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

Tarjetas de este mazo

1. Tarjeta 1

   Pregunta

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

   Respuesta

   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. Tarjeta 2

   Pregunta

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

   Respuesta

   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. Tarjeta 3

   Pregunta

   When does direct substitution evaluate a limit?

   Respuesta

   When the function is continuous at the target input. Then

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

4. Tarjeta 4

   Pregunta

   Three conditions for continuity at $x=a$?

   Respuesta

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

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

5. Tarjeta 5

   Pregunta

   Intermediate Value Theorem: hypotheses and conclusion?

   Respuesta

   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. Tarjeta 6

   Pregunta

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

   Respuesta

   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. Tarjeta 7

   Pregunta

   How do you read a finite limit from a graph?

   Respuesta

   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. Tarjeta 8

   Pregunta

   Limit law for a sum or difference?

   Respuesta

   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. Tarjeta 9

   Pregunta

   What makes a discontinuity removable?

   Respuesta

   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. Tarjeta 10

    Pregunta

    Squeeze Theorem: usable form?

    Respuesta

    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. Tarjeta 11

    Pregunta

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

    Respuesta

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

12. Tarjeta 12

    Pregunta

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

    Respuesta

    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. Tarjeta 13

    Pregunta

    Limit law for a product?

    Respuesta

    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. Tarjeta 14

    Pregunta

    Graph signature of a jump discontinuity?

    Respuesta

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

15. Tarjeta 15

    Pregunta

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

    Respuesta

    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. Tarjeta 16

    Pregunta

    Horizontal asymptote from a limit at infinity?

    Respuesta

    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. Tarjeta 17

    Pregunta

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

    Respuesta

    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. Tarjeta 18

    Pregunta

    Limit law for a quotient—and its condition?

    Respuesta

    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. Tarjeta 19

    Pregunta

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

    Respuesta

    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. Tarjeta 20

    Pregunta

    When is the Squeeze Theorem a natural choice?

    Respuesta

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

21. Tarjeta 21

    Pregunta

    Vertical asymptote from one-sided behavior?

    Respuesta

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

22. Tarjeta 22

    Pregunta

    Limit at infinity of equal-degree rational functions?

    Respuesta

    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. Tarjeta 23

    Pregunta

    When can a limit pass through a continuous outer function?

    Respuesta

    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. Tarjeta 24

    Pregunta

    What makes a discontinuity infinite?

    Respuesta

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

25. Tarjeta 25

    Pregunta

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

    Respuesta

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

26. Tarjeta 26

    Pregunta

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

    Respuesta

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Tarjeta 27

    Pregunta

    Continuity of a composition?

    Respuesta

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

28. Tarjeta 28

    Pregunta

    Limit at infinity when a rational numerator has lower degree?

    Respuesta

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

29. Tarjeta 29

    Pregunta

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

    Respuesta

    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. Tarjeta 30

    Pregunta

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

    Respuesta

    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. Tarjeta 31

    Pregunta

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

    Respuesta

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

32. Tarjeta 32

    Pregunta

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

    Respuesta

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

33. Tarjeta 33

    Pregunta

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

    Respuesta

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

34. Tarjeta 34

    Pregunta

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

    Respuesta

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

35. Tarjeta 35

    Pregunta

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

    Respuesta

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

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

36. Tarjeta 36

    Pregunta

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

    Respuesta

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

    The derivative exists only if this finite limit exists.

37. Tarjeta 37

    Pregunta

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

    Respuesta

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

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

38. Tarjeta 38

    Pregunta

    What does differentiability imply about continuity?

    Respuesta

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

39. Tarjeta 39

    Pregunta

    Power rule for derivatives?

    Respuesta

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

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

40. Tarjeta 40

    Pregunta

    Units of $f'(x)$?

    Respuesta

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

41. Tarjeta 41

    Pregunta

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

    Respuesta

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

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

42. Tarjeta 42

    Pregunta

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

    Respuesta

    $f'(x)>0$ 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. Tarjeta 43

    Pregunta

    Derivative of a constant?

    Respuesta

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

    A constant function has zero rate of change.

44. Tarjeta 44

    Pregunta

    Derivative of $\sin x$?

    Respuesta

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

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

45. Tarjeta 45

    Pregunta

    Product rule?

    Respuesta

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

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

46. Tarjeta 46

    Pregunta

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

    Respuesta

    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. Tarjeta 47

    Pregunta

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

    Respuesta

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

48. Tarjeta 48

    Pregunta

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

    Respuesta

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

49. Tarjeta 49

    Pregunta

    Derivative of a sum or difference?

    Respuesta

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

50. Tarjeta 50

    Pregunta

    Derivative of $\cos x$?

    Respuesta

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

    The standard formula assumes radians.

51. Tarjeta 51

    Pregunta

    Quotient rule?

    Respuesta

    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. Tarjeta 52

    Pregunta

    Common notations for the first derivative?

    Respuesta

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

53. Tarjeta 53

    Pregunta

    Derivative of $e^x$?

    Respuesta

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

54. Tarjeta 54

    Pregunta

    What graph features can make $f$ nondifferentiable?

    Respuesta

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

55. Tarjeta 55

    Pregunta

    Derivative of $\tan x$?

    Respuesta

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Tarjeta 56

    Pregunta

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

    Respuesta

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

57. Tarjeta 57

    Pregunta

    Derivative of $\ln x$?

    Respuesta

    For $x>0$,

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

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

58. Tarjeta 58

    Pregunta

    How does the power rule handle roots or negative powers?

    Respuesta

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

59. Tarjeta 59

    Pregunta

    Derivative of $\csc x$?

    Respuesta

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Tarjeta 60

    Pregunta

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

    Respuesta

    $f$ is increasing on that interval.

61. Tarjeta 61

    Pregunta

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

    Respuesta

    For $a>0$,

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

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

62. Tarjeta 62

    Pregunta

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

    Respuesta

    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. Tarjeta 63

    Pregunta

    Derivative of $\sec x$?

    Respuesta

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Tarjeta 64

    Pregunta

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

    Respuesta

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

65. Tarjeta 65

    Pregunta

    Derivative of $\log_a x$?

    Respuesta

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

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

66. Tarjeta 66

    Pregunta

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

    Respuesta

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Tarjeta 67

    Pregunta

    Derivative of $\cot x$?

    Respuesta

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Tarjeta 68

    Pregunta

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

    Respuesta

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

69. Tarjeta 69

    Pregunta

    Constant-multiple rule?

    Respuesta

    For a constant $c$,

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

70. Tarjeta 70

    Pregunta

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

    Respuesta

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

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

71. Tarjeta 71

    Pregunta

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

    Respuesta

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

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

72. Tarjeta 72

    Pregunta

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

    Respuesta

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

73. Tarjeta 73

    Pregunta

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

    Respuesta

    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. Tarjeta 74

    Pregunta

    Derivative of an inverse function at $x$?

    Respuesta

    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. Tarjeta 75

    Pregunta

    Derivative of $\arcsin x$?

    Respuesta

    For $|x|<1$,

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

76. Tarjeta 76

    Pregunta

    Notation for the third derivative of $f$?

    Respuesta

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

77. Tarjeta 77

    Pregunta

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

    Respuesta

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

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

78. Tarjeta 78

    Pregunta

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

    Respuesta

    Where $y\ne0$,

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

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

79. Tarjeta 79

    Pregunta

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

    Respuesta

    Provided $f'(a)\ne0$,

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

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

80. Tarjeta 80

    Pregunta

    Derivative of $\arctan x$?

    Respuesta

    For every real $x$,

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

81. Tarjeta 81

    Pregunta

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

    Respuesta

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

    The extra factor is the chain rule.

82. Tarjeta 82

    Pregunta

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

    Respuesta

    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. Tarjeta 83

    Pregunta

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

    Respuesta

    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. Tarjeta 84

    Pregunta

    Derivative of $\arccos x$?

    Respuesta

    For $|x|<1$,

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

85. Tarjeta 85

    Pregunta

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

    Respuesta

    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. Tarjeta 86

    Pregunta

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

    Respuesta

    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. Tarjeta 87

    Pregunta

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

    Respuesta

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

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

88. Tarjeta 88

    Pregunta

    How are tangent slopes of inverse graphs related?

    Respuesta

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

89. Tarjeta 89

    Pregunta

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

    Respuesta

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

90. Tarjeta 90

    Pregunta

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

    Respuesta

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

91. Tarjeta 91

    Pregunta

    Horizontal tangent on an implicit curve: derivative condition?

    Respuesta

    $dy/dx=0$ 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. Tarjeta 92

    Pregunta

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

    Respuesta

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

93. Tarjeta 93

    Pregunta

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

    Respuesta

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

94. Tarjeta 94

    Pregunta

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

    Respuesta

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

    This combines the power rule with the chain rule.

95. Tarjeta 95

    Pregunta

    Vertical tangent on an implicit curve: derivative clue?

    Respuesta

    $dy/dx$ 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. Tarjeta 96

    Pregunta

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

    Respuesta

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

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

97. Tarjeta 97

    Pregunta

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

    Respuesta

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

98. Tarjeta 98

    Pregunta

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

    Respuesta

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

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

99. Tarjeta 99

    Pregunta

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

    Respuesta

    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. Tarjeta 100

     Pregunta

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

     Respuesta

     When the functions are differentiable, the chain rule gives

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

101. Tarjeta 101

     Pregunta

     What local property lets a function have an inverse derivative?

     Respuesta

     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. Tarjeta 102

     Pregunta

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

     Respuesta

     For $a>0$,

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

103. Tarjeta 103

     Pregunta

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

     Respuesta

     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. Tarjeta 104

     Pregunta

     Position, velocity, and acceleration relationships?

     Respuesta

     For position $s(t)$,

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

105. Tarjeta 105

     Pregunta

     Central idea of a related-rates problem?

     Respuesta

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

106. Tarjeta 106

     Pregunta

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

     Respuesta

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

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

107. Tarjeta 107

     Pregunta

     L’Hospital’s Rule: basic conditions?

     Respuesta

     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. Tarjeta 108

     Pregunta

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

     Respuesta

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

109. Tarjeta 109

     Pregunta

     Speed in terms of velocity?

     Respuesta

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Tarjeta 110

     Pregunta

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

     Respuesta

     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. Tarjeta 111

     Pregunta

     Differential approximation connecting $dx$ and $dy$?

     Respuesta

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

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

112. Tarjeta 112

     Pregunta

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

     Respuesta

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

113. Tarjeta 113

     Pregunta

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

     Respuesta

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

114. Tarjeta 114

     Pregunta

     What does positive acceleration say about velocity?

     Respuesta

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

115. Tarjeta 115

     Pregunta

     Related rates: when should numerical values be substituted?

     Respuesta

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

116. Tarjeta 116

     Pregunta

     How does concavity predict linearization error?

     Respuesta

     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. Tarjeta 117

     Pregunta

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

     Respuesta

     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. Tarjeta 118

     Pregunta

     When is a particle moving in the positive direction?

     Respuesta

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

119. Tarjeta 119

     Pregunta

     How can velocity show a change of direction?

     Respuesta

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

120. Tarjeta 120

     Pregunta

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

     Respuesta

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

121. Tarjeta 121

     Pregunta

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

     Respuesta

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

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

122. Tarjeta 122

     Pregunta

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

     Respuesta

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

123. Tarjeta 123

     Pregunta

     What must a contextual derivative sentence include?

     Respuesta

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

124. Tarjeta 124

     Pregunta

     Velocity negative and acceleration positive: what happens?

     Respuesta

     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. Tarjeta 125

     Pregunta

     How should a negative related rate be interpreted?

     Respuesta

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

126. Tarjeta 126

     Pregunta

     When is local linearity a sound approximation tool?

     Respuesta

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

127. Tarjeta 127

     Pregunta

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

     Respuesta

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

128. Tarjeta 128

     Pregunta

     When is speed increasing?

     Respuesta

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

129. Tarjeta 129

     Pregunta

     Volume changes with time: notation for its rate?

     Respuesta

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

130. Tarjeta 130

     Pregunta

     Why are similar triangles useful in related rates?

     Respuesta

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

131. Tarjeta 131

     Pregunta

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

     Respuesta

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

132. Tarjeta 132

     Pregunta

     What conclusion does L’Hospital’s Rule permit?

     Respuesta

     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. Tarjeta 133

     Pregunta

     When is speed decreasing?

     Respuesta

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

134. Tarjeta 134

     Pregunta

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

     Respuesta

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

135. Tarjeta 135

     Pregunta

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

     Respuesta

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

136. Tarjeta 136

     Pregunta

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

     Respuesta

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

137. Tarjeta 137

     Pregunta

     Extreme Value Theorem: hypothesis and conclusion?

     Respuesta

     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. Tarjeta 138

     Pregunta

     What is a critical number of $f$?

     Respuesta

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

139. Tarjeta 139

     Pregunta

     First derivative test for a local maximum?

     Respuesta

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

140. Tarjeta 140

     Pregunta

     Second-derivative sign for concave up?

     Respuesta

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

141. Tarjeta 141

     Pregunta

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

     Respuesta

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

142. Tarjeta 142

     Pregunta

     First step in an optimization model?

     Respuesta

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

143. Tarjeta 143

     Pregunta

     Mean Value Theorem: hypotheses and conclusion?

     Respuesta

     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. Tarjeta 144

     Pregunta

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

     Respuesta

     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. Tarjeta 145

     Pregunta

     First derivative test for a local minimum?

     Respuesta

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

146. Tarjeta 146

     Pregunta

     What must happen at an inflection point?

     Respuesta

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

147. Tarjeta 147

     Pregunta

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

     Respuesta

     $f''$ 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. Tarjeta 148

     Pregunta

     How do you confirm an optimization answer is absolute?

     Respuesta

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

149. Tarjeta 149

     Pregunta

     Rolle’s Theorem: hypotheses and conclusion?

     Respuesta

     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. Tarjeta 150

     Pregunta

     Difference between absolute and relative extrema?

     Respuesta

     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. Tarjeta 151

     Pregunta

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

     Respuesta

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

152. Tarjeta 152

     Pregunta

     Second derivative test for a local minimum?

     Respuesta

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

153. Tarjeta 153

     Pregunta

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

     Respuesta

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

154. Tarjeta 154

     Pregunta

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

     Respuesta

     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. Tarjeta 155

     Pregunta

     Which theorem links an average slope to an instantaneous slope?

     Respuesta

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

156. Tarjeta 156

     Pregunta

     How can an implicit derivative locate a horizontal tangent?

     Respuesta

     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. Tarjeta 157

     Pregunta

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

     Respuesta

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

158. Tarjeta 158

     Pregunta

     Second derivative test for a local maximum?

     Respuesta

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

159. Tarjeta 159

     Pregunta

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

     Respuesta

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

160. Tarjeta 160

     Pregunta

     Why must an optimization domain be stated?

     Respuesta

     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. Tarjeta 161

     Pregunta

     Which theorem guarantees absolute extrema, not where they occur?

     Respuesta

     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. Tarjeta 162

     Pregunta

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

     Respuesta

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

163. Tarjeta 163

     Pregunta

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

     Respuesta

     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. Tarjeta 164

     Pregunta

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

     Respuesta

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

165. Tarjeta 165

     Pregunta

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

     Respuesta

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

166. Tarjeta 166

     Pregunta

     How can an implicit derivative locate a vertical tangent?

     Respuesta

     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. Tarjeta 167

     Pregunta

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

     Respuesta

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

168. Tarjeta 168

     Pregunta

     Why are endpoints included in the candidates test?

     Respuesta

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

169. Tarjeta 169

     Pregunta

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

     Respuesta

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

170. Tarjeta 170

     Pregunta

     Second-derivative sign for concave down?

     Respuesta

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

171. Tarjeta 171

     Pregunta

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

     Respuesta

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

172. Tarjeta 172

     Pregunta

     What should the final line of an optimization solution state?

     Respuesta

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

173. Tarjeta 173

     Pregunta

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

     Respuesta

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

174. Tarjeta 174

     Pregunta

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

     Respuesta

     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. Tarjeta 175

     Pregunta

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

     Respuesta

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

176. Tarjeta 176

     Pregunta

     Left Riemann sum on equal subintervals?

     Respuesta

     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. Tarjeta 177

     Pregunta

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

     Respuesta

     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. Tarjeta 178

     Pregunta

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Respuesta

     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. Tarjeta 179

     Pregunta

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

     Respuesta

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Tarjeta 180

     Pregunta

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

     Respuesta

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

181. Tarjeta 181

     Pregunta

     Right Riemann sum on equal subintervals?

     Respuesta

     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. Tarjeta 182

     Pregunta

     How does reversing integral bounds change the value?

     Respuesta

     It changes the sign:

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

183. Tarjeta 183

     Pregunta

     Net Change Theorem?

     Respuesta

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

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

184. Tarjeta 184

     Pregunta

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

     Respuesta

     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. Tarjeta 185

     Pregunta

     Power rule for antiderivatives?

     Respuesta

     For $n\ne-1$,

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

186. Tarjeta 186

     Pregunta

     Midpoint Riemann sum on equal subintervals?

     Respuesta

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

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

187. Tarjeta 187

     Pregunta

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

     Respuesta

     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. Tarjeta 188

     Pregunta

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

     Respuesta

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Tarjeta 189

     Pregunta

     Antiderivative of $1/x$?

     Respuesta

     On any interval not crossing zero,

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

190. Tarjeta 190

     Pregunta

     Trapezoidal approximation on equal subintervals?

     Respuesta

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

191. Tarjeta 191

     Pregunta

     How do geometric regions help evaluate a definite integral?

     Respuesta

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

192. Tarjeta 192

     Pregunta

     Basic antiderivatives of sine and cosine?

     Respuesta

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

193. Tarjeta 193

     Pregunta

     Definite integral as a limit of Riemann sums?

     Respuesta

     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. Tarjeta 194

     Pregunta

     Constant-multiple rule for integrals?

     Respuesta

     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. Tarjeta 195

     Pregunta

     What pattern suggests $u$-substitution?

     Respuesta

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

196. Tarjeta 196

     Pregunta

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

     Respuesta

     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. Tarjeta 197

     Pregunta

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

     Respuesta

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

198. Tarjeta 198

     Pregunta

     Sum-and-difference rule for definite integrals?

     Respuesta

     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. Tarjeta 199

     Pregunta

     Basic antiderivative of $e^x$?

     Respuesta

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

200. Tarjeta 200

     Pregunta

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

     Respuesta

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

201. Tarjeta 201

     Pregunta

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

     Respuesta

     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. Tarjeta 202

     Pregunta

     How does concavity predict trapezoidal and midpoint error?

     Respuesta

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

203. Tarjeta 203

     Pregunta

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

     Respuesta

     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. Tarjeta 204

     Pregunta

     What denominator pattern suggests an arctangent antiderivative?

     Respuesta

     After completing the square and scaling, a form like

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

205. Tarjeta 205

     Pregunta

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

     Respuesta

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

206. Tarjeta 206

     Pregunta

     How does an initial condition determine an antiderivative?

     Respuesta

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

207. Tarjeta 207

     Pregunta

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

     Respuesta

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

208. Tarjeta 208

     Pregunta

     Why does an indefinite integral include $+C$?

     Respuesta

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

209. Tarjeta 209

     Pregunta

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

     Respuesta

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

210. Tarjeta 210

     Pregunta

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

     Respuesta

     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. Tarjeta 211

     Pregunta

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

     Respuesta

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

212. Tarjeta 212

     Pregunta

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

     Respuesta

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

213. Tarjeta 213

     Pregunta

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

     Respuesta

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

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

214. Tarjeta 214

     Pregunta

     Riemann sum for unequal subinterval widths?

     Respuesta

     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. Tarjeta 215

     Pregunta

     Does continuity guarantee integrability on a closed interval?

     Respuesta

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

216. Tarjeta 216

     Pregunta

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

     Respuesta

     Where $g(x)\ne0$,

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

217. Tarjeta 217

     Pregunta

     What algebraic rewrites often reveal a basic antiderivative?

     Respuesta

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

218. Tarjeta 218

     Pregunta

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

     Respuesta

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

219. Tarjeta 219

     Pregunta

     What is a differential equation?

     Respuesta

     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. Tarjeta 220

     Pregunta

     How does a verbal rate statement become a differential equation?

     Respuesta

     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. Tarjeta 221

     Pregunta

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

     Respuesta

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

222. Tarjeta 222

     Pregunta

     General solution versus particular solution?

     Respuesta

     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. Tarjeta 223

     Pregunta

     What does one segment in a slope field show?

     Respuesta

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

224. Tarjeta 224

     Pregunta

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

     Respuesta

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

225. Tarjeta 225

     Pregunta

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

     Respuesta

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

226. Tarjeta 226

     Pregunta

     What makes a first-order differential equation separable?

     Respuesta

     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. Tarjeta 227

     Pregunta

     What is an initial value problem?

     Respuesta

     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. Tarjeta 228

     Pregunta

     What is an isocline in a slope field?

     Respuesta

     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. Tarjeta 229

     Pregunta

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

     Respuesta

     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. Tarjeta 230

     Pregunta

     General solution of $dy/dt=ky$?

     Respuesta

     $$y=Ce^{kt}$$

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

231. Tarjeta 231

     Pregunta

     Core method for solving a separable differential equation?

     Respuesta

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

232. Tarjeta 232

     Pregunta

     How should a solution curve follow a slope field?

     Respuesta

     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. Tarjeta 233

     Pregunta

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

     Respuesta

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

234. Tarjeta 234

     Pregunta

     Why is one integration constant enough after integrating both sides?

     Respuesta

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

235. Tarjeta 235

     Pregunta

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

     Respuesta

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

236. Tarjeta 236

     Pregunta

     Can one differential equation have infinitely many solutions?

     Respuesta

     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. Tarjeta 237

     Pregunta

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

     Respuesta

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

238. Tarjeta 238

     Pregunta

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

     Respuesta

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

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

239. Tarjeta 239

     Pregunta

     What can be lost when dividing to separate variables?

     Respuesta

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

240. Tarjeta 240

     Pregunta

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

     Respuesta

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

241. Tarjeta 241

     Pregunta

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

     Respuesta

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

242. Tarjeta 242

     Pregunta

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

     Respuesta

     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. Tarjeta 243

     Pregunta

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

     Respuesta

     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. Tarjeta 244

     Pregunta

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

     Respuesta

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

245. Tarjeta 245

     Pregunta

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

     Respuesta

     For $k>0$,

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

     It is independent of the initial amount.

246. Tarjeta 246

     Pregunta

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

     Respuesta

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

247. Tarjeta 247

     Pregunta

     How is an initial condition used after separation?

     Respuesta

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

248. Tarjeta 248

     Pregunta

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

     Respuesta

     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. Tarjeta 249

     Pregunta

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

     Respuesta

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

250. Tarjeta 250

     Pregunta

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

     Respuesta

     For $k<0$,

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

251. Tarjeta 251

     Pregunta

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

     Respuesta

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

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

252. Tarjeta 252

     Pregunta

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

     Respuesta

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

     Velocity below zero contributes negative displacement.

253. Tarjeta 253

     Pregunta

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

     Respuesta

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

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

     Think top minus bottom.

254. Tarjeta 254

     Pregunta

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

     Respuesta

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

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

255. Tarjeta 255

     Pregunta

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Respuesta

     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. Tarjeta 256

     Pregunta

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

     Respuesta

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

257. Tarjeta 257

     Pregunta

     Cross-sectional area when each slice is a square?

     Respuesta

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

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

258. Tarjeta 258

     Pregunta

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

     Respuesta

     Integrate the net rate:

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

259. Tarjeta 259

     Pregunta

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

     Respuesta

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

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

     Think right minus left.

260. Tarjeta 260

     Pregunta

     Disc-method volume formula?

     Respuesta

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

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

261. Tarjeta 261

     Pregunta

     What units does average value have?

     Respuesta

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

262. Tarjeta 262

     Pregunta

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

     Respuesta

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

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

263. Tarjeta 263

     Pregunta

     Cross-sectional area when each slice is a rectangle?

     Respuesta

     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. Tarjeta 264

     Pregunta

     How do you determine bounds for area between curves?

     Respuesta

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

265. Tarjeta 265

     Pregunta

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

     Respuesta

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

266. Tarjeta 266

     Pregunta

     When is a particle moving to the right or left?

     Respuesta

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

267. Tarjeta 267

     Pregunta

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

     Respuesta

     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. Tarjeta 268

     Pregunta

     Why must an area integral be split where curves intersect?

     Respuesta

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

269. Tarjeta 269

     Pregunta

     How can a velocity table approximate displacement?

     Respuesta

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

270. Tarjeta 270

     Pregunta

     Washer-method volume formula?

     Respuesta

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

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

271. Tarjeta 271

     Pregunta

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

     Respuesta

     If $s(a)$ is known,

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

272. Tarjeta 272

     Pregunta

     How do you choose between vertical and horizontal area slices?

     Respuesta

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

273. Tarjeta 273

     Pregunta

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

     Respuesta

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

274. Tarjeta 274

     Pregunta

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

     Respuesta

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

275. Tarjeta 275

     Pregunta

     Single expression for area between two curves?

     Respuesta

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Tarjeta 276

     Pregunta

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

     Respuesta

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

277. Tarjeta 277

     Pregunta

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

     Respuesta

     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. Tarjeta 278

     Pregunta

     When should a volume integral use $dy$?

     Respuesta

     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. Tarjeta 279

     Pregunta

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

     Respuesta

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

280. Tarjeta 280

     Pregunta

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

     Respuesta

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

281. Tarjeta 281

     Pregunta

     Why must total distance split at velocity sign changes?

     Respuesta

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

282. Tarjeta 282

     Pregunta

     When does an accumulated quantity reach a local maximum?

     Respuesta

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

283. Tarjeta 283

     Pregunta

     What distinguishes area from a definite integral?

     Respuesta

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

284. Tarjeta 284

     Pregunta

     How do position, velocity, and acceleration graphs correspond?

     Respuesta

     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. Tarjeta 285

     Pregunta

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

     Respuesta

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

286. Tarjeta 286

     Pregunta

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

     Respuesta

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

287. Tarjeta 287

     Pregunta

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

     Respuesta

     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. Tarjeta 288

     Pregunta

     Why should a contextual integral answer include a sentence?

     Respuesta

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