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

Картки в цій колоді

1. Картка 1

   Питання

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

   Відповідь

   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. Картка 2

   Питання

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

   Відповідь

   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. Картка 3

   Питання

   When does direct substitution evaluate a limit?

   Відповідь

   When the function is continuous at the target input. Then

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

4. Картка 4

   Питання

   Three conditions for continuity at $x=a$?

   Відповідь

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

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

5. Картка 5

   Питання

   Intermediate Value Theorem: hypotheses and conclusion?

   Відповідь

   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. Картка 6

   Питання

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

   Відповідь

   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. Картка 7

   Питання

   How do you read a finite limit from a graph?

   Відповідь

   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. Картка 8

   Питання

   Limit law for a sum or difference?

   Відповідь

   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. Картка 9

   Питання

   What makes a discontinuity removable?

   Відповідь

   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. Картка 10

    Питання

    Squeeze Theorem: usable form?

    Відповідь

    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. Картка 11

    Питання

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

    Відповідь

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

12. Картка 12

    Питання

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

    Відповідь

    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. Картка 13

    Питання

    Limit law for a product?

    Відповідь

    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. Картка 14

    Питання

    Graph signature of a jump discontinuity?

    Відповідь

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

15. Картка 15

    Питання

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

    Відповідь

    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. Картка 16

    Питання

    Horizontal asymptote from a limit at infinity?

    Відповідь

    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. Картка 17

    Питання

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

    Відповідь

    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. Картка 18

    Питання

    Limit law for a quotient—and its condition?

    Відповідь

    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. Картка 19

    Питання

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

    Відповідь

    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. Картка 20

    Питання

    When is the Squeeze Theorem a natural choice?

    Відповідь

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

21. Картка 21

    Питання

    Vertical asymptote from one-sided behavior?

    Відповідь

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

22. Картка 22

    Питання

    Limit at infinity of equal-degree rational functions?

    Відповідь

    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. Картка 23

    Питання

    When can a limit pass through a continuous outer function?

    Відповідь

    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. Картка 24

    Питання

    What makes a discontinuity infinite?

    Відповідь

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

25. Картка 25

    Питання

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

    Відповідь

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

26. Картка 26

    Питання

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

    Відповідь

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Картка 27

    Питання

    Continuity of a composition?

    Відповідь

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

28. Картка 28

    Питання

    Limit at infinity when a rational numerator has lower degree?

    Відповідь

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

29. Картка 29

    Питання

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

    Відповідь

    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. Картка 30

    Питання

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

    Відповідь

    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. Картка 31

    Питання

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

    Відповідь

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

32. Картка 32

    Питання

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

    Відповідь

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

33. Картка 33

    Питання

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

    Відповідь

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

34. Картка 34

    Питання

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

    Відповідь

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

35. Картка 35

    Питання

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

    Відповідь

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

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

36. Картка 36

    Питання

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

    Відповідь

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

    The derivative exists only if this finite limit exists.

37. Картка 37

    Питання

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

    Відповідь

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

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

38. Картка 38

    Питання

    What does differentiability imply about continuity?

    Відповідь

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

39. Картка 39

    Питання

    Power rule for derivatives?

    Відповідь

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

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

40. Картка 40

    Питання

    Units of $f'(x)$?

    Відповідь

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

41. Картка 41

    Питання

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

    Відповідь

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

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

42. Картка 42

    Питання

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

    Відповідь

    $f'(x)>0$ 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. Картка 43

    Питання

    Derivative of a constant?

    Відповідь

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

    A constant function has zero rate of change.

44. Картка 44

    Питання

    Derivative of $\sin x$?

    Відповідь

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

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

45. Картка 45

    Питання

    Product rule?

    Відповідь

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

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

46. Картка 46

    Питання

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

    Відповідь

    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. Картка 47

    Питання

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

    Відповідь

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

48. Картка 48

    Питання

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

    Відповідь

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

49. Картка 49

    Питання

    Derivative of a sum or difference?

    Відповідь

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

50. Картка 50

    Питання

    Derivative of $\cos x$?

    Відповідь

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

    The standard formula assumes radians.

51. Картка 51

    Питання

    Quotient rule?

    Відповідь

    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. Картка 52

    Питання

    Common notations for the first derivative?

    Відповідь

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

53. Картка 53

    Питання

    Derivative of $e^x$?

    Відповідь

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

54. Картка 54

    Питання

    What graph features can make $f$ nondifferentiable?

    Відповідь

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

55. Картка 55

    Питання

    Derivative of $\tan x$?

    Відповідь

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Картка 56

    Питання

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

    Відповідь

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

57. Картка 57

    Питання

    Derivative of $\ln x$?

    Відповідь

    For $x>0$,

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

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

58. Картка 58

    Питання

    How does the power rule handle roots or negative powers?

    Відповідь

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

59. Картка 59

    Питання

    Derivative of $\csc x$?

    Відповідь

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Картка 60

    Питання

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

    Відповідь

    $f$ is increasing on that interval.

61. Картка 61

    Питання

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

    Відповідь

    For $a>0$,

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

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

62. Картка 62

    Питання

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

    Відповідь

    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. Картка 63

    Питання

    Derivative of $\sec x$?

    Відповідь

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Картка 64

    Питання

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

    Відповідь

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

65. Картка 65

    Питання

    Derivative of $\log_a x$?

    Відповідь

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

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

66. Картка 66

    Питання

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

    Відповідь

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Картка 67

    Питання

    Derivative of $\cot x$?

    Відповідь

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Картка 68

    Питання

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

    Відповідь

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

69. Картка 69

    Питання

    Constant-multiple rule?

    Відповідь

    For a constant $c$,

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

70. Картка 70

    Питання

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

    Відповідь

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

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

71. Картка 71

    Питання

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

    Відповідь

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

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

72. Картка 72

    Питання

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

    Відповідь

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

73. Картка 73

    Питання

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

    Відповідь

    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. Картка 74

    Питання

    Derivative of an inverse function at $x$?

    Відповідь

    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. Картка 75

    Питання

    Derivative of $\arcsin x$?

    Відповідь

    For $|x|<1$,

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

76. Картка 76

    Питання

    Notation for the third derivative of $f$?

    Відповідь

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

77. Картка 77

    Питання

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

    Відповідь

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

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

78. Картка 78

    Питання

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

    Відповідь

    Where $y\ne0$,

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

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

79. Картка 79

    Питання

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

    Відповідь

    Provided $f'(a)\ne0$,

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

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

80. Картка 80

    Питання

    Derivative of $\arctan x$?

    Відповідь

    For every real $x$,

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

81. Картка 81

    Питання

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

    Відповідь

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

    The extra factor is the chain rule.

82. Картка 82

    Питання

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

    Відповідь

    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. Картка 83

    Питання

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

    Відповідь

    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. Картка 84

    Питання

    Derivative of $\arccos x$?

    Відповідь

    For $|x|<1$,

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

85. Картка 85

    Питання

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

    Відповідь

    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. Картка 86

    Питання

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

    Відповідь

    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. Картка 87

    Питання

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

    Відповідь

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

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

88. Картка 88

    Питання

    How are tangent slopes of inverse graphs related?

    Відповідь

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

89. Картка 89

    Питання

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

    Відповідь

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

90. Картка 90

    Питання

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

    Відповідь

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

91. Картка 91

    Питання

    Horizontal tangent on an implicit curve: derivative condition?

    Відповідь

    $dy/dx=0$ 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. Картка 92

    Питання

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

    Відповідь

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

93. Картка 93

    Питання

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

    Відповідь

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

94. Картка 94

    Питання

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

    Відповідь

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

    This combines the power rule with the chain rule.

95. Картка 95

    Питання

    Vertical tangent on an implicit curve: derivative clue?

    Відповідь

    $dy/dx$ 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. Картка 96

    Питання

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

    Відповідь

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

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

97. Картка 97

    Питання

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

    Відповідь

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

98. Картка 98

    Питання

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

    Відповідь

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

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

99. Картка 99

    Питання

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

    Відповідь

    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. Картка 100

     Питання

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

     Відповідь

     When the functions are differentiable, the chain rule gives

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

101. Картка 101

     Питання

     What local property lets a function have an inverse derivative?

     Відповідь

     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. Картка 102

     Питання

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

     Відповідь

     For $a>0$,

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

103. Картка 103

     Питання

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

     Відповідь

     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. Картка 104

     Питання

     Position, velocity, and acceleration relationships?

     Відповідь

     For position $s(t)$,

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

105. Картка 105

     Питання

     Central idea of a related-rates problem?

     Відповідь

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

106. Картка 106

     Питання

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

     Відповідь

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

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

107. Картка 107

     Питання

     L’Hospital’s Rule: basic conditions?

     Відповідь

     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. Картка 108

     Питання

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

     Відповідь

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

109. Картка 109

     Питання

     Speed in terms of velocity?

     Відповідь

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Картка 110

     Питання

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

     Відповідь

     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. Картка 111

     Питання

     Differential approximation connecting $dx$ and $dy$?

     Відповідь

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

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

112. Картка 112

     Питання

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

     Відповідь

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

113. Картка 113

     Питання

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

     Відповідь

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

114. Картка 114

     Питання

     What does positive acceleration say about velocity?

     Відповідь

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

115. Картка 115

     Питання

     Related rates: when should numerical values be substituted?

     Відповідь

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

116. Картка 116

     Питання

     How does concavity predict linearization error?

     Відповідь

     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. Картка 117

     Питання

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

     Відповідь

     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. Картка 118

     Питання

     When is a particle moving in the positive direction?

     Відповідь

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

119. Картка 119

     Питання

     How can velocity show a change of direction?

     Відповідь

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

120. Картка 120

     Питання

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

     Відповідь

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

121. Картка 121

     Питання

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

     Відповідь

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

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

122. Картка 122

     Питання

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

     Відповідь

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

123. Картка 123

     Питання

     What must a contextual derivative sentence include?

     Відповідь

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

124. Картка 124

     Питання

     Velocity negative and acceleration positive: what happens?

     Відповідь

     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. Картка 125

     Питання

     How should a negative related rate be interpreted?

     Відповідь

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

126. Картка 126

     Питання

     When is local linearity a sound approximation tool?

     Відповідь

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

127. Картка 127

     Питання

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

     Відповідь

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

128. Картка 128

     Питання

     When is speed increasing?

     Відповідь

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

129. Картка 129

     Питання

     Volume changes with time: notation for its rate?

     Відповідь

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

130. Картка 130

     Питання

     Why are similar triangles useful in related rates?

     Відповідь

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

131. Картка 131

     Питання

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

     Відповідь

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

132. Картка 132

     Питання

     What conclusion does L’Hospital’s Rule permit?

     Відповідь

     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. Картка 133

     Питання

     When is speed decreasing?

     Відповідь

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

134. Картка 134

     Питання

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

     Відповідь

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

135. Картка 135

     Питання

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

     Відповідь

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

136. Картка 136

     Питання

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

     Відповідь

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

137. Картка 137

     Питання

     Extreme Value Theorem: hypothesis and conclusion?

     Відповідь

     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. Картка 138

     Питання

     What is a critical number of $f$?

     Відповідь

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

139. Картка 139

     Питання

     First derivative test for a local maximum?

     Відповідь

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

140. Картка 140

     Питання

     Second-derivative sign for concave up?

     Відповідь

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

141. Картка 141

     Питання

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

     Відповідь

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

142. Картка 142

     Питання

     First step in an optimization model?

     Відповідь

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

143. Картка 143

     Питання

     Mean Value Theorem: hypotheses and conclusion?

     Відповідь

     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. Картка 144

     Питання

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

     Відповідь

     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 карток

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

     Вивчати цю колоду безкоштовно

     Відкриється Nibomo, і ви зможете почати навчання.

145. Картка 145

     Питання

     First derivative test for a local minimum?

     Відповідь

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

146. Картка 146

     Питання

     What must happen at an inflection point?

     Відповідь

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

147. Картка 147

     Питання

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

     Відповідь

     $f''$ 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. Картка 148

     Питання

     How do you confirm an optimization answer is absolute?

     Відповідь

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

149. Картка 149

     Питання

     Rolle’s Theorem: hypotheses and conclusion?

     Відповідь

     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. Картка 150

     Питання

     Difference between absolute and relative extrema?

     Відповідь

     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. Картка 151

     Питання

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

     Відповідь

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

152. Картка 152

     Питання

     Second derivative test for a local minimum?

     Відповідь

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

153. Картка 153

     Питання

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

     Відповідь

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

154. Картка 154

     Питання

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

     Відповідь

     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. Картка 155

     Питання

     Which theorem links an average slope to an instantaneous slope?

     Відповідь

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

156. Картка 156

     Питання

     How can an implicit derivative locate a horizontal tangent?

     Відповідь

     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. Картка 157

     Питання

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

     Відповідь

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

158. Картка 158

     Питання

     Second derivative test for a local maximum?

     Відповідь

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

159. Картка 159

     Питання

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

     Відповідь

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

160. Картка 160

     Питання

     Why must an optimization domain be stated?

     Відповідь

     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. Картка 161

     Питання

     Which theorem guarantees absolute extrema, not where they occur?

     Відповідь

     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. Картка 162

     Питання

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

     Відповідь

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

163. Картка 163

     Питання

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

     Відповідь

     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. Картка 164

     Питання

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

     Відповідь

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

165. Картка 165

     Питання

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

     Відповідь

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

166. Картка 166

     Питання

     How can an implicit derivative locate a vertical tangent?

     Відповідь

     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. Картка 167

     Питання

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

     Відповідь

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

168. Картка 168

     Питання

     Why are endpoints included in the candidates test?

     Відповідь

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

169. Картка 169

     Питання

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

     Відповідь

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

170. Картка 170

     Питання

     Second-derivative sign for concave down?

     Відповідь

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

171. Картка 171

     Питання

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

     Відповідь

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

172. Картка 172

     Питання

     What should the final line of an optimization solution state?

     Відповідь

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

173. Картка 173

     Питання

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

     Відповідь

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

174. Картка 174

     Питання

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

     Відповідь

     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. Картка 175

     Питання

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

     Відповідь

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

176. Картка 176

     Питання

     Left Riemann sum on equal subintervals?

     Відповідь

     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. Картка 177

     Питання

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

     Відповідь

     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. Картка 178

     Питання

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Відповідь

     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. Картка 179

     Питання

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

     Відповідь

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Картка 180

     Питання

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

     Відповідь

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

181. Картка 181

     Питання

     Right Riemann sum on equal subintervals?

     Відповідь

     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. Картка 182

     Питання

     How does reversing integral bounds change the value?

     Відповідь

     It changes the sign:

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

183. Картка 183

     Питання

     Net Change Theorem?

     Відповідь

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

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

184. Картка 184

     Питання

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

     Відповідь

     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. Картка 185

     Питання

     Power rule for antiderivatives?

     Відповідь

     For $n\ne-1$,

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

186. Картка 186

     Питання

     Midpoint Riemann sum on equal subintervals?

     Відповідь

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

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

187. Картка 187

     Питання

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

     Відповідь

     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. Картка 188

     Питання

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

     Відповідь

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Картка 189

     Питання

     Antiderivative of $1/x$?

     Відповідь

     On any interval not crossing zero,

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

190. Картка 190

     Питання

     Trapezoidal approximation on equal subintervals?

     Відповідь

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

191. Картка 191

     Питання

     How do geometric regions help evaluate a definite integral?

     Відповідь

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

192. Картка 192

     Питання

     Basic antiderivatives of sine and cosine?

     Відповідь

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

193. Картка 193

     Питання

     Definite integral as a limit of Riemann sums?

     Відповідь

     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. Картка 194

     Питання

     Constant-multiple rule for integrals?

     Відповідь

     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. Картка 195

     Питання

     What pattern suggests $u$-substitution?

     Відповідь

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

196. Картка 196

     Питання

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

     Відповідь

     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. Картка 197

     Питання

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

     Відповідь

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

198. Картка 198

     Питання

     Sum-and-difference rule for definite integrals?

     Відповідь

     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. Картка 199

     Питання

     Basic antiderivative of $e^x$?

     Відповідь

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

200. Картка 200

     Питання

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

     Відповідь

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

201. Картка 201

     Питання

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

     Відповідь

     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. Картка 202

     Питання

     How does concavity predict trapezoidal and midpoint error?

     Відповідь

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

203. Картка 203

     Питання

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

     Відповідь

     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. Картка 204

     Питання

     What denominator pattern suggests an arctangent antiderivative?

     Відповідь

     After completing the square and scaling, a form like

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

205. Картка 205

     Питання

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

     Відповідь

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

206. Картка 206

     Питання

     How does an initial condition determine an antiderivative?

     Відповідь

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

207. Картка 207

     Питання

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

     Відповідь

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

208. Картка 208

     Питання

     Why does an indefinite integral include $+C$?

     Відповідь

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

209. Картка 209

     Питання

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

     Відповідь

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

210. Картка 210

     Питання

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

     Відповідь

     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. Картка 211

     Питання

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

     Відповідь

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

212. Картка 212

     Питання

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

     Відповідь

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

213. Картка 213

     Питання

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

     Відповідь

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

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

214. Картка 214

     Питання

     Riemann sum for unequal subinterval widths?

     Відповідь

     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. Картка 215

     Питання

     Does continuity guarantee integrability on a closed interval?

     Відповідь

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

216. Картка 216

     Питання

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

     Відповідь

     Where $g(x)\ne0$,

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

217. Картка 217

     Питання

     What algebraic rewrites often reveal a basic antiderivative?

     Відповідь

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

218. Картка 218

     Питання

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

     Відповідь

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

219. Картка 219

     Питання

     What is a differential equation?

     Відповідь

     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. Картка 220

     Питання

     How does a verbal rate statement become a differential equation?

     Відповідь

     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. Картка 221

     Питання

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

     Відповідь

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

222. Картка 222

     Питання

     General solution versus particular solution?

     Відповідь

     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. Картка 223

     Питання

     What does one segment in a slope field show?

     Відповідь

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

224. Картка 224

     Питання

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

     Відповідь

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

225. Картка 225

     Питання

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

     Відповідь

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

226. Картка 226

     Питання

     What makes a first-order differential equation separable?

     Відповідь

     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. Картка 227

     Питання

     What is an initial value problem?

     Відповідь

     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. Картка 228

     Питання

     What is an isocline in a slope field?

     Відповідь

     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. Картка 229

     Питання

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

     Відповідь

     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. Картка 230

     Питання

     General solution of $dy/dt=ky$?

     Відповідь

     $$y=Ce^{kt}$$

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

231. Картка 231

     Питання

     Core method for solving a separable differential equation?

     Відповідь

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

232. Картка 232

     Питання

     How should a solution curve follow a slope field?

     Відповідь

     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. Картка 233

     Питання

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

     Відповідь

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

234. Картка 234

     Питання

     Why is one integration constant enough after integrating both sides?

     Відповідь

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

235. Картка 235

     Питання

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

     Відповідь

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

236. Картка 236

     Питання

     Can one differential equation have infinitely many solutions?

     Відповідь

     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. Картка 237

     Питання

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

     Відповідь

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

238. Картка 238

     Питання

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

     Відповідь

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

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

239. Картка 239

     Питання

     What can be lost when dividing to separate variables?

     Відповідь

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

240. Картка 240

     Питання

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

     Відповідь

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

241. Картка 241

     Питання

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

     Відповідь

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

242. Картка 242

     Питання

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

     Відповідь

     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. Картка 243

     Питання

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

     Відповідь

     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. Картка 244

     Питання

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

     Відповідь

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

245. Картка 245

     Питання

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

     Відповідь

     For $k>0$,

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

     It is independent of the initial amount.

246. Картка 246

     Питання

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

     Відповідь

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

247. Картка 247

     Питання

     How is an initial condition used after separation?

     Відповідь

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

248. Картка 248

     Питання

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

     Відповідь

     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. Картка 249

     Питання

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

     Відповідь

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

250. Картка 250

     Питання

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

     Відповідь

     For $k<0$,

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

251. Картка 251

     Питання

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

     Відповідь

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

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

252. Картка 252

     Питання

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

     Відповідь

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

     Velocity below zero contributes negative displacement.

253. Картка 253

     Питання

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

     Відповідь

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

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

     Think top minus bottom.

254. Картка 254

     Питання

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

     Відповідь

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

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

255. Картка 255

     Питання

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Відповідь

     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. Картка 256

     Питання

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

     Відповідь

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

257. Картка 257

     Питання

     Cross-sectional area when each slice is a square?

     Відповідь

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

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

258. Картка 258

     Питання

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

     Відповідь

     Integrate the net rate:

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

259. Картка 259

     Питання

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

     Відповідь

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

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

     Think right minus left.

260. Картка 260

     Питання

     Disc-method volume formula?

     Відповідь

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

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

261. Картка 261

     Питання

     What units does average value have?

     Відповідь

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

262. Картка 262

     Питання

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

     Відповідь

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

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

263. Картка 263

     Питання

     Cross-sectional area when each slice is a rectangle?

     Відповідь

     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. Картка 264

     Питання

     How do you determine bounds for area between curves?

     Відповідь

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

265. Картка 265

     Питання

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

     Відповідь

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

266. Картка 266

     Питання

     When is a particle moving to the right or left?

     Відповідь

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

267. Картка 267

     Питання

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

     Відповідь

     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. Картка 268

     Питання

     Why must an area integral be split where curves intersect?

     Відповідь

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

269. Картка 269

     Питання

     How can a velocity table approximate displacement?

     Відповідь

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

270. Картка 270

     Питання

     Washer-method volume formula?

     Відповідь

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

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

271. Картка 271

     Питання

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

     Відповідь

     If $s(a)$ is known,

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

272. Картка 272

     Питання

     How do you choose between vertical and horizontal area slices?

     Відповідь

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

273. Картка 273

     Питання

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

     Відповідь

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

274. Картка 274

     Питання

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

     Відповідь

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

275. Картка 275

     Питання

     Single expression for area between two curves?

     Відповідь

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Картка 276

     Питання

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

     Відповідь

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

277. Картка 277

     Питання

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

     Відповідь

     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. Картка 278

     Питання

     When should a volume integral use $dy$?

     Відповідь

     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. Картка 279

     Питання

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

     Відповідь

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

280. Картка 280

     Питання

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

     Відповідь

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

281. Картка 281

     Питання

     Why must total distance split at velocity sign changes?

     Відповідь

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

282. Картка 282

     Питання

     When does an accumulated quantity reach a local maximum?

     Відповідь

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

283. Картка 283

     Питання

     What distinguishes area from a definite integral?

     Відповідь

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

284. Картка 284

     Питання

     How do position, velocity, and acceleration graphs correspond?

     Відповідь

     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. Картка 285

     Питання

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

     Відповідь

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

286. Картка 286

     Питання

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

     Відповідь

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

287. Картка 287

     Питання

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

     Відповідь

     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. Картка 288

     Питання

     Why should a contextual integral answer include a sentence?

     Відповідь

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