка 98

    Вопрос

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

    Ответ

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

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

99. Карточка 99

    Вопрос

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

    Ответ

    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. Карточка 100

     Вопрос

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

     Ответ

     When the functions are differentiable, the chain rule gives

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

101. Карточка 101

     Вопрос

     What local property lets a function have an inverse derivative?

     Ответ

     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. Карточка 102

     Вопрос

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

     Ответ

     For $a>0$,

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

103. Карточка 103

     Вопрос

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

     Ответ

     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. Карточка 104

     Вопрос

     Position, velocity, and acceleration relationships?

     Ответ

     For position $s(t)$,

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

105. Карточка 105

     Вопрос

     Central idea of a related-rates problem?

     Ответ

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

106. Карточка 106

     Вопрос

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

     Ответ

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

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

107. Карточка 107

     Вопрос

     L’Hospital’s Rule: basic conditions?

     Ответ

     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. Карточка 108

     Вопрос

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

     Ответ

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

109. Карточка 109

     Вопрос

     Speed in terms of velocity?

     Ответ

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Карточка 110

     Вопрос

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

     Ответ

     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. Карточка 111

     Вопрос

     Differential approximation connecting $dx$ and $dy$?

     Ответ

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

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

112. Карточка 112

     Вопрос

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

     Ответ

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

113. Карточка 113

     Вопрос

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

     Ответ

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

114. Карточка 114

     Вопрос

     What does positive acceleration say about velocity?

     Ответ

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

115. Карточка 115

     Вопрос

     Related rates: when should numerical values be substituted?

     Ответ

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

116. Карточка 116

     Вопрос

     How does concavity predict linearization error?

     Ответ

     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. Карточка 117

     Вопрос

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

     Ответ

     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. Карточка 118

     Вопрос

     When is a particle moving in the positive direction?

     Ответ

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

119. Карточка 119

     Вопрос

     How can velocity show a change of direction?

     Ответ

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

120. Карточка 120

     Вопрос

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

     Ответ

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

121. Карточка 121

     Вопрос

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

     Ответ

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

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

122. Карточка 122

     Вопрос

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

     Ответ

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

123. Карточка 123

     Вопрос

     What must a contextual derivative sentence include?

     Ответ

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

124. Карточка 124

     Вопрос

     Velocity negative and acceleration positive: what happens?

     Ответ

     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. Карточка 125

     Вопрос

     How should a negative related rate be interpreted?

     Ответ

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

126. Карточка 126

     Вопрос

     When is local linearity a sound approximation tool?

     Ответ

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

127. Карточка 127

     Вопрос

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

     Ответ

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

128. Карточка 128

     Вопрос

     When is speed increasing?

     Ответ

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

129. Карточка 129

     Вопрос

     Volume changes with time: notation for its rate?

     Ответ

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

130. Карточка 130

     Вопрос

     Why are similar triangles useful in related rates?

     Ответ

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

131. Карточка 131

     Вопрос

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

     Ответ

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

132. Карточка 132

     Вопрос

     What conclusion does L’Hospital’s Rule permit?

     Ответ

     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. Карточка 133

     Вопрос

     When is speed decreasing?

     Ответ

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

134. Карточка 134

     Вопрос

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

     Ответ

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

135. Карточка 135

     Вопрос

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

     Ответ

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

136. Карточка 136

     Вопрос

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

     Ответ

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

137. Карточка 137

     Вопрос

     Extreme Value Theorem: hypothesis and conclusion?

     Ответ

     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. Карточка 138

     Вопрос

     What is a critical number of $f$?

     Ответ

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

139. Карточка 139

     Вопрос

     First derivative test for a local maximum?

     Ответ

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

140. Карточка 140

     Вопрос

     Second-derivative sign for concave up?

     Ответ

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

141. Карточка 141

     Вопрос

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

     Ответ

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

142. Карточка 142

     Вопрос

     First step in an optimization model?

     Ответ

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

143. Карточка 143

     Вопрос

     Mean Value Theorem: hypotheses and conclusion?

     Ответ

     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. Карточка 144

     Вопрос

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

     Ответ

     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. Карточка 145

     Вопрос

     First derivative test for a local minimum?

     Ответ

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

146. Карточка 146

     Вопрос

     What must happen at an inflection point?

     Ответ

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

147. Карточка 147

     Вопрос

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

     Ответ

     $f''$ 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. Карточка 148

     Вопрос

     How do you confirm an optimization answer is absolute?

     Ответ

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

149. Карточка 149

     Вопрос

     Rolle’s Theorem: hypotheses and conclusion?

     Ответ

     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. Карточка 150

     Вопрос

     Difference between absolute and relative extrema?

     Ответ

     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. Карточка 151

     Вопрос

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

     Ответ

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

152. Карточка 152

     Вопрос

     Second derivative test for a local minimum?

     Ответ

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

153. Карточка 153

     Вопрос

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

     Ответ

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

154. Карточка 154

     Вопрос

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

     Ответ

     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. Карточка 155

     Вопрос

     Which theorem links an average slope to an instantaneous slope?

     Ответ

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

156. Карточка 156

     Вопрос

     How can an implicit derivative locate a horizontal tangent?

     Ответ

     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. Карточка 157

     Вопрос

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

     Ответ

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

158. Карточка 158

     Вопрос

     Second derivative test for a local maximum?

     Ответ

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

159. Карточка 159

     Вопрос

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

     Ответ

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

160. Карточка 160

     Вопрос

     Why must an optimization domain be stated?

     Ответ

     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. Карточка 161

     Вопрос

     Which theorem guarantees absolute extrema, not where they occur?

     Ответ

     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. Карточка 162

     Вопрос

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

     Ответ

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

163. Карточка 163

     Вопрос

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

     Ответ

     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. Карточка 164

     Вопрос

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

     Ответ

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

165. Карточка 165

     Вопрос

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

     Ответ

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

166. Карточка 166

     Вопрос

     How can an implicit derivative locate a vertical tangent?

     Ответ

     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. Карточка 167

     Вопрос

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

     Ответ

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

168. Карточка 168

     Вопрос

     Why are endpoints included in the candidates test?

     Ответ

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

169. Карточка 169

     Вопрос

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

     Ответ

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

170. Карточка 170

     Вопрос

     Second-derivative sign for concave down?

     Ответ

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

171. Карточка 171

     Вопрос

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

     Ответ

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

172. Карточка 172

     Вопрос

     What should the final line of an optimization solution state?

     Ответ

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

173. Карточка 173

     Вопрос

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

     Ответ

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

174. Карточка 174

     Вопрос

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

     Ответ

     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. Карточка 175

     Вопрос

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

     Ответ

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

176. Карточка 176

     Вопрос

     Left Riemann sum on equal subintervals?

     Ответ

     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. Карточка 177

     Вопрос

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

     Ответ

     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. Карточка 178

     Вопрос

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Ответ

     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. Карточка 179

     Вопрос

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

     Ответ

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Карточка 180

     Вопрос

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

     Ответ

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

181. Карточка 181

     Вопрос

     Right Riemann sum on equal subintervals?

     Ответ

     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. Карточка 182

     Вопрос

     How does reversing integral bounds change the value?

     Ответ

     It changes the sign:

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

183. Карточка 183

     Вопрос

     Net Change Theorem?

     Ответ

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

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

184. Карточка 184

     Вопрос

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

     Ответ

     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. Карточка 185

     Вопрос

     Power rule for antiderivatives?

     Ответ

     For $n\ne-1$,

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

186. Карточка 186

     Вопрос

     Midpoint Riemann sum on equal subintervals?

     Ответ

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

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

187. Карточка 187

     Вопрос

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

     Ответ

     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. Карточка 188

     Вопрос

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

     Ответ

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Карточка 189

     Вопрос

     Antiderivative of $1/x$?

     Ответ

     On any interval not crossing zero,

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

190. Карточка 190

     Вопрос

     Trapezoidal approximation on equal subintervals?

     Ответ

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

191. Карточка 191

     Вопрос

     How do geometric regions help evaluate a definite integral?

     Ответ

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

192. Карточка 192

     Вопрос

     Basic antiderivatives of sine and cosine?

     Ответ

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

193. Карточка 193

     Вопрос

     Definite integral as a limit of Riemann sums?

     Ответ

     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. Карточка 194

     Вопрос

     Constant-multiple rule for integrals?

     Ответ

     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. Карточка 195

     Вопрос

     What pattern suggests $u$-substitution?

     Ответ

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

196. Карточка 196

     Вопрос

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

     Ответ

     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. Карточка 197

     Вопрос

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

     Ответ

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

198. Карточка 198

     Вопрос

     Sum-and-difference rule for definite integrals?

     Ответ

     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. Карточка 199

     Вопрос

     Basic antiderivative of $e^x$?

     Ответ

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

200. Карточка 200

     Вопрос

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

     Ответ

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

201. Карточка 201

     Вопрос

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

     Ответ

     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. Карточка 202

     Вопрос

     How does concavity predict trapezoidal and midpoint error?

     Ответ

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

203. Карточка 203

     Вопрос

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

     Ответ

     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. Карточка 204

     Вопрос

     What denominator pattern suggests an arctangent antiderivative?

     Ответ

     After completing the square and scaling, a form like

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

205. Карточка 205

     Вопрос

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

     Ответ

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

206. Карточка 206

     Вопрос

     How does an initial condition determine an antiderivative?

     Ответ

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

207. Карточка 207

     Вопрос

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

     Ответ

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

208. Карточка 208

     Вопрос

     Why does an indefinite integral include $+C$?

     Ответ

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

209. Карточка 209

     Вопрос

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

     Ответ

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

210. Карточка 210

     Вопрос

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

     Ответ

     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. Карточка 211

     Вопрос

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

     Ответ

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

212. Карточка 212

     Вопрос

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

     Ответ

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

213. Карточка 213

     Вопрос

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

     Ответ

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

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

214. Карточка 214

     Вопрос

     Riemann sum for unequal subinterval widths?

     Ответ

     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. Карточка 215

     Вопрос

     Does continuity guarantee integrability on a closed interval?

     Ответ

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

216. Карточка 216

     Вопрос

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

     Ответ

     Where $g(x)\ne0$,

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

217. Карточка 217

     Вопрос

     What algebraic rewrites often reveal a basic antiderivative?

     Ответ

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

218. Карточка 218

     Вопрос

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

     Ответ

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

219. Карточка 219

     Вопрос

     What is a differential equation?

     Ответ

     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. Карточка 220

     Вопрос

     How does a verbal rate statement become a differential equation?

     Ответ

     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. Карточка 221

     Вопрос

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

     Ответ

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

222. Карточка 222

     Вопрос

     General solution versus particular solution?

     Ответ

     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. Карточка 223

     Вопрос

     What does one segment in a slope field show?

     Ответ

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

224. Карточка 224

     Вопрос

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

     Ответ

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

225. Карточка 225

     Вопрос

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

     Ответ

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

226. Карточка 226

     Вопрос

     What makes a first-order differential equation separable?

     Ответ

     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. Карточка 227

     Вопрос

     What is an initial value problem?

     Ответ

     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. Карточка 228

     Вопрос

     What is an isocline in a slope field?

     Ответ

     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. Карточка 229

     Вопрос

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

     Ответ

     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. Карточка 230

     Вопрос

     General solution of $dy/dt=ky$?

     Ответ

     $$y=Ce^{kt}$$

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

231. Карточка 231

     Вопрос

     Core method for solving a separable differential equation?

     Ответ

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

232. Карточка 232

     Вопрос

     How should a solution curve follow a slope field?

     Ответ

     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. Карточка 233

     Вопрос

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

     Ответ

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

234. Карточка 234

     Вопрос

     Why is one integration constant enough after integrating both sides?

     Ответ

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

235. Карточка 235

     Вопрос

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

     Ответ

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

236. Карточка 236

     Вопрос

     Can one differential equation have infinitely many solutions?

     Ответ

     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. Карточка 237

     Вопрос

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

     Ответ

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

238. Карточка 238

     Вопрос

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

     Ответ

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

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

239. Карточка 239

     Вопрос

     What can be lost when dividing to separate variables?

     Ответ

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

240. Карточка 240

     Вопрос

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

     Ответ

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

241. Карточка 241

     Вопрос

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

     Ответ

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

242. Карточка 242

     Вопрос

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

     Ответ

     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. Карточка 243

     Вопрос

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

     Ответ

     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. Карточка 244

     Вопрос

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

     Ответ

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

245. Карточка 245

     Вопрос

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

     Ответ

     For $k>0$,

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

     It is independent of the initial amount.

246. Карточка 246

     Вопрос

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

     Ответ

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

247. Карточка 247

     Вопрос

     How is an initial condition used after separation?

     Ответ

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

248. Карточка 248

     Вопрос

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

     Ответ

     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. Карточка 249

     Вопрос

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

     Ответ

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

250. Карточка 250

     Вопрос

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

     Ответ

     For $k<0$,

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

251. Карточка 251

     Вопрос

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

     Ответ

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

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

252. Карточка 252

     Вопрос

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

     Ответ

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

     Velocity below zero contributes negative displacement.

253. Карточка 253

     Вопрос

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

     Ответ

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

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

     Think top minus bottom.

254. Карточка 254

     Вопрос

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

     Ответ

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

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

255. Карточка 255

     Вопрос

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Ответ

     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. Карточка 256

     Вопрос

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

     Ответ

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

257. Карточка 257

     Вопрос

     Cross-sectional area when each slice is a square?

     Ответ

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

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

258. Карточка 258

     Вопрос

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

     Ответ

     Integrate the net rate:

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

259. Карточка 259

     Вопрос

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

     Ответ

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

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

     Think right minus left.

260. Карточка 260

     Вопрос

     Disc-method volume formula?

     Ответ

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

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

261. Карточка 261

     Вопрос

     What units does average value have?

     Ответ

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

262. Карточка 262

     Вопрос

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

     Ответ

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

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

263. Карточка 263

     Вопрос

     Cross-sectional area when each slice is a rectangle?

     Ответ

     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. Карточка 264

     Вопрос

     How do you determine bounds for area between curves?

     Ответ

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

265. Карточка 265

     Вопрос

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

     Ответ

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

266. Карточка 266

     Вопрос

     When is a particle moving to the right or left?

     Ответ

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

267. Карточка 267

     Вопрос

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

     Ответ

     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. Карточка 268

     Вопрос

     Why must an area integral be split where curves intersect?

     Ответ

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

269. Карточка 269

     Вопрос

     How can a velocity table approximate displacement?

     Ответ

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

270. Карточка 270

     Вопрос

     Washer-method volume formula?

     Ответ

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

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

271. Карточка 271

     Вопрос

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

     Ответ

     If $s(a)$ is known,

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

272. Карточка 272

     Вопрос

     How do you choose between vertical and horizontal area slices?

     Ответ

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

273. Карточка 273

     Вопрос

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

     Ответ

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

274. Карточка 274

     Вопрос

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

     Ответ

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

275. Карточка 275

     Вопрос

     Single expression for area between two curves?

     Ответ

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Карточка 276

     Вопрос

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

     Ответ

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

277. Карточка 277

     Вопрос

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

     Ответ

     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. Карточка 278

     Вопрос

     When should a volume integral use $dy$?

     Ответ

     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. Карточка 279

     Вопрос

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

     Ответ

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

280. Карточка 280

     Вопрос

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

     Ответ

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

281. Карточка 281

     Вопрос

     Why must total distance split at velocity sign changes?

     Ответ

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

282. Карточка 282

     Вопрос

     When does an accumulated quantity reach a local maximum?

     Ответ

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

283. Карточка 283

     Вопрос

     What distinguishes area from a definite integral?

     Ответ

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

284. Карточка 284

     Вопрос

     How do position, velocity, and acceleration graphs correspond?

     Ответ

     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. Карточка 285

     Вопрос

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

     Ответ

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

286. Карточка 286

     Вопрос

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

     Ответ

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

287. Карточка 287

     Вопрос

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

     Ответ

     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. Карточка 288

     Вопрос

     Why should a contextual integral answer include a sentence?

     Ответ

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