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

Карти в това тесте

1. Карта 1

   Въпрос

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

   Отговор

   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. Карта 2

   Въпрос

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

   Отговор

   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. Карта 3

   Въпрос

   When does direct substitution evaluate a limit?

   Отговор

   When the function is continuous at the target input. Then

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

4. Карта 4

   Въпрос

   Three conditions for continuity at $x=a$?

   Отговор

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

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

5. Карта 5

   Въпрос

   Intermediate Value Theorem: hypotheses and conclusion?

   Отговор

   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. Карта 6

   Въпрос

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

   Отговор

   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. Карта 7

   Въпрос

   How do you read a finite limit from a graph?

   Отговор

   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. Карта 8

   Въпрос

   Limit law for a sum or difference?

   Отговор

   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. Карта 9

   Въпрос

   What makes a discontinuity removable?

   Отговор

   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. Карта 10

    Въпрос

    Squeeze Theorem: usable form?

    Отговор

    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. Карта 11

    Въпрос

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

    Отговор

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

12. Карта 12

    Въпрос

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

    Отговор

    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. Карта 13

    Въпрос

    Limit law for a product?

    Отговор

    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. Карта 14

    Въпрос

    Graph signature of a jump discontinuity?

    Отговор

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

15. Карта 15

    Въпрос

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

    Отговор

    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. Карта 16

    Въпрос

    Horizontal asymptote from a limit at infinity?

    Отговор

    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. Карта 17

    Въпрос

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

    Отговор

    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. Карта 18

    Въпрос

    Limit law for a quotient—and its condition?

    Отговор

    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. Карта 19

    Въпрос

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

    Отговор

    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. Карта 20

    Въпрос

    When is the Squeeze Theorem a natural choice?

    Отговор

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

21. Карта 21

    Въпрос

    Vertical asymptote from one-sided behavior?

    Отговор

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

22. Карта 22

    Въпрос

    Limit at infinity of equal-degree rational functions?

    Отговор

    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. Карта 23

    Въпрос

    When can a limit pass through a continuous outer function?

    Отговор

    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. Карта 24

    Въпрос

    What makes a discontinuity infinite?

    Отговор

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

25. Карта 25

    Въпрос

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

    Отговор

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

26. Карта 26

    Въпрос

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

    Отговор

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Карта 27

    Въпрос

    Continuity of a composition?

    Отговор

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

28. Карта 28

    Въпрос

    Limit at infinity when a rational numerator has lower degree?

    Отговор

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

29. Карта 29

    Въпрос

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

    Отговор

    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. Карта 30

    Въпрос

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

    Отговор

    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. Карта 31

    Въпрос

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

    Отговор

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

32. Карта 32

    Въпрос

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

    Отговор

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

33. Карта 33

    Въпрос

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

    Отговор

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

34. Карта 34

    Въпрос

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

    Отговор

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

35. Карта 35

    Въпрос

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

    Отговор

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

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

36. Карта 36

    Въпрос

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

    Отговор

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

    The derivative exists only if this finite limit exists.

37. Карта 37

    Въпрос

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

    Отговор

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

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

38. Карта 38

    Въпрос

    What does differentiability imply about continuity?

    Отговор

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

39. Карта 39

    Въпрос

    Power rule for derivatives?

    Отговор

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

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

40. Карта 40

    Въпрос

    Units of $f'(x)$?

    Отговор

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

41. Карта 41

    Въпрос

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

    Отговор

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

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

42. Карта 42

    Въпрос

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

    Отговор

    $f'(x)>0$ 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. Карта 43

    Въпрос

    Derivative of a constant?

    Отговор

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

    A constant function has zero rate of change.

44. Карта 44

    Въпрос

    Derivative of $\sin x$?

    Отговор

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

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

45. Карта 45

    Въпрос

    Product rule?

    Отговор

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

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

46. Карта 46

    Въпрос

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

    Отговор

    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. Карта 47

    Въпрос

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

    Отговор

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

48. Карта 48

    Въпрос

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

    Отговор

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

49. Карта 49

    Въпрос

    Derivative of a sum or difference?

    Отговор

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

50. Карта 50

    Въпрос

    Derivative of $\cos x$?

    Отговор

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

    The standard formula assumes radians.

51. Карта 51

    Въпрос

    Quotient rule?

    Отговор

    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. Карта 52

    Въпрос

    Common notations for the first derivative?

    Отговор

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

53. Карта 53

    Въпрос

    Derivative of $e^x$?

    Отговор

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

54. Карта 54

    Въпрос

    What graph features can make $f$ nondifferentiable?

    Отговор

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

55. Карта 55

    Въпрос

    Derivative of $\tan x$?

    Отговор

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Карта 56

    Въпрос

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

    Отговор

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

57. Карта 57

    Въпрос

    Derivative of $\ln x$?

    Отговор

    For $x>0$,

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

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

58. Карта 58

    Въпрос

    How does the power rule handle roots or negative powers?

    Отговор

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

59. Карта 59

    Въпрос

    Derivative of $\csc x$?

    Отговор

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Карта 60

    Въпрос

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

    Отговор

    $f$ is increasing on that interval.

61. Карта 61

    Въпрос

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

    Отговор

    For $a>0$,

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

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

62. Карта 62

    Въпрос

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

    Отговор

    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. Карта 63

    Въпрос

    Derivative of $\sec x$?

    Отговор

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Карта 64

    Въпрос

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

    Отговор

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

65. Карта 65

    Въпрос

    Derivative of $\log_a x$?

    Отговор

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

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

66. Карта 66

    Въпрос

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

    Отговор

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Карта 67

    Въпрос

    Derivative of $\cot x$?

    Отговор

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Карта 68

    Въпрос

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

    Отговор

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

69. Карта 69

    Въпрос

    Constant-multiple rule?

    Отговор

    For a constant $c$,

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

70. Карта 70

    Въпрос

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

    Отговор

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

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

71. Карта 71

    Въпрос

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

    Отговор

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

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

72. Карта 72

    Въпрос

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

    Отговор

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

73. Карта 73

    Въпрос

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

    Отговор

    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. Карта 74

    Въпрос

    Derivative of an inverse function at $x$?

    Отговор

    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. Карта 75

    Въпрос

    Derivative of $\arcsin x$?

    Отговор

    For $|x|<1$,

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

76. Карта 76

    Въпрос

    Notation for the third derivative of $f$?

    Отговор

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

77. Карта 77

    Въпрос

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

    Отговор

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

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

78. Карта 78

    Въпрос

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

    Отговор

    Where $y\ne0$,

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

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

79. Карта 79

    Въпрос

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

    Отговор

    Provided $f'(a)\ne0$,

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

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

80. Карта 80

    Въпрос

    Derivative of $\arctan x$?

    Отговор

    For every real $x$,

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

81. Карта 81

    Въпрос

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

    Отговор

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

    The extra factor is the chain rule.

82. Карта 82

    Въпрос

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

    Отговор

    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. Карта 83

    Въпрос

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

    Отговор

    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. Карта 84

    Въпрос

    Derivative of $\arccos x$?

    Отговор

    For $|x|<1$,

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

85. Карта 85

    Въпрос

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

    Отговор

    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. Карта 86

    Въпрос

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

    Отговор

    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. Карта 87

    Въпрос

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

    Отговор

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

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

88. Карта 88

    Въпрос

    How are tangent slopes of inverse graphs related?

    Отговор

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

89. Карта 89

    Въпрос

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

    Отговор

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

90. Карта 90

    Въпрос

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

    Отговор

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

91. Карта 91

    Въпрос

    Horizontal tangent on an implicit curve: derivative condition?

    Отговор

    $dy/dx=0$ 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. Карта 92

    Въпрос

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

    Отговор

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

93. Карта 93

    Въпрос

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

    Отговор

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

94. Карта 94

    Въпрос

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

    Отговор

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

    This combines the power rule with the chain rule.

95. Карта 95

    Въпрос

    Vertical tangent on an implicit curve: derivative clue?

    Отговор

    $dy/dx$ 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. Карта 96

    Въпрос

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

    Отговор

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

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

97. Карта 97

    Въпрос

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

    Отговор

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

98. Карта 98

    Въпрос

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

    Отговор

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

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

99. Карта 99

    Въпрос

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

    Отговор

    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. Карта 100

     Въпрос

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

     Отговор

     When the functions are differentiable, the chain rule gives

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

101. Карта 101

     Въпрос

     What local property lets a function have an inverse derivative?

     Отговор

     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. Карта 102

     Въпрос

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

     Отговор

     For $a>0$,

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

103. Карта 103

     Въпрос

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

     Отговор

     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. Карта 104

     Въпрос

     Position, velocity, and acceleration relationships?

     Отговор

     For position $s(t)$,

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

105. Карта 105

     Въпрос

     Central idea of a related-rates problem?

     Отговор

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

106. Карта 106

     Въпрос

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

     Отговор

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

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

107. Карта 107

     Въпрос

     L’Hospital’s Rule: basic conditions?

     Отговор

     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. Карта 108

     Въпрос

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

     Отговор

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

109. Карта 109

     Въпрос

     Speed in terms of velocity?

     Отговор

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Карта 110

     Въпрос

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

     Отговор

     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. Карта 111

     Въпрос

     Differential approximation connecting $dx$ and $dy$?

     Отговор

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

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

112. Карта 112

     Въпрос

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

     Отговор

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

113. Карта 113

     Въпрос

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

     Отговор

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

114. Карта 114

     Въпрос

     What does positive acceleration say about velocity?

     Отговор

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

115. Карта 115

     Въпрос

     Related rates: when should numerical values be substituted?

     Отговор

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

116. Карта 116

     Въпрос

     How does concavity predict linearization error?

     Отговор

     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. Карта 117

     Въпрос

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

     Отговор

     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. Карта 118

     Въпрос

     When is a particle moving in the positive direction?

     Отговор

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

119. Карта 119

     Въпрос

     How can velocity show a change of direction?

     Отговор

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

120. Карта 120

     Въпрос

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

     Отговор

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

121. Карта 121

     Въпрос

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

     Отговор

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

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

122. Карта 122

     Въпрос

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

     Отговор

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

123. Карта 123

     Въпрос

     What must a contextual derivative sentence include?

     Отговор

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

124. Карта 124

     Въпрос

     Velocity negative and acceleration positive: what happens?

     Отговор

     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. Карта 125

     Въпрос

     How should a negative related rate be interpreted?

     Отговор

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

126. Карта 126

     Въпрос

     When is local linearity a sound approximation tool?

     Отговор

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

127. Карта 127

     Въпрос

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

     Отговор

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

128. Карта 128

     Въпрос

     When is speed increasing?

     Отговор

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

129. Карта 129

     Въпрос

     Volume changes with time: notation for its rate?

     Отговор

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

130. Карта 130

     Въпрос

     Why are similar triangles useful in related rates?

     Отговор

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

131. Карта 131

     Въпрос

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

     Отговор

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

132. Карта 132

     Въпрос

     What conclusion does L’Hospital’s Rule permit?

     Отговор

     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. Карта 133

     Въпрос

     When is speed decreasing?

     Отговор

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

134. Карта 134

     Въпрос

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

     Отговор

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

135. Карта 135

     Въпрос

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

     Отговор

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

136. Карта 136

     Въпрос

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

     Отговор

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

137. Карта 137

     Въпрос

     Extreme Value Theorem: hypothesis and conclusion?

     Отговор

     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. Карта 138

     Въпрос

     What is a critical number of $f$?

     Отговор

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

139. Карта 139

     Въпрос

     First derivative test for a local maximum?

     Отговор

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

140. Карта 140

     Въпрос

     Second-derivative sign for concave up?

     Отговор

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

141. Карта 141

     Въпрос

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

     Отговор

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

142. Карта 142

     Въпрос

     First step in an optimization model?

     Отговор

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

143. Карта 143

     Въпрос

     Mean Value Theorem: hypotheses and conclusion?

     Отговор

     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. Карта 144

     Въпрос

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

     Отговор

     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.

     

     288 карти

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

     Учете с това тесте безплатно

     Nibomo се отваря, за да започнете да учите.

145. Карта 145

     Въпрос

     First derivative test for a local minimum?

     Отговор

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

146. Карта 146

     Въпрос

     What must happen at an inflection point?

     Отговор

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

147. Карта 147

     Въпрос

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

     Отговор

     $f''$ 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. Карта 148

     Въпрос

     How do you confirm an optimization answer is absolute?

     Отговор

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

149. Карта 149

     Въпрос

     Rolle’s Theorem: hypotheses and conclusion?

     Отговор

     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. Карта 150

     Въпрос

     Difference between absolute and relative extrema?

     Отговор

     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. Карта 151

     Въпрос

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

     Отговор

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

152. Карта 152

     Въпрос

     Second derivative test for a local minimum?

     Отговор

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

153. Карта 153

     Въпрос

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

     Отговор

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

154. Карта 154

     Въпрос

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

     Отговор

     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. Карта 155

     Въпрос

     Which theorem links an average slope to an instantaneous slope?

     Отговор

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

156. Карта 156

     Въпрос

     How can an implicit derivative locate a horizontal tangent?

     Отговор

     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. Карта 157

     Въпрос

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

     Отговор

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

158. Карта 158

     Въпрос

     Second derivative test for a local maximum?

     Отговор

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

159. Карта 159

     Въпрос

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

     Отговор

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

160. Карта 160

     Въпрос

     Why must an optimization domain be stated?

     Отговор

     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. Карта 161

     Въпрос

     Which theorem guarantees absolute extrema, not where they occur?

     Отговор

     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. Карта 162

     Въпрос

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

     Отговор

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

163. Карта 163

     Въпрос

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

     Отговор

     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. Карта 164

     Въпрос

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

     Отговор

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

165. Карта 165

     Въпрос

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

     Отговор

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

166. Карта 166

     Въпрос

     How can an implicit derivative locate a vertical tangent?

     Отговор

     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. Карта 167

     Въпрос

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

     Отговор

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

168. Карта 168

     Въпрос

     Why are endpoints included in the candidates test?

     Отговор

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

169. Карта 169

     Въпрос

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

     Отговор

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

170. Карта 170

     Въпрос

     Second-derivative sign for concave down?

     Отговор

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

171. Карта 171

     Въпрос

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

     Отговор

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

172. Карта 172

     Въпрос

     What should the final line of an optimization solution state?

     Отговор

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

173. Карта 173

     Въпрос

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

     Отговор

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

174. Карта 174

     Въпрос

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

     Отговор

     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. Карта 175

     Въпрос

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

     Отговор

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

176. Карта 176

     Въпрос

     Left Riemann sum on equal subintervals?

     Отговор

     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. Карта 177

     Въпрос

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

     Отговор

     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. Карта 178

     Въпрос

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Отговор

     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. Карта 179

     Въпрос

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

     Отговор

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Карта 180

     Въпрос

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

     Отговор

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

181. Карта 181

     Въпрос

     Right Riemann sum on equal subintervals?

     Отговор

     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. Карта 182

     Въпрос

     How does reversing integral bounds change the value?

     Отговор

     It changes the sign:

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

183. Карта 183

     Въпрос

     Net Change Theorem?

     Отговор

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

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

184. Карта 184

     Въпрос

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

     Отговор

     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. Карта 185

     Въпрос

     Power rule for antiderivatives?

     Отговор

     For $n\ne-1$,

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

186. Карта 186

     Въпрос

     Midpoint Riemann sum on equal subintervals?

     Отговор

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

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

187. Карта 187

     Въпрос

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

     Отговор

     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. Карта 188

     Въпрос

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

     Отговор

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Карта 189

     Въпрос

     Antiderivative of $1/x$?

     Отговор

     On any interval not crossing zero,

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

190. Карта 190

     Въпрос

     Trapezoidal approximation on equal subintervals?

     Отговор

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

191. Карта 191

     Въпрос

     How do geometric regions help evaluate a definite integral?

     Отговор

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

192. Карта 192

     Въпрос

     Basic antiderivatives of sine and cosine?

     Отговор

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

193. Карта 193

     Въпрос

     Definite integral as a limit of Riemann sums?

     Отговор

     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. Карта 194

     Въпрос

     Constant-multiple rule for integrals?

     Отговор

     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. Карта 195

     Въпрос

     What pattern suggests $u$-substitution?

     Отговор

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

196. Карта 196

     Въпрос

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

     Отговор

     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. Карта 197

     Въпрос

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

     Отговор

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

198. Карта 198

     Въпрос

     Sum-and-difference rule for definite integrals?

     Отговор

     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. Карта 199

     Въпрос

     Basic antiderivative of $e^x$?

     Отговор

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

200. Карта 200

     Въпрос

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

     Отговор

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

201. Карта 201

     Въпрос

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

     Отговор

     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. Карта 202

     Въпрос

     How does concavity predict trapezoidal and midpoint error?

     Отговор

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

203. Карта 203

     Въпрос

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

     Отговор

     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. Карта 204

     Въпрос

     What denominator pattern suggests an arctangent antiderivative?

     Отговор

     After completing the square and scaling, a form like

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

205. Карта 205

     Въпрос

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

     Отговор

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

206. Карта 206

     Въпрос

     How does an initial condition determine an antiderivative?

     Отговор

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

207. Карта 207

     Въпрос

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

     Отговор

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

208. Карта 208

     Въпрос

     Why does an indefinite integral include $+C$?

     Отговор

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

209. Карта 209

     Въпрос

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

     Отговор

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

210. Карта 210

     Въпрос

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

     Отговор

     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. Карта 211

     Въпрос

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

     Отговор

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

212. Карта 212

     Въпрос

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

     Отговор

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

213. Карта 213

     Въпрос

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

     Отговор

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

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

214. Карта 214

     Въпрос

     Riemann sum for unequal subinterval widths?

     Отговор

     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. Карта 215

     Въпрос

     Does continuity guarantee integrability on a closed interval?

     Отговор

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

216. Карта 216

     Въпрос

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

     Отговор

     Where $g(x)\ne0$,

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

217. Карта 217

     Въпрос

     What algebraic rewrites often reveal a basic antiderivative?

     Отговор

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

218. Карта 218

     Въпрос

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

     Отговор

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

219. Карта 219

     Въпрос

     What is a differential equation?

     Отговор

     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. Карта 220

     Въпрос

     How does a verbal rate statement become a differential equation?

     Отговор

     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. Карта 221

     Въпрос

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

     Отговор

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

222. Карта 222

     Въпрос

     General solution versus particular solution?

     Отговор

     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. Карта 223

     Въпрос

     What does one segment in a slope field show?

     Отговор

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

224. Карта 224

     Въпрос

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

     Отговор

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

225. Карта 225

     Въпрос

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

     Отговор

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

226. Карта 226

     Въпрос

     What makes a first-order differential equation separable?

     Отговор

     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. Карта 227

     Въпрос

     What is an initial value problem?

     Отговор

     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. Карта 228

     Въпрос

     What is an isocline in a slope field?

     Отговор

     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. Карта 229

     Въпрос

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

     Отговор

     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. Карта 230

     Въпрос

     General solution of $dy/dt=ky$?

     Отговор

     $$y=Ce^{kt}$$

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

231. Карта 231

     Въпрос

     Core method for solving a separable differential equation?

     Отговор

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

232. Карта 232

     Въпрос

     How should a solution curve follow a slope field?

     Отговор

     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. Карта 233

     Въпрос

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

     Отговор

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

234. Карта 234

     Въпрос

     Why is one integration constant enough after integrating both sides?

     Отговор

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

235. Карта 235

     Въпрос

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

     Отговор

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

236. Карта 236

     Въпрос

     Can one differential equation have infinitely many solutions?

     Отговор

     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. Карта 237

     Въпрос

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

     Отговор

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

238. Карта 238

     Въпрос

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

     Отговор

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

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

239. Карта 239

     Въпрос

     What can be lost when dividing to separate variables?

     Отговор

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

240. Карта 240

     Въпрос

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

     Отговор

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

241. Карта 241

     Въпрос

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

     Отговор

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

242. Карта 242

     Въпрос

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

     Отговор

     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. Карта 243

     Въпрос

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

     Отговор

     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. Карта 244

     Въпрос

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

     Отговор

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

245. Карта 245

     Въпрос

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

     Отговор

     For $k>0$,

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

     It is independent of the initial amount.

246. Карта 246

     Въпрос

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

     Отговор

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

247. Карта 247

     Въпрос

     How is an initial condition used after separation?

     Отговор

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

248. Карта 248

     Въпрос

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

     Отговор

     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. Карта 249

     Въпрос

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

     Отговор

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

250. Карта 250

     Въпрос

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

     Отговор

     For $k<0$,

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

251. Карта 251

     Въпрос

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

     Отговор

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

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

252. Карта 252

     Въпрос

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

     Отговор

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

     Velocity below zero contributes negative displacement.

253. Карта 253

     Въпрос

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

     Отговор

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

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

     Think top minus bottom.

254. Карта 254

     Въпрос

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

     Отговор

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

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

255. Карта 255

     Въпрос

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Отговор

     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. Карта 256

     Въпрос

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

     Отговор

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

257. Карта 257

     Въпрос

     Cross-sectional area when each slice is a square?

     Отговор

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

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

258. Карта 258

     Въпрос

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

     Отговор

     Integrate the net rate:

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

259. Карта 259

     Въпрос

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

     Отговор

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

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

     Think right minus left.

260. Карта 260

     Въпрос

     Disc-method volume formula?

     Отговор

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

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

261. Карта 261

     Въпрос

     What units does average value have?

     Отговор

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

262. Карта 262

     Въпрос

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

     Отговор

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

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

263. Карта 263

     Въпрос

     Cross-sectional area when each slice is a rectangle?

     Отговор

     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. Карта 264

     Въпрос

     How do you determine bounds for area between curves?

     Отговор

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

265. Карта 265

     Въпрос

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

     Отговор

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

266. Карта 266

     Въпрос

     When is a particle moving to the right or left?

     Отговор

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

267. Карта 267

     Въпрос

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

     Отговор

     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. Карта 268

     Въпрос

     Why must an area integral be split where curves intersect?

     Отговор

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

269. Карта 269

     Въпрос

     How can a velocity table approximate displacement?

     Отговор

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

270. Карта 270

     Въпрос

     Washer-method volume formula?

     Отговор

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

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

271. Карта 271

     Въпрос

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

     Отговор

     If $s(a)$ is known,

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

272. Карта 272

     Въпрос

     How do you choose between vertical and horizontal area slices?

     Отговор

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

273. Карта 273

     Въпрос

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

     Отговор

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

274. Карта 274

     Въпрос

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

     Отговор

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

275. Карта 275

     Въпрос

     Single expression for area between two curves?

     Отговор

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Карта 276

     Въпрос

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

     Отговор

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

277. Карта 277

     Въпрос

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

     Отговор

     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. Карта 278

     Въпрос

     When should a volume integral use $dy$?

     Отговор

     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. Карта 279

     Въпрос

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

     Отговор

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

280. Карта 280

     Въпрос

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

     Отговор

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

281. Карта 281

     Въпрос

     Why must total distance split at velocity sign changes?

     Отговор

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

282. Карта 282

     Въпрос

     When does an accumulated quantity reach a local maximum?

     Отговор

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

283. Карта 283

     Въпрос

     What distinguishes area from a definite integral?

     Отговор

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

284. Карта 284

     Въпрос

     How do position, velocity, and acceleration graphs correspond?

     Отговор

     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. Карта 285

     Въпрос

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

     Отговор

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

286. Карта 286

     Въпрос

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

     Отговор

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

287. Карта 287

     Въпрос

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

     Отговор

     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. Карта 288

     Въпрос

     Why should a contextual integral answer include a sentence?

     Отговор

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