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

Cartões deste baralho

1. Cartão 1

   Pergunta

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

   Resposta

   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. Cartão 2

   Pergunta

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

   Resposta

   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. Cartão 3

   Pergunta

   When does direct substitution evaluate a limit?

   Resposta

   When the function is continuous at the target input. Then

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

4. Cartão 4

   Pergunta

   Three conditions for continuity at $x=a$?

   Resposta

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

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

5. Cartão 5

   Pergunta

   Intermediate Value Theorem: hypotheses and conclusion?

   Resposta

   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. Cartão 6

   Pergunta

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

   Resposta

   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. Cartão 7

   Pergunta

   How do you read a finite limit from a graph?

   Resposta

   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. Cartão 8

   Pergunta

   Limit law for a sum or difference?

   Resposta

   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. Cartão 9

   Pergunta

   What makes a discontinuity removable?

   Resposta

   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. Cartão 10

    Pergunta

    Squeeze Theorem: usable form?

    Resposta

    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. Cartão 11

    Pergunta

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

    Resposta

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

12. Cartão 12

    Pergunta

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

    Resposta

    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. Cartão 13

    Pergunta

    Limit law for a product?

    Resposta

    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. Cartão 14

    Pergunta

    Graph signature of a jump discontinuity?

    Resposta

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

15. Cartão 15

    Pergunta

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

    Resposta

    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. Cartão 16

    Pergunta

    Horizontal asymptote from a limit at infinity?

    Resposta

    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. Cartão 17

    Pergunta

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

    Resposta

    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. Cartão 18

    Pergunta

    Limit law for a quotient—and its condition?

    Resposta

    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. Cartão 19

    Pergunta

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

    Resposta

    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. Cartão 20

    Pergunta

    When is the Squeeze Theorem a natural choice?

    Resposta

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

21. Cartão 21

    Pergunta

    Vertical asymptote from one-sided behavior?

    Resposta

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

22. Cartão 22

    Pergunta

    Limit at infinity of equal-degree rational functions?

    Resposta

    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. Cartão 23

    Pergunta

    When can a limit pass through a continuous outer function?

    Resposta

    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. Cartão 24

    Pergunta

    What makes a discontinuity infinite?

    Resposta

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

25. Cartão 25

    Pergunta

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

    Resposta

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

26. Cartão 26

    Pergunta

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

    Resposta

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Cartão 27

    Pergunta

    Continuity of a composition?

    Resposta

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

28. Cartão 28

    Pergunta

    Limit at infinity when a rational numerator has lower degree?

    Resposta

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

29. Cartão 29

    Pergunta

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

    Resposta

    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. Cartão 30

    Pergunta

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

    Resposta

    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. Cartão 31

    Pergunta

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

    Resposta

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

32. Cartão 32

    Pergunta

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

    Resposta

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

33. Cartão 33

    Pergunta

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

    Resposta

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

34. Cartão 34

    Pergunta

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

    Resposta

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

35. Cartão 35

    Pergunta

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

    Resposta

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

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

36. Cartão 36

    Pergunta

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

    Resposta

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

    The derivative exists only if this finite limit exists.

37. Cartão 37

    Pergunta

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

    Resposta

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

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

38. Cartão 38

    Pergunta

    What does differentiability imply about continuity?

    Resposta

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

39. Cartão 39

    Pergunta

    Power rule for derivatives?

    Resposta

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

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

40. Cartão 40

    Pergunta

    Units of $f'(x)$?

    Resposta

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

41. Cartão 41

    Pergunta

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

    Resposta

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

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

42. Cartão 42

    Pergunta

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

    Resposta

    $f'(x)>0$ 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. Cartão 43

    Pergunta

    Derivative of a constant?

    Resposta

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

    A constant function has zero rate of change.

44. Cartão 44

    Pergunta

    Derivative of $\sin x$?

    Resposta

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

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

45. Cartão 45

    Pergunta

    Product rule?

    Resposta

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

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

46. Cartão 46

    Pergunta

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

    Resposta

    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. Cartão 47

    Pergunta

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

    Resposta

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

48. Cartão 48

    Pergunta

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

    Resposta

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

49. Cartão 49

    Pergunta

    Derivative of a sum or difference?

    Resposta

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

50. Cartão 50

    Pergunta

    Derivative of $\cos x$?

    Resposta

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

    The standard formula assumes radians.

51. Cartão 51

    Pergunta

    Quotient rule?

    Resposta

    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. Cartão 52

    Pergunta

    Common notations for the first derivative?

    Resposta

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

53. Cartão 53

    Pergunta

    Derivative of $e^x$?

    Resposta

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

54. Cartão 54

    Pergunta

    What graph features can make $f$ nondifferentiable?

    Resposta

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

55. Cartão 55

    Pergunta

    Derivative of $\tan x$?

    Resposta

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Cartão 56

    Pergunta

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

    Resposta

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

57. Cartão 57

    Pergunta

    Derivative of $\ln x$?

    Resposta

    For $x>0$,

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

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

58. Cartão 58

    Pergunta

    How does the power rule handle roots or negative powers?

    Resposta

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

59. Cartão 59

    Pergunta

    Derivative of $\csc x$?

    Resposta

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Cartão 60

    Pergunta

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

    Resposta

    $f$ is increasing on that interval.

61. Cartão 61

    Pergunta

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

    Resposta

    For $a>0$,

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

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

62. Cartão 62

    Pergunta

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

    Resposta

    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. Cartão 63

    Pergunta

    Derivative of $\sec x$?

    Resposta

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Cartão 64

    Pergunta

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

    Resposta

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

65. Cartão 65

    Pergunta

    Derivative of $\log_a x$?

    Resposta

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

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

66. Cartão 66

    Pergunta

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

    Resposta

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Cartão 67

    Pergunta

    Derivative of $\cot x$?

    Resposta

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Cartão 68

    Pergunta

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

    Resposta

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

69. Cartão 69

    Pergunta

    Constant-multiple rule?

    Resposta

    For a constant $c$,

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

70. Cartão 70

    Pergunta

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

    Resposta

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

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

71. Cartão 71

    Pergunta

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

    Resposta

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

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

72. Cartão 72

    Pergunta

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

    Resposta

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

73. Cartão 73

    Pergunta

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

    Resposta

    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. Cartão 74

    Pergunta

    Derivative of an inverse function at $x$?

    Resposta

    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. Cartão 75

    Pergunta

    Derivative of $\arcsin x$?

    Resposta

    For $|x|<1$,

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

76. Cartão 76

    Pergunta

    Notation for the third derivative of $f$?

    Resposta

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

77. Cartão 77

    Pergunta

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

    Resposta

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

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

78. Cartão 78

    Pergunta

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

    Resposta

    Where $y\ne0$,

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

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

79. Cartão 79

    Pergunta

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

    Resposta

    Provided $f'(a)\ne0$,

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

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

80. Cartão 80

    Pergunta

    Derivative of $\arctan x$?

    Resposta

    For every real $x$,

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

81. Cartão 81

    Pergunta

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

    Resposta

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

    The extra factor is the chain rule.

82. Cartão 82

    Pergunta

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

    Resposta

    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. Cartão 83

    Pergunta

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

    Resposta

    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. Cartão 84

    Pergunta

    Derivative of $\arccos x$?

    Resposta

    For $|x|<1$,

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

85. Cartão 85

    Pergunta

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

    Resposta

    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. Cartão 86

    Pergunta

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

    Resposta

    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. Cartão 87

    Pergunta

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

    Resposta

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

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

88. Cartão 88

    Pergunta

    How are tangent slopes of inverse graphs related?

    Resposta

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

89. Cartão 89

    Pergunta

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

    Resposta

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

90. Cartão 90

    Pergunta

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

    Resposta

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

91. Cartão 91

    Pergunta

    Horizontal tangent on an implicit curve: derivative condition?

    Resposta

    $dy/dx=0$ 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. Cartão 92

    Pergunta

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

    Resposta

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

93. Cartão 93

    Pergunta

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

    Resposta

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

94. Cartão 94

    Pergunta

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

    Resposta

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

    This combines the power rule with the chain rule.

95. Cartão 95

    Pergunta

    Vertical tangent on an implicit curve: derivative clue?

    Resposta

    $dy/dx$ 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. Cartão 96

    Pergunta

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

    Resposta

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

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

97. Cartão 97

    Pergunta

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

    Resposta

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

98. Cartão 98

    Pergunta

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

    Resposta

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

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

99. Cartão 99

    Pergunta

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

    Resposta

    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. Cartão 100

     Pergunta

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

     Resposta

     When the functions are differentiable, the chain rule gives

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

101. Cartão 101

     Pergunta

     What local property lets a function have an inverse derivative?

     Resposta

     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. Cartão 102

     Pergunta

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

     Resposta

     For $a>0$,

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

103. Cartão 103

     Pergunta

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

     Resposta

     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. Cartão 104

     Pergunta

     Position, velocity, and acceleration relationships?

     Resposta

     For position $s(t)$,

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

105. Cartão 105

     Pergunta

     Central idea of a related-rates problem?

     Resposta

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

106. Cartão 106

     Pergunta

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

     Resposta

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

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

107. Cartão 107

     Pergunta

     L’Hospital’s Rule: basic conditions?

     Resposta

     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. Cartão 108

     Pergunta

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

     Resposta

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

109. Cartão 109

     Pergunta

     Speed in terms of velocity?

     Resposta

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Cartão 110

     Pergunta

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

     Resposta

     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. Cartão 111

     Pergunta

     Differential approximation connecting $dx$ and $dy$?

     Resposta

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

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

112. Cartão 112

     Pergunta

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

     Resposta

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

113. Cartão 113

     Pergunta

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

     Resposta

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

114. Cartão 114

     Pergunta

     What does positive acceleration say about velocity?

     Resposta

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

115. Cartão 115

     Pergunta

     Related rates: when should numerical values be substituted?

     Resposta

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

116. Cartão 116

     Pergunta

     How does concavity predict linearization error?

     Resposta

     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. Cartão 117

     Pergunta

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

     Resposta

     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. Cartão 118

     Pergunta

     When is a particle moving in the positive direction?

     Resposta

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

119. Cartão 119

     Pergunta

     How can velocity show a change of direction?

     Resposta

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

120. Cartão 120

     Pergunta

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

     Resposta

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

121. Cartão 121

     Pergunta

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

     Resposta

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

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

122. Cartão 122

     Pergunta

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

     Resposta

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

123. Cartão 123

     Pergunta

     What must a contextual derivative sentence include?

     Resposta

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

124. Cartão 124

     Pergunta

     Velocity negative and acceleration positive: what happens?

     Resposta

     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. Cartão 125

     Pergunta

     How should a negative related rate be interpreted?

     Resposta

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

126. Cartão 126

     Pergunta

     When is local linearity a sound approximation tool?

     Resposta

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

127. Cartão 127

     Pergunta

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

     Resposta

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

128. Cartão 128

     Pergunta

     When is speed increasing?

     Resposta

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

129. Cartão 129

     Pergunta

     Volume changes with time: notation for its rate?

     Resposta

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

130. Cartão 130

     Pergunta

     Why are similar triangles useful in related rates?

     Resposta

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

131. Cartão 131

     Pergunta

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

     Resposta

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

132. Cartão 132

     Pergunta

     What conclusion does L’Hospital’s Rule permit?

     Resposta

     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. Cartão 133

     Pergunta

     When is speed decreasing?

     Resposta

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

134. Cartão 134

     Pergunta

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

     Resposta

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

135. Cartão 135

     Pergunta

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

     Resposta

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

136. Cartão 136

     Pergunta

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

     Resposta

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

137. Cartão 137

     Pergunta

     Extreme Value Theorem: hypothesis and conclusion?

     Resposta

     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. Cartão 138

     Pergunta

     What is a critical number of $f$?

     Resposta

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

139. Cartão 139

     Pergunta

     First derivative test for a local maximum?

     Resposta

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

140. Cartão 140

     Pergunta

     Second-derivative sign for concave up?

     Resposta

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

141. Cartão 141

     Pergunta

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

     Resposta

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

142. Cartão 142

     Pergunta

     First step in an optimization model?

     Resposta

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

143. Cartão 143

     Pergunta

     Mean Value Theorem: hypotheses and conclusion?

     Resposta

     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. Cartão 144

     Pergunta

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

     Resposta

     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. Cartão 145

     Pergunta

     First derivative test for a local minimum?

     Resposta

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

146. Cartão 146

     Pergunta

     What must happen at an inflection point?

     Resposta

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

147. Cartão 147

     Pergunta

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

     Resposta

     $f''$ 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. Cartão 148

     Pergunta

     How do you confirm an optimization answer is absolute?

     Resposta

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

149. Cartão 149

     Pergunta

     Rolle’s Theorem: hypotheses and conclusion?

     Resposta

     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. Cartão 150

     Pergunta

     Difference between absolute and relative extrema?

     Resposta

     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. Cartão 151

     Pergunta

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

     Resposta

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

152. Cartão 152

     Pergunta

     Second derivative test for a local minimum?

     Resposta

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

153. Cartão 153

     Pergunta

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

     Resposta

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

154. Cartão 154

     Pergunta

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

     Resposta

     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. Cartão 155

     Pergunta

     Which theorem links an average slope to an instantaneous slope?

     Resposta

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

156. Cartão 156

     Pergunta

     How can an implicit derivative locate a horizontal tangent?

     Resposta

     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. Cartão 157

     Pergunta

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

     Resposta

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

158. Cartão 158

     Pergunta

     Second derivative test for a local maximum?

     Resposta

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

159. Cartão 159

     Pergunta

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

     Resposta

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

160. Cartão 160

     Pergunta

     Why must an optimization domain be stated?

     Resposta

     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. Cartão 161

     Pergunta

     Which theorem guarantees absolute extrema, not where they occur?

     Resposta

     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. Cartão 162

     Pergunta

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

     Resposta

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

163. Cartão 163

     Pergunta

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

     Resposta

     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. Cartão 164

     Pergunta

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

     Resposta

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

165. Cartão 165

     Pergunta

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

     Resposta

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

166. Cartão 166

     Pergunta

     How can an implicit derivative locate a vertical tangent?

     Resposta

     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. Cartão 167

     Pergunta

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

     Resposta

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

168. Cartão 168

     Pergunta

     Why are endpoints included in the candidates test?

     Resposta

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

169. Cartão 169

     Pergunta

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

     Resposta

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

170. Cartão 170

     Pergunta

     Second-derivative sign for concave down?

     Resposta

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

171. Cartão 171

     Pergunta

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

     Resposta

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

172. Cartão 172

     Pergunta

     What should the final line of an optimization solution state?

     Resposta

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

173. Cartão 173

     Pergunta

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

     Resposta

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

174. Cartão 174

     Pergunta

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

     Resposta

     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. Cartão 175

     Pergunta

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

     Resposta

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

176. Cartão 176

     Pergunta

     Left Riemann sum on equal subintervals?

     Resposta

     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. Cartão 177

     Pergunta

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

     Resposta

     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. Cartão 178

     Pergunta

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Resposta

     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. Cartão 179

     Pergunta

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

     Resposta

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Cartão 180

     Pergunta

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

     Resposta

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

181. Cartão 181

     Pergunta

     Right Riemann sum on equal subintervals?

     Resposta

     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. Cartão 182

     Pergunta

     How does reversing integral bounds change the value?

     Resposta

     It changes the sign:

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

183. Cartão 183

     Pergunta

     Net Change Theorem?

     Resposta

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

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

184. Cartão 184

     Pergunta

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

     Resposta

     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. Cartão 185

     Pergunta

     Power rule for antiderivatives?

     Resposta

     For $n\ne-1$,

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

186. Cartão 186

     Pergunta

     Midpoint Riemann sum on equal subintervals?

     Resposta

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

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

187. Cartão 187

     Pergunta

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

     Resposta

     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. Cartão 188

     Pergunta

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

     Resposta

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Cartão 189

     Pergunta

     Antiderivative of $1/x$?

     Resposta

     On any interval not crossing zero,

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

190. Cartão 190

     Pergunta

     Trapezoidal approximation on equal subintervals?

     Resposta

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

191. Cartão 191

     Pergunta

     How do geometric regions help evaluate a definite integral?

     Resposta

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

192. Cartão 192

     Pergunta

     Basic antiderivatives of sine and cosine?

     Resposta

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

193. Cartão 193

     Pergunta

     Definite integral as a limit of Riemann sums?

     Resposta

     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. Cartão 194

     Pergunta

     Constant-multiple rule for integrals?

     Resposta

     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. Cartão 195

     Pergunta

     What pattern suggests $u$-substitution?

     Resposta

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

196. Cartão 196

     Pergunta

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

     Resposta

     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. Cartão 197

     Pergunta

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

     Resposta

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

198. Cartão 198

     Pergunta

     Sum-and-difference rule for definite integrals?

     Resposta

     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. Cartão 199

     Pergunta

     Basic antiderivative of $e^x$?

     Resposta

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

200. Cartão 200

     Pergunta

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

     Resposta

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

201. Cartão 201

     Pergunta

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

     Resposta

     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. Cartão 202

     Pergunta

     How does concavity predict trapezoidal and midpoint error?

     Resposta

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

203. Cartão 203

     Pergunta

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

     Resposta

     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. Cartão 204

     Pergunta

     What denominator pattern suggests an arctangent antiderivative?

     Resposta

     After completing the square and scaling, a form like

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

205. Cartão 205

     Pergunta

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

     Resposta

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

206. Cartão 206

     Pergunta

     How does an initial condition determine an antiderivative?

     Resposta

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

207. Cartão 207

     Pergunta

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

     Resposta

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

208. Cartão 208

     Pergunta

     Why does an indefinite integral include $+C$?

     Resposta

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

209. Cartão 209

     Pergunta

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

     Resposta

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

210. Cartão 210

     Pergunta

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

     Resposta

     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. Cartão 211

     Pergunta

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

     Resposta

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

212. Cartão 212

     Pergunta

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

     Resposta

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

213. Cartão 213

     Pergunta

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

     Resposta

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

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

214. Cartão 214

     Pergunta

     Riemann sum for unequal subinterval widths?

     Resposta

     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. Cartão 215

     Pergunta

     Does continuity guarantee integrability on a closed interval?

     Resposta

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

216. Cartão 216

     Pergunta

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

     Resposta

     Where $g(x)\ne0$,

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

217. Cartão 217

     Pergunta

     What algebraic rewrites often reveal a basic antiderivative?

     Resposta

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

218. Cartão 218

     Pergunta

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

     Resposta

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

219. Cartão 219

     Pergunta

     What is a differential equation?

     Resposta

     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. Cartão 220

     Pergunta

     How does a verbal rate statement become a differential equation?

     Resposta

     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. Cartão 221

     Pergunta

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

     Resposta

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

222. Cartão 222

     Pergunta

     General solution versus particular solution?

     Resposta

     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. Cartão 223

     Pergunta

     What does one segment in a slope field show?

     Resposta

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

224. Cartão 224

     Pergunta

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

     Resposta

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

225. Cartão 225

     Pergunta

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

     Resposta

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

226. Cartão 226

     Pergunta

     What makes a first-order differential equation separable?

     Resposta

     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. Cartão 227

     Pergunta

     What is an initial value problem?

     Resposta

     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. Cartão 228

     Pergunta

     What is an isocline in a slope field?

     Resposta

     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. Cartão 229

     Pergunta

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

     Resposta

     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. Cartão 230

     Pergunta

     General solution of $dy/dt=ky$?

     Resposta

     $$y=Ce^{kt}$$

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

231. Cartão 231

     Pergunta

     Core method for solving a separable differential equation?

     Resposta

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

232. Cartão 232

     Pergunta

     How should a solution curve follow a slope field?

     Resposta

     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. Cartão 233

     Pergunta

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

     Resposta

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

234. Cartão 234

     Pergunta

     Why is one integration constant enough after integrating both sides?

     Resposta

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

235. Cartão 235

     Pergunta

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

     Resposta

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

236. Cartão 236

     Pergunta

     Can one differential equation have infinitely many solutions?

     Resposta

     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. Cartão 237

     Pergunta

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

     Resposta

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

238. Cartão 238

     Pergunta

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

     Resposta

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

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

239. Cartão 239

     Pergunta

     What can be lost when dividing to separate variables?

     Resposta

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

240. Cartão 240

     Pergunta

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

     Resposta

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

241. Cartão 241

     Pergunta

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

     Resposta

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

242. Cartão 242

     Pergunta

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

     Resposta

     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. Cartão 243

     Pergunta

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

     Resposta

     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. Cartão 244

     Pergunta

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

     Resposta

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

245. Cartão 245

     Pergunta

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

     Resposta

     For $k>0$,

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

     It is independent of the initial amount.

246. Cartão 246

     Pergunta

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

     Resposta

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

247. Cartão 247

     Pergunta

     How is an initial condition used after separation?

     Resposta

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

248. Cartão 248

     Pergunta

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

     Resposta

     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. Cartão 249

     Pergunta

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

     Resposta

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

250. Cartão 250

     Pergunta

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

     Resposta

     For $k<0$,

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

251. Cartão 251

     Pergunta

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

     Resposta

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

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

252. Cartão 252

     Pergunta

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

     Resposta

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

     Velocity below zero contributes negative displacement.

253. Cartão 253

     Pergunta

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

     Resposta

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

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

     Think top minus bottom.

254. Cartão 254

     Pergunta

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

     Resposta

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

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

255. Cartão 255

     Pergunta

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Resposta

     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. Cartão 256

     Pergunta

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

     Resposta

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

257. Cartão 257

     Pergunta

     Cross-sectional area when each slice is a square?

     Resposta

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

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

258. Cartão 258

     Pergunta

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

     Resposta

     Integrate the net rate:

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

259. Cartão 259

     Pergunta

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

     Resposta

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

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

     Think right minus left.

260. Cartão 260

     Pergunta

     Disc-method volume formula?

     Resposta

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

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

261. Cartão 261

     Pergunta

     What units does average value have?

     Resposta

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

262. Cartão 262

     Pergunta

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

     Resposta

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

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

263. Cartão 263

     Pergunta

     Cross-sectional area when each slice is a rectangle?

     Resposta

     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. Cartão 264

     Pergunta

     How do you determine bounds for area between curves?

     Resposta

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

265. Cartão 265

     Pergunta

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

     Resposta

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

266. Cartão 266

     Pergunta

     When is a particle moving to the right or left?

     Resposta

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

267. Cartão 267

     Pergunta

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

     Resposta

     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. Cartão 268

     Pergunta

     Why must an area integral be split where curves intersect?

     Resposta

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

269. Cartão 269

     Pergunta

     How can a velocity table approximate displacement?

     Resposta

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

270. Cartão 270

     Pergunta

     Washer-method volume formula?

     Resposta

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

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

271. Cartão 271

     Pergunta

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

     Resposta

     If $s(a)$ is known,

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

272. Cartão 272

     Pergunta

     How do you choose between vertical and horizontal area slices?

     Resposta

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

273. Cartão 273

     Pergunta

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

     Resposta

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

274. Cartão 274

     Pergunta

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

     Resposta

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

275. Cartão 275

     Pergunta

     Single expression for area between two curves?

     Resposta

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Cartão 276

     Pergunta

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

     Resposta

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

277. Cartão 277

     Pergunta

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

     Resposta

     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. Cartão 278

     Pergunta

     When should a volume integral use $dy$?

     Resposta

     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. Cartão 279

     Pergunta

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

     Resposta

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

280. Cartão 280

     Pergunta

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

     Resposta

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

281. Cartão 281

     Pergunta

     Why must total distance split at velocity sign changes?

     Resposta

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

282. Cartão 282

     Pergunta

     When does an accumulated quantity reach a local maximum?

     Resposta

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

283. Cartão 283

     Pergunta

     What distinguishes area from a definite integral?

     Resposta

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

284. Cartão 284

     Pergunta

     How do position, velocity, and acceleration graphs correspond?

     Resposta

     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. Cartão 285

     Pergunta

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

     Resposta

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

286. Cartão 286

     Pergunta

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

     Resposta

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

287. Cartão 287

     Pergunta

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

     Resposta

     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. Cartão 288

     Pergunta

     Why should a contextual integral answer include a sentence?

     Resposta

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