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

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1. Carta 1

   Domanda

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

   Risposta

   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. Carta 2

   Domanda

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

   Risposta

   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. Carta 3

   Domanda

   When does direct substitution evaluate a limit?

   Risposta

   When the function is continuous at the target input. Then

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

4. Carta 4

   Domanda

   Three conditions for continuity at $x=a$?

   Risposta

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

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

5. Carta 5

   Domanda

   Intermediate Value Theorem: hypotheses and conclusion?

   Risposta

   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. Carta 6

   Domanda

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

   Risposta

   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. Carta 7

   Domanda

   How do you read a finite limit from a graph?

   Risposta

   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. Carta 8

   Domanda

   Limit law for a sum or difference?

   Risposta

   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. Carta 9

   Domanda

   What makes a discontinuity removable?

   Risposta

   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. Carta 10

    Domanda

    Squeeze Theorem: usable form?

    Risposta

    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. Carta 11

    Domanda

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

    Risposta

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

12. Carta 12

    Domanda

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

    Risposta

    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. Carta 13

    Domanda

    Limit law for a product?

    Risposta

    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. Carta 14

    Domanda

    Graph signature of a jump discontinuity?

    Risposta

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

15. Carta 15

    Domanda

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

    Risposta

    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. Carta 16

    Domanda

    Horizontal asymptote from a limit at infinity?

    Risposta

    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. Carta 17

    Domanda

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

    Risposta

    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. Carta 18

    Domanda

    Limit law for a quotient—and its condition?

    Risposta

    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. Carta 19

    Domanda

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

    Risposta

    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. Carta 20

    Domanda

    When is the Squeeze Theorem a natural choice?

    Risposta

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

21. Carta 21

    Domanda

    Vertical asymptote from one-sided behavior?

    Risposta

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

22. Carta 22

    Domanda

    Limit at infinity of equal-degree rational functions?

    Risposta

    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. Carta 23

    Domanda

    When can a limit pass through a continuous outer function?

    Risposta

    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. Carta 24

    Domanda

    What makes a discontinuity infinite?

    Risposta

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

25. Carta 25

    Domanda

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

    Risposta

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

26. Carta 26

    Domanda

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

    Risposta

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Carta 27

    Domanda

    Continuity of a composition?

    Risposta

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

28. Carta 28

    Domanda

    Limit at infinity when a rational numerator has lower degree?

    Risposta

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

29. Carta 29

    Domanda

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

    Risposta

    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. Carta 30

    Domanda

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

    Risposta

    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. Carta 31

    Domanda

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

    Risposta

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

32. Carta 32

    Domanda

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

    Risposta

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

33. Carta 33

    Domanda

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

    Risposta

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

34. Carta 34

    Domanda

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

    Risposta

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

35. Carta 35

    Domanda

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

    Risposta

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

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

36. Carta 36

    Domanda

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

    Risposta

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

    The derivative exists only if this finite limit exists.

37. Carta 37

    Domanda

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

    Risposta

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

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

38. Carta 38

    Domanda

    What does differentiability imply about continuity?

    Risposta

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

39. Carta 39

    Domanda

    Power rule for derivatives?

    Risposta

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

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

40. Carta 40

    Domanda

    Units of $f'(x)$?

    Risposta

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

41. Carta 41

    Domanda

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

    Risposta

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

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

42. Carta 42

    Domanda

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

    Risposta

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

    Domanda

    Derivative of a constant?

    Risposta

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

    A constant function has zero rate of change.

44. Carta 44

    Domanda

    Derivative of $\sin x$?

    Risposta

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

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

45. Carta 45

    Domanda

    Product rule?

    Risposta

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

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

46. Carta 46

    Domanda

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

    Risposta

    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. Carta 47

    Domanda

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

    Risposta

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

48. Carta 48

    Domanda

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

    Risposta

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

49. Carta 49

    Domanda

    Derivative of a sum or difference?

    Risposta

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

50. Carta 50

    Domanda

    Derivative of $\cos x$?

    Risposta

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

    The standard formula assumes radians.

51. Carta 51

    Domanda

    Quotient rule?

    Risposta

    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. Carta 52

    Domanda

    Common notations for the first derivative?

    Risposta

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

53. Carta 53

    Domanda

    Derivative of $e^x$?

    Risposta

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

54. Carta 54

    Domanda

    What graph features can make $f$ nondifferentiable?

    Risposta

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

55. Carta 55

    Domanda

    Derivative of $\tan x$?

    Risposta

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Carta 56

    Domanda

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

    Risposta

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

57. Carta 57

    Domanda

    Derivative of $\ln x$?

    Risposta

    For $x>0$,

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

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

58. Carta 58

    Domanda

    How does the power rule handle roots or negative powers?

    Risposta

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

59. Carta 59

    Domanda

    Derivative of $\csc x$?

    Risposta

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Carta 60

    Domanda

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

    Risposta

    $f$ is increasing on that interval.

61. Carta 61

    Domanda

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

    Risposta

    For $a>0$,

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

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

62. Carta 62

    Domanda

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

    Risposta

    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. Carta 63

    Domanda

    Derivative of $\sec x$?

    Risposta

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Carta 64

    Domanda

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

    Risposta

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

65. Carta 65

    Domanda

    Derivative of $\log_a x$?

    Risposta

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

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

66. Carta 66

    Domanda

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

    Risposta

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Carta 67

    Domanda

    Derivative of $\cot x$?

    Risposta

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Carta 68

    Domanda

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

    Risposta

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

69. Carta 69

    Domanda

    Constant-multiple rule?

    Risposta

    For a constant $c$,

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

70. Carta 70

    Domanda

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

    Risposta

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

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

71. Carta 71

    Domanda

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

    Risposta

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

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

72. Carta 72

    Domanda

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

    Risposta

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

73. Carta 73

    Domanda

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

    Risposta

    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. Carta 74

    Domanda

    Derivative of an inverse function at $x$?

    Risposta

    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. Carta 75

    Domanda

    Derivative of $\arcsin x$?

    Risposta

    For $|x|<1$,

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

76. Carta 76

    Domanda

    Notation for the third derivative of $f$?

    Risposta

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

77. Carta 77

    Domanda

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

    Risposta

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

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

78. Carta 78

    Domanda

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

    Risposta

    Where $y\ne0$,

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

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

79. Carta 79

    Domanda

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

    Risposta

    Provided $f'(a)\ne0$,

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

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

80. Carta 80

    Domanda

    Derivative of $\arctan x$?

    Risposta

    For every real $x$,

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

81. Carta 81

    Domanda

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

    Risposta

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

    The extra factor is the chain rule.

82. Carta 82

    Domanda

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

    Risposta

    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. Carta 83

    Domanda

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

    Risposta

    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. Carta 84

    Domanda

    Derivative of $\arccos x$?

    Risposta

    For $|x|<1$,

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

85. Carta 85

    Domanda

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

    Risposta

    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. Carta 86

    Domanda

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

    Risposta

    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. Carta 87

    Domanda

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

    Risposta

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

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

88. Carta 88

    Domanda

    How are tangent slopes of inverse graphs related?

    Risposta

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

89. Carta 89

    Domanda

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

    Risposta

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

90. Carta 90

    Domanda

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

    Risposta

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

91. Carta 91

    Domanda

    Horizontal tangent on an implicit curve: derivative condition?

    Risposta

    $dy/dx=0$ 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. Carta 92

    Domanda

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

    Risposta

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

93. Carta 93

    Domanda

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

    Risposta

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

94. Carta 94

    Domanda

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

    Risposta

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

    This combines the power rule with the chain rule.

95. Carta 95

    Domanda

    Vertical tangent on an implicit curve: derivative clue?

    Risposta

    $dy/dx$ 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. Carta 96

    Domanda

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

    Risposta

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

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

97. Carta 97

    Domanda

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

    Risposta

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

98. Carta 98

    Domanda

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

    Risposta

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

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

99. Carta 99

    Domanda

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

    Risposta

    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. Carta 100

     Domanda

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

     Risposta

     When the functions are differentiable, the chain rule gives

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

101. Carta 101

     Domanda

     What local property lets a function have an inverse derivative?

     Risposta

     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. Carta 102

     Domanda

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

     Risposta

     For $a>0$,

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

103. Carta 103

     Domanda

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

     Risposta

     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. Carta 104

     Domanda

     Position, velocity, and acceleration relationships?

     Risposta

     For position $s(t)$,

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

105. Carta 105

     Domanda

     Central idea of a related-rates problem?

     Risposta

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

106. Carta 106

     Domanda

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

     Risposta

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

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

107. Carta 107

     Domanda

     L’Hospital’s Rule: basic conditions?

     Risposta

     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. Carta 108

     Domanda

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

     Risposta

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

109. Carta 109

     Domanda

     Speed in terms of velocity?

     Risposta

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Carta 110

     Domanda

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

     Risposta

     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. Carta 111

     Domanda

     Differential approximation connecting $dx$ and $dy$?

     Risposta

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

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

112. Carta 112

     Domanda

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

     Risposta

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

113. Carta 113

     Domanda

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

     Risposta

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

114. Carta 114

     Domanda

     What does positive acceleration say about velocity?

     Risposta

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

115. Carta 115

     Domanda

     Related rates: when should numerical values be substituted?

     Risposta

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

116. Carta 116

     Domanda

     How does concavity predict linearization error?

     Risposta

     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. Carta 117

     Domanda

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

     Risposta

     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. Carta 118

     Domanda

     When is a particle moving in the positive direction?

     Risposta

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

119. Carta 119

     Domanda

     How can velocity show a change of direction?

     Risposta

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

120. Carta 120

     Domanda

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

     Risposta

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

121. Carta 121

     Domanda

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

     Risposta

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

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

122. Carta 122

     Domanda

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

     Risposta

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

123. Carta 123

     Domanda

     What must a contextual derivative sentence include?

     Risposta

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

124. Carta 124

     Domanda

     Velocity negative and acceleration positive: what happens?

     Risposta

     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. Carta 125

     Domanda

     How should a negative related rate be interpreted?

     Risposta

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

126. Carta 126

     Domanda

     When is local linearity a sound approximation tool?

     Risposta

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

127. Carta 127

     Domanda

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

     Risposta

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

128. Carta 128

     Domanda

     When is speed increasing?

     Risposta

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

129. Carta 129

     Domanda

     Volume changes with time: notation for its rate?

     Risposta

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

130. Carta 130

     Domanda

     Why are similar triangles useful in related rates?

     Risposta

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

131. Carta 131

     Domanda

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

     Risposta

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

132. Carta 132

     Domanda

     What conclusion does L’Hospital’s Rule permit?

     Risposta

     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. Carta 133

     Domanda

     When is speed decreasing?

     Risposta

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

134. Carta 134

     Domanda

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

     Risposta

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

135. Carta 135

     Domanda

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

     Risposta

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

136. Carta 136

     Domanda

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

     Risposta

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

137. Carta 137

     Domanda

     Extreme Value Theorem: hypothesis and conclusion?

     Risposta

     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. Carta 138

     Domanda

     What is a critical number of $f$?

     Risposta

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

139. Carta 139

     Domanda

     First derivative test for a local maximum?

     Risposta

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

140. Carta 140

     Domanda

     Second-derivative sign for concave up?

     Risposta

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

141. Carta 141

     Domanda

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

     Risposta

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

142. Carta 142

     Domanda

     First step in an optimization model?

     Risposta

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

143. Carta 143

     Domanda

     Mean Value Theorem: hypotheses and conclusion?

     Risposta

     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. Carta 144

     Domanda

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

     Risposta

     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. Carta 145

     Domanda

     First derivative test for a local minimum?

     Risposta

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

146. Carta 146

     Domanda

     What must happen at an inflection point?

     Risposta

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

147. Carta 147

     Domanda

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

     Risposta

     $f''$ 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. Carta 148

     Domanda

     How do you confirm an optimization answer is absolute?

     Risposta

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

149. Carta 149

     Domanda

     Rolle’s Theorem: hypotheses and conclusion?

     Risposta

     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. Carta 150

     Domanda

     Difference between absolute and relative extrema?

     Risposta

     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. Carta 151

     Domanda

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

     Risposta

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

152. Carta 152

     Domanda

     Second derivative test for a local minimum?

     Risposta

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

153. Carta 153

     Domanda

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

     Risposta

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

154. Carta 154

     Domanda

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

     Risposta

     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. Carta 155

     Domanda

     Which theorem links an average slope to an instantaneous slope?

     Risposta

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

156. Carta 156

     Domanda

     How can an implicit derivative locate a horizontal tangent?

     Risposta

     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. Carta 157

     Domanda

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

     Risposta

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

158. Carta 158

     Domanda

     Second derivative test for a local maximum?

     Risposta

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

159. Carta 159

     Domanda

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

     Risposta

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

160. Carta 160

     Domanda

     Why must an optimization domain be stated?

     Risposta

     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. Carta 161

     Domanda

     Which theorem guarantees absolute extrema, not where they occur?

     Risposta

     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. Carta 162

     Domanda

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

     Risposta

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

163. Carta 163

     Domanda

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

     Risposta

     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. Carta 164

     Domanda

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

     Risposta

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

165. Carta 165

     Domanda

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

     Risposta

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

166. Carta 166

     Domanda

     How can an implicit derivative locate a vertical tangent?

     Risposta

     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. Carta 167

     Domanda

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

     Risposta

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

168. Carta 168

     Domanda

     Why are endpoints included in the candidates test?

     Risposta

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

169. Carta 169

     Domanda

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

     Risposta

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

170. Carta 170

     Domanda

     Second-derivative sign for concave down?

     Risposta

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

171. Carta 171

     Domanda

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

     Risposta

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

172. Carta 172

     Domanda

     What should the final line of an optimization solution state?

     Risposta

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

173. Carta 173

     Domanda

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

     Risposta

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

174. Carta 174

     Domanda

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

     Risposta

     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. Carta 175

     Domanda

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

     Risposta

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

176. Carta 176

     Domanda

     Left Riemann sum on equal subintervals?

     Risposta

     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. Carta 177

     Domanda

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

     Risposta

     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. Carta 178

     Domanda

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Risposta

     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. Carta 179

     Domanda

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

     Risposta

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Carta 180

     Domanda

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

     Risposta

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

181. Carta 181

     Domanda

     Right Riemann sum on equal subintervals?

     Risposta

     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. Carta 182

     Domanda

     How does reversing integral bounds change the value?

     Risposta

     It changes the sign:

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

183. Carta 183

     Domanda

     Net Change Theorem?

     Risposta

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

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

184. Carta 184

     Domanda

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

     Risposta

     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. Carta 185

     Domanda

     Power rule for antiderivatives?

     Risposta

     For $n\ne-1$,

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

186. Carta 186

     Domanda

     Midpoint Riemann sum on equal subintervals?

     Risposta

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

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

187. Carta 187

     Domanda

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

     Risposta

     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. Carta 188

     Domanda

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

     Risposta

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Carta 189

     Domanda

     Antiderivative of $1/x$?

     Risposta

     On any interval not crossing zero,

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

190. Carta 190

     Domanda

     Trapezoidal approximation on equal subintervals?

     Risposta

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

191. Carta 191

     Domanda

     How do geometric regions help evaluate a definite integral?

     Risposta

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

192. Carta 192

     Domanda

     Basic antiderivatives of sine and cosine?

     Risposta

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

193. Carta 193

     Domanda

     Definite integral as a limit of Riemann sums?

     Risposta

     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. Carta 194

     Domanda

     Constant-multiple rule for integrals?

     Risposta

     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. Carta 195

     Domanda

     What pattern suggests $u$-substitution?

     Risposta

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

196. Carta 196

     Domanda

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

     Risposta

     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. Carta 197

     Domanda

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

     Risposta

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

198. Carta 198

     Domanda

     Sum-and-difference rule for definite integrals?

     Risposta

     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. Carta 199

     Domanda

     Basic antiderivative of $e^x$?

     Risposta

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

200. Carta 200

     Domanda

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

     Risposta

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

201. Carta 201

     Domanda

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

     Risposta

     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. Carta 202

     Domanda

     How does concavity predict trapezoidal and midpoint error?

     Risposta

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

203. Carta 203

     Domanda

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

     Risposta

     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. Carta 204

     Domanda

     What denominator pattern suggests an arctangent antiderivative?

     Risposta

     After completing the square and scaling, a form like

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

205. Carta 205

     Domanda

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

     Risposta

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

206. Carta 206

     Domanda

     How does an initial condition determine an antiderivative?

     Risposta

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

207. Carta 207

     Domanda

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

     Risposta

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

208. Carta 208

     Domanda

     Why does an indefinite integral include $+C$?

     Risposta

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

209. Carta 209

     Domanda

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

     Risposta

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

210. Carta 210

     Domanda

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

     Risposta

     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. Carta 211

     Domanda

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

     Risposta

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

212. Carta 212

     Domanda

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

     Risposta

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

213. Carta 213

     Domanda

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

     Risposta

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

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

214. Carta 214

     Domanda

     Riemann sum for unequal subinterval widths?

     Risposta

     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. Carta 215

     Domanda

     Does continuity guarantee integrability on a closed interval?

     Risposta

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

216. Carta 216

     Domanda

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

     Risposta

     Where $g(x)\ne0$,

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

217. Carta 217

     Domanda

     What algebraic rewrites often reveal a basic antiderivative?

     Risposta

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

218. Carta 218

     Domanda

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

     Risposta

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

219. Carta 219

     Domanda

     What is a differential equation?

     Risposta

     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. Carta 220

     Domanda

     How does a verbal rate statement become a differential equation?

     Risposta

     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. Carta 221

     Domanda

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

     Risposta

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

222. Carta 222

     Domanda

     General solution versus particular solution?

     Risposta

     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. Carta 223

     Domanda

     What does one segment in a slope field show?

     Risposta

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

224. Carta 224

     Domanda

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

     Risposta

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

225. Carta 225

     Domanda

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

     Risposta

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

226. Carta 226

     Domanda

     What makes a first-order differential equation separable?

     Risposta

     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. Carta 227

     Domanda

     What is an initial value problem?

     Risposta

     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. Carta 228

     Domanda

     What is an isocline in a slope field?

     Risposta

     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. Carta 229

     Domanda

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

     Risposta

     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. Carta 230

     Domanda

     General solution of $dy/dt=ky$?

     Risposta

     $$y=Ce^{kt}$$

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

231. Carta 231

     Domanda

     Core method for solving a separable differential equation?

     Risposta

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

232. Carta 232

     Domanda

     How should a solution curve follow a slope field?

     Risposta

     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. Carta 233

     Domanda

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

     Risposta

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

234. Carta 234

     Domanda

     Why is one integration constant enough after integrating both sides?

     Risposta

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

235. Carta 235

     Domanda

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

     Risposta

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

236. Carta 236

     Domanda

     Can one differential equation have infinitely many solutions?

     Risposta

     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. Carta 237

     Domanda

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

     Risposta

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

238. Carta 238

     Domanda

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

     Risposta

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

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

239. Carta 239

     Domanda

     What can be lost when dividing to separate variables?

     Risposta

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

240. Carta 240

     Domanda

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

     Risposta

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

241. Carta 241

     Domanda

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

     Risposta

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

242. Carta 242

     Domanda

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

     Risposta

     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. Carta 243

     Domanda

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

     Risposta

     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. Carta 244

     Domanda

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

     Risposta

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

245. Carta 245

     Domanda

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

     Risposta

     For $k>0$,

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

     It is independent of the initial amount.

246. Carta 246

     Domanda

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

     Risposta

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

247. Carta 247

     Domanda

     How is an initial condition used after separation?

     Risposta

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

248. Carta 248

     Domanda

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

     Risposta

     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. Carta 249

     Domanda

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

     Risposta

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

250. Carta 250

     Domanda

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

     Risposta

     For $k<0$,

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

251. Carta 251

     Domanda

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

     Risposta

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

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

252. Carta 252

     Domanda

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

     Risposta

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

     Velocity below zero contributes negative displacement.

253. Carta 253

     Domanda

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

     Risposta

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

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

     Think top minus bottom.

254. Carta 254

     Domanda

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

     Risposta

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

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

255. Carta 255

     Domanda

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Risposta

     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. Carta 256

     Domanda

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

     Risposta

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

257. Carta 257

     Domanda

     Cross-sectional area when each slice is a square?

     Risposta

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

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

258. Carta 258

     Domanda

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

     Risposta

     Integrate the net rate:

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

259. Carta 259

     Domanda

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

     Risposta

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

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

     Think right minus left.

260. Carta 260

     Domanda

     Disc-method volume formula?

     Risposta

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

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

261. Carta 261

     Domanda

     What units does average value have?

     Risposta

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

262. Carta 262

     Domanda

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

     Risposta

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

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

263. Carta 263

     Domanda

     Cross-sectional area when each slice is a rectangle?

     Risposta

     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. Carta 264

     Domanda

     How do you determine bounds for area between curves?

     Risposta

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

265. Carta 265

     Domanda

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

     Risposta

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

266. Carta 266

     Domanda

     When is a particle moving to the right or left?

     Risposta

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

267. Carta 267

     Domanda

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

     Risposta

     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. Carta 268

     Domanda

     Why must an area integral be split where curves intersect?

     Risposta

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

269. Carta 269

     Domanda

     How can a velocity table approximate displacement?

     Risposta

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

270. Carta 270

     Domanda

     Washer-method volume formula?

     Risposta

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

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

271. Carta 271

     Domanda

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

     Risposta

     If $s(a)$ is known,

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

272. Carta 272

     Domanda

     How do you choose between vertical and horizontal area slices?

     Risposta

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

273. Carta 273

     Domanda

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

     Risposta

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

274. Carta 274

     Domanda

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

     Risposta

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

275. Carta 275

     Domanda

     Single expression for area between two curves?

     Risposta

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Carta 276

     Domanda

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

     Risposta

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

277. Carta 277

     Domanda

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

     Risposta

     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. Carta 278

     Domanda

     When should a volume integral use $dy$?

     Risposta

     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. Carta 279

     Domanda

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

     Risposta

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

280. Carta 280

     Domanda

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

     Risposta

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

281. Carta 281

     Domanda

     Why must total distance split at velocity sign changes?

     Risposta

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

282. Carta 282

     Domanda

     When does an accumulated quantity reach a local maximum?

     Risposta

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

283. Carta 283

     Domanda

     What distinguishes area from a definite integral?

     Risposta

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

284. Carta 284

     Domanda

     How do position, velocity, and acceleration graphs correspond?

     Risposta

     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. Carta 285

     Domanda

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

     Risposta

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

286. Carta 286

     Domanda

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

     Risposta

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

287. Carta 287

     Domanda

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

     Risposta

     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. Carta 288

     Domanda

     Why should a contextual integral answer include a sentence?

     Risposta

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