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

Cartes de ce paquet

1. Carte 1

   Question

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

   Réponse

   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. Carte 2

   Question

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

   Réponse

   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. Carte 3

   Question

   When does direct substitution evaluate a limit?

   Réponse

   When the function is continuous at the target input. Then

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

4. Carte 4

   Question

   Three conditions for continuity at $x=a$?

   Réponse

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

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

5. Carte 5

   Question

   Intermediate Value Theorem: hypotheses and conclusion?

   Réponse

   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. Carte 6

   Question

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

   Réponse

   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. Carte 7

   Question

   How do you read a finite limit from a graph?

   Réponse

   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. Carte 8

   Question

   Limit law for a sum or difference?

   Réponse

   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. Carte 9

   Question

   What makes a discontinuity removable?

   Réponse

   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. Carte 10

    Question

    Squeeze Theorem: usable form?

    Réponse

    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. Carte 11

    Question

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

    Réponse

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

12. Carte 12

    Question

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

    Réponse

    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. Carte 13

    Question

    Limit law for a product?

    Réponse

    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. Carte 14

    Question

    Graph signature of a jump discontinuity?

    Réponse

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

15. Carte 15

    Question

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

    Réponse

    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. Carte 16

    Question

    Horizontal asymptote from a limit at infinity?

    Réponse

    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. Carte 17

    Question

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

    Réponse

    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. Carte 18

    Question

    Limit law for a quotient—and its condition?

    Réponse

    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. Carte 19

    Question

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

    Réponse

    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. Carte 20

    Question

    When is the Squeeze Theorem a natural choice?

    Réponse

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

21. Carte 21

    Question

    Vertical asymptote from one-sided behavior?

    Réponse

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

22. Carte 22

    Question

    Limit at infinity of equal-degree rational functions?

    Réponse

    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. Carte 23

    Question

    When can a limit pass through a continuous outer function?

    Réponse

    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. Carte 24

    Question

    What makes a discontinuity infinite?

    Réponse

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

25. Carte 25

    Question

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

    Réponse

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

26. Carte 26

    Question

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

    Réponse

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Carte 27

    Question

    Continuity of a composition?

    Réponse

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

28. Carte 28

    Question

    Limit at infinity when a rational numerator has lower degree?

    Réponse

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

29. Carte 29

    Question

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

    Réponse

    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. Carte 30

    Question

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

    Réponse

    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. Carte 31

    Question

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

    Réponse

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

32. Carte 32

    Question

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

    Réponse

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

33. Carte 33

    Question

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

    Réponse

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

34. Carte 34

    Question

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

    Réponse

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

35. Carte 35

    Question

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

    Réponse

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

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

36. Carte 36

    Question

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

    Réponse

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

    The derivative exists only if this finite limit exists.

37. Carte 37

    Question

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

    Réponse

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

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

38. Carte 38

    Question

    What does differentiability imply about continuity?

    Réponse

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

39. Carte 39

    Question

    Power rule for derivatives?

    Réponse

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

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

40. Carte 40

    Question

    Units of $f'(x)$?

    Réponse

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

41. Carte 41

    Question

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

    Réponse

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

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

42. Carte 42

    Question

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

    Réponse

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

    Question

    Derivative of a constant?

    Réponse

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

    A constant function has zero rate of change.

44. Carte 44

    Question

    Derivative of $\sin x$?

    Réponse

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

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

45. Carte 45

    Question

    Product rule?

    Réponse

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

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

46. Carte 46

    Question

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

    Réponse

    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. Carte 47

    Question

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

    Réponse

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

48. Carte 48

    Question

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

    Réponse

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

49. Carte 49

    Question

    Derivative of a sum or difference?

    Réponse

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

50. Carte 50

    Question

    Derivative of $\cos x$?

    Réponse

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

    The standard formula assumes radians.

51. Carte 51

    Question

    Quotient rule?

    Réponse

    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. Carte 52

    Question

    Common notations for the first derivative?

    Réponse

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

53. Carte 53

    Question

    Derivative of $e^x$?

    Réponse

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

54. Carte 54

    Question

    What graph features can make $f$ nondifferentiable?

    Réponse

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

55. Carte 55

    Question

    Derivative of $\tan x$?

    Réponse

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Carte 56

    Question

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

    Réponse

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

57. Carte 57

    Question

    Derivative of $\ln x$?

    Réponse

    For $x>0$,

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

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

58. Carte 58

    Question

    How does the power rule handle roots or negative powers?

    Réponse

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

59. Carte 59

    Question

    Derivative of $\csc x$?

    Réponse

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Carte 60

    Question

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

    Réponse

    $f$ is increasing on that interval.

61. Carte 61

    Question

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

    Réponse

    For $a>0$,

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

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

62. Carte 62

    Question

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

    Réponse

    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. Carte 63

    Question

    Derivative of $\sec x$?

    Réponse

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Carte 64

    Question

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

    Réponse

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

65. Carte 65

    Question

    Derivative of $\log_a x$?

    Réponse

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

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

66. Carte 66

    Question

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

    Réponse

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Carte 67

    Question

    Derivative of $\cot x$?

    Réponse

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Carte 68

    Question

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

    Réponse

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

69. Carte 69

    Question

    Constant-multiple rule?

    Réponse

    For a constant $c$,

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

70. Carte 70

    Question

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

    Réponse

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

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

71. Carte 71

    Question

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

    Réponse

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

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

72. Carte 72

    Question

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

    Réponse

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

73. Carte 73

    Question

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

    Réponse

    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. Carte 74

    Question

    Derivative of an inverse function at $x$?

    Réponse

    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. Carte 75

    Question

    Derivative of $\arcsin x$?

    Réponse

    For $|x|<1$,

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

76. Carte 76

    Question

    Notation for the third derivative of $f$?

    Réponse

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

77. Carte 77

    Question

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

    Réponse

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

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

78. Carte 78

    Question

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

    Réponse

    Where $y\ne0$,

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

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

79. Carte 79

    Question

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

    Réponse

    Provided $f'(a)\ne0$,

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

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

80. Carte 80

    Question

    Derivative of $\arctan x$?

    Réponse

    For every real $x$,

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

81. Carte 81

    Question

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

    Réponse

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

    The extra factor is the chain rule.

82. Carte 82

    Question

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

    Réponse

    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. Carte 83

    Question

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

    Réponse

    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. Carte 84

    Question

    Derivative of $\arccos x$?

    Réponse

    For $|x|<1$,

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

85. Carte 85

    Question

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

    Réponse

    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. Carte 86

    Question

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

    Réponse

    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. Carte 87

    Question

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

    Réponse

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

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

88. Carte 88

    Question

    How are tangent slopes of inverse graphs related?

    Réponse

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

89. Carte 89

    Question

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

    Réponse

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

90. Carte 90

    Question

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

    Réponse

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

91. Carte 91

    Question

    Horizontal tangent on an implicit curve: derivative condition?

    Réponse

    $dy/dx=0$ 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. Carte 92

    Question

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

    Réponse

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

93. Carte 93

    Question

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

    Réponse

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

94. Carte 94

    Question

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

    Réponse

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

    This combines the power rule with the chain rule.

95. Carte 95

    Question

    Vertical tangent on an implicit curve: derivative clue?

    Réponse

    $dy/dx$ 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. Carte 96

    Question

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

    Réponse

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

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

97. Carte 97

    Question

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

    Réponse

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

98. Carte 98

    Question

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

    Réponse

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

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

99. Carte 99

    Question

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

    Réponse

    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. Carte 100

     Question

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

     Réponse

     When the functions are differentiable, the chain rule gives

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

101. Carte 101

     Question

     What local property lets a function have an inverse derivative?

     Réponse

     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. Carte 102

     Question

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

     Réponse

     For $a>0$,

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

103. Carte 103

     Question

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

     Réponse

     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. Carte 104

     Question

     Position, velocity, and acceleration relationships?

     Réponse

     For position $s(t)$,

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

105. Carte 105

     Question

     Central idea of a related-rates problem?

     Réponse

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

106. Carte 106

     Question

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

     Réponse

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

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

107. Carte 107

     Question

     L’Hospital’s Rule: basic conditions?

     Réponse

     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. Carte 108

     Question

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

     Réponse

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

109. Carte 109

     Question

     Speed in terms of velocity?

     Réponse

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Carte 110

     Question

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

     Réponse

     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. Carte 111

     Question

     Differential approximation connecting $dx$ and $dy$?

     Réponse

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

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

112. Carte 112

     Question

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

     Réponse

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

113. Carte 113

     Question

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

     Réponse

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

114. Carte 114

     Question

     What does positive acceleration say about velocity?

     Réponse

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

115. Carte 115

     Question

     Related rates: when should numerical values be substituted?

     Réponse

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

116. Carte 116

     Question

     How does concavity predict linearization error?

     Réponse

     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. Carte 117

     Question

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

     Réponse

     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. Carte 118

     Question

     When is a particle moving in the positive direction?

     Réponse

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

119. Carte 119

     Question

     How can velocity show a change of direction?

     Réponse

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

120. Carte 120

     Question

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

     Réponse

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

121. Carte 121

     Question

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

     Réponse

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

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

122. Carte 122

     Question

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

     Réponse

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

123. Carte 123

     Question

     What must a contextual derivative sentence include?

     Réponse

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

124. Carte 124

     Question

     Velocity negative and acceleration positive: what happens?

     Réponse

     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. Carte 125

     Question

     How should a negative related rate be interpreted?

     Réponse

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

126. Carte 126

     Question

     When is local linearity a sound approximation tool?

     Réponse

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

127. Carte 127

     Question

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

     Réponse

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

128. Carte 128

     Question

     When is speed increasing?

     Réponse

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

129. Carte 129

     Question

     Volume changes with time: notation for its rate?

     Réponse

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

130. Carte 130

     Question

     Why are similar triangles useful in related rates?

     Réponse

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

131. Carte 131

     Question

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

     Réponse

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

132. Carte 132

     Question

     What conclusion does L’Hospital’s Rule permit?

     Réponse

     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. Carte 133

     Question

     When is speed decreasing?

     Réponse

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

134. Carte 134

     Question

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

     Réponse

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

135. Carte 135

     Question

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

     Réponse

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

136. Carte 136

     Question

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

     Réponse

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

137. Carte 137

     Question

     Extreme Value Theorem: hypothesis and conclusion?

     Réponse

     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. Carte 138

     Question

     What is a critical number of $f$?

     Réponse

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

139. Carte 139

     Question

     First derivative test for a local maximum?

     Réponse

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

140. Carte 140

     Question

     Second-derivative sign for concave up?

     Réponse

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

141. Carte 141

     Question

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

     Réponse

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

142. Carte 142

     Question

     First step in an optimization model?

     Réponse

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

143. Carte 143

     Question

     Mean Value Theorem: hypotheses and conclusion?

     Réponse

     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. Carte 144

     Question

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

     Réponse

     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. Carte 145

     Question

     First derivative test for a local minimum?

     Réponse

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

146. Carte 146

     Question

     What must happen at an inflection point?

     Réponse

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

147. Carte 147

     Question

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

     Réponse

     $f''$ 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. Carte 148

     Question

     How do you confirm an optimization answer is absolute?

     Réponse

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

149. Carte 149

     Question

     Rolle’s Theorem: hypotheses and conclusion?

     Réponse

     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. Carte 150

     Question

     Difference between absolute and relative extrema?

     Réponse

     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. Carte 151

     Question

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

     Réponse

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

152. Carte 152

     Question

     Second derivative test for a local minimum?

     Réponse

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

153. Carte 153

     Question

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

     Réponse

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

154. Carte 154

     Question

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

     Réponse

     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. Carte 155

     Question

     Which theorem links an average slope to an instantaneous slope?

     Réponse

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

156. Carte 156

     Question

     How can an implicit derivative locate a horizontal tangent?

     Réponse

     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. Carte 157

     Question

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

     Réponse

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

158. Carte 158

     Question

     Second derivative test for a local maximum?

     Réponse

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

159. Carte 159

     Question

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

     Réponse

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

160. Carte 160

     Question

     Why must an optimization domain be stated?

     Réponse

     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. Carte 161

     Question

     Which theorem guarantees absolute extrema, not where they occur?

     Réponse

     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. Carte 162

     Question

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

     Réponse

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

163. Carte 163

     Question

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

     Réponse

     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. Carte 164

     Question

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

     Réponse

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

165. Carte 165

     Question

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

     Réponse

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

166. Carte 166

     Question

     How can an implicit derivative locate a vertical tangent?

     Réponse

     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. Carte 167

     Question

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

     Réponse

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

168. Carte 168

     Question

     Why are endpoints included in the candidates test?

     Réponse

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

169. Carte 169

     Question

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

     Réponse

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

170. Carte 170

     Question

     Second-derivative sign for concave down?

     Réponse

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

171. Carte 171

     Question

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

     Réponse

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

172. Carte 172

     Question

     What should the final line of an optimization solution state?

     Réponse

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

173. Carte 173

     Question

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

     Réponse

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

174. Carte 174

     Question

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

     Réponse

     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. Carte 175

     Question

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

     Réponse

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

176. Carte 176

     Question

     Left Riemann sum on equal subintervals?

     Réponse

     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. Carte 177

     Question

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

     Réponse

     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. Carte 178

     Question

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Réponse

     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. Carte 179

     Question

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

     Réponse

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Carte 180

     Question

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

     Réponse

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

181. Carte 181

     Question

     Right Riemann sum on equal subintervals?

     Réponse

     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. Carte 182

     Question

     How does reversing integral bounds change the value?

     Réponse

     It changes the sign:

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

183. Carte 183

     Question

     Net Change Theorem?

     Réponse

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

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

184. Carte 184

     Question

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

     Réponse

     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. Carte 185

     Question

     Power rule for antiderivatives?

     Réponse

     For $n\ne-1$,

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

186. Carte 186

     Question

     Midpoint Riemann sum on equal subintervals?

     Réponse

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

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

187. Carte 187

     Question

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

     Réponse

     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. Carte 188

     Question

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

     Réponse

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Carte 189

     Question

     Antiderivative of $1/x$?

     Réponse

     On any interval not crossing zero,

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

190. Carte 190

     Question

     Trapezoidal approximation on equal subintervals?

     Réponse

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

191. Carte 191

     Question

     How do geometric regions help evaluate a definite integral?

     Réponse

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

192. Carte 192

     Question

     Basic antiderivatives of sine and cosine?

     Réponse

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

193. Carte 193

     Question

     Definite integral as a limit of Riemann sums?

     Réponse

     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. Carte 194

     Question

     Constant-multiple rule for integrals?

     Réponse

     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. Carte 195

     Question

     What pattern suggests $u$-substitution?

     Réponse

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

196. Carte 196

     Question

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

     Réponse

     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. Carte 197

     Question

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

     Réponse

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

198. Carte 198

     Question

     Sum-and-difference rule for definite integrals?

     Réponse

     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. Carte 199

     Question

     Basic antiderivative of $e^x$?

     Réponse

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

200. Carte 200

     Question

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

     Réponse

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

201. Carte 201

     Question

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

     Réponse

     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. Carte 202

     Question

     How does concavity predict trapezoidal and midpoint error?

     Réponse

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

203. Carte 203

     Question

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

     Réponse

     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. Carte 204

     Question

     What denominator pattern suggests an arctangent antiderivative?

     Réponse

     After completing the square and scaling, a form like

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

205. Carte 205

     Question

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

     Réponse

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

206. Carte 206

     Question

     How does an initial condition determine an antiderivative?

     Réponse

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

207. Carte 207

     Question

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

     Réponse

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

208. Carte 208

     Question

     Why does an indefinite integral include $+C$?

     Réponse

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

209. Carte 209

     Question

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

     Réponse

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

210. Carte 210

     Question

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

     Réponse

     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. Carte 211

     Question

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

     Réponse

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

212. Carte 212

     Question

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

     Réponse

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

213. Carte 213

     Question

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

     Réponse

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

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

214. Carte 214

     Question

     Riemann sum for unequal subinterval widths?

     Réponse

     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. Carte 215

     Question

     Does continuity guarantee integrability on a closed interval?

     Réponse

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

216. Carte 216

     Question

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

     Réponse

     Where $g(x)\ne0$,

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

217. Carte 217

     Question

     What algebraic rewrites often reveal a basic antiderivative?

     Réponse

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

218. Carte 218

     Question

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

     Réponse

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

219. Carte 219

     Question

     What is a differential equation?

     Réponse

     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. Carte 220

     Question

     How does a verbal rate statement become a differential equation?

     Réponse

     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. Carte 221

     Question

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

     Réponse

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

222. Carte 222

     Question

     General solution versus particular solution?

     Réponse

     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. Carte 223

     Question

     What does one segment in a slope field show?

     Réponse

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

224. Carte 224

     Question

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

     Réponse

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

225. Carte 225

     Question

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

     Réponse

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

226. Carte 226

     Question

     What makes a first-order differential equation separable?

     Réponse

     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. Carte 227

     Question

     What is an initial value problem?

     Réponse

     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. Carte 228

     Question

     What is an isocline in a slope field?

     Réponse

     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. Carte 229

     Question

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

     Réponse

     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. Carte 230

     Question

     General solution of $dy/dt=ky$?

     Réponse

     $$y=Ce^{kt}$$

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

231. Carte 231

     Question

     Core method for solving a separable differential equation?

     Réponse

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

232. Carte 232

     Question

     How should a solution curve follow a slope field?

     Réponse

     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. Carte 233

     Question

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

     Réponse

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

234. Carte 234

     Question

     Why is one integration constant enough after integrating both sides?

     Réponse

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

235. Carte 235

     Question

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

     Réponse

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

236. Carte 236

     Question

     Can one differential equation have infinitely many solutions?

     Réponse

     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. Carte 237

     Question

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

     Réponse

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

238. Carte 238

     Question

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

     Réponse

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

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

239. Carte 239

     Question

     What can be lost when dividing to separate variables?

     Réponse

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

240. Carte 240

     Question

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

     Réponse

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

241. Carte 241

     Question

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

     Réponse

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

242. Carte 242

     Question

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

     Réponse

     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. Carte 243

     Question

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

     Réponse

     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. Carte 244

     Question

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

     Réponse

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

245. Carte 245

     Question

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

     Réponse

     For $k>0$,

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

     It is independent of the initial amount.

246. Carte 246

     Question

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

     Réponse

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

247. Carte 247

     Question

     How is an initial condition used after separation?

     Réponse

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

248. Carte 248

     Question

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

     Réponse

     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. Carte 249

     Question

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

     Réponse

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

250. Carte 250

     Question

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

     Réponse

     For $k<0$,

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

251. Carte 251

     Question

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

     Réponse

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

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

252. Carte 252

     Question

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

     Réponse

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

     Velocity below zero contributes negative displacement.

253. Carte 253

     Question

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

     Réponse

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

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

     Think top minus bottom.

254. Carte 254

     Question

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

     Réponse

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

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

255. Carte 255

     Question

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Réponse

     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. Carte 256

     Question

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

     Réponse

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

257. Carte 257

     Question

     Cross-sectional area when each slice is a square?

     Réponse

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

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

258. Carte 258

     Question

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

     Réponse

     Integrate the net rate:

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

259. Carte 259

     Question

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

     Réponse

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

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

     Think right minus left.

260. Carte 260

     Question

     Disc-method volume formula?

     Réponse

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

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

261. Carte 261

     Question

     What units does average value have?

     Réponse

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

262. Carte 262

     Question

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

     Réponse

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

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

263. Carte 263

     Question

     Cross-sectional area when each slice is a rectangle?

     Réponse

     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. Carte 264

     Question

     How do you determine bounds for area between curves?

     Réponse

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

265. Carte 265

     Question

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

     Réponse

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

266. Carte 266

     Question

     When is a particle moving to the right or left?

     Réponse

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

267. Carte 267

     Question

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

     Réponse

     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. Carte 268

     Question

     Why must an area integral be split where curves intersect?

     Réponse

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

269. Carte 269

     Question

     How can a velocity table approximate displacement?

     Réponse

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

270. Carte 270

     Question

     Washer-method volume formula?

     Réponse

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

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

271. Carte 271

     Question

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

     Réponse

     If $s(a)$ is known,

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

272. Carte 272

     Question

     How do you choose between vertical and horizontal area slices?

     Réponse

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

273. Carte 273

     Question

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

     Réponse

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

274. Carte 274

     Question

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

     Réponse

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

275. Carte 275

     Question

     Single expression for area between two curves?

     Réponse

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Carte 276

     Question

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

     Réponse

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

277. Carte 277

     Question

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

     Réponse

     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. Carte 278

     Question

     When should a volume integral use $dy$?

     Réponse

     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. Carte 279

     Question

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

     Réponse

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

280. Carte 280

     Question

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

     Réponse

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

281. Carte 281

     Question

     Why must total distance split at velocity sign changes?

     Réponse

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

282. Carte 282

     Question

     When does an accumulated quantity reach a local maximum?

     Réponse

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

283. Carte 283

     Question

     What distinguishes area from a definite integral?

     Réponse

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

284. Carte 284

     Question

     How do position, velocity, and acceleration graphs correspond?

     Réponse

     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. Carte 285

     Question

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

     Réponse

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

286. Carte 286

     Question

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

     Réponse

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

287. Carte 287

     Question

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

     Réponse

     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. Carte 288

     Question

     Why should a contextual integral answer include a sentence?

     Réponse

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