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

Kartu dalam dek ini

1. Kartu 1

   Pertanyaan

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

   Jawaban

   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. Kartu 2

   Pertanyaan

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

   Jawaban

   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. Kartu 3

   Pertanyaan

   When does direct substitution evaluate a limit?

   Jawaban

   When the function is continuous at the target input. Then

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

4. Kartu 4

   Pertanyaan

   Three conditions for continuity at $x=a$?

   Jawaban

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

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

5. Kartu 5

   Pertanyaan

   Intermediate Value Theorem: hypotheses and conclusion?

   Jawaban

   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. Kartu 6

   Pertanyaan

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

   Jawaban

   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. Kartu 7

   Pertanyaan

   How do you read a finite limit from a graph?

   Jawaban

   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. Kartu 8

   Pertanyaan

   Limit law for a sum or difference?

   Jawaban

   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. Kartu 9

   Pertanyaan

   What makes a discontinuity removable?

   Jawaban

   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. Kartu 10

    Pertanyaan

    Squeeze Theorem: usable form?

    Jawaban

    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. Kartu 11

    Pertanyaan

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

    Jawaban

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

12. Kartu 12

    Pertanyaan

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

    Jawaban

    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. Kartu 13

    Pertanyaan

    Limit law for a product?

    Jawaban

    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. Kartu 14

    Pertanyaan

    Graph signature of a jump discontinuity?

    Jawaban

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

15. Kartu 15

    Pertanyaan

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

    Jawaban

    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. Kartu 16

    Pertanyaan

    Horizontal asymptote from a limit at infinity?

    Jawaban

    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. Kartu 17

    Pertanyaan

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

    Jawaban

    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. Kartu 18

    Pertanyaan

    Limit law for a quotient—and its condition?

    Jawaban

    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. Kartu 19

    Pertanyaan

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

    Jawaban

    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. Kartu 20

    Pertanyaan

    When is the Squeeze Theorem a natural choice?

    Jawaban

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

21. Kartu 21

    Pertanyaan

    Vertical asymptote from one-sided behavior?

    Jawaban

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

22. Kartu 22

    Pertanyaan

    Limit at infinity of equal-degree rational functions?

    Jawaban

    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. Kartu 23

    Pertanyaan

    When can a limit pass through a continuous outer function?

    Jawaban

    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. Kartu 24

    Pertanyaan

    What makes a discontinuity infinite?

    Jawaban

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

25. Kartu 25

    Pertanyaan

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

    Jawaban

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

26. Kartu 26

    Pertanyaan

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

    Jawaban

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kartu 27

    Pertanyaan

    Continuity of a composition?

    Jawaban

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

28. Kartu 28

    Pertanyaan

    Limit at infinity when a rational numerator has lower degree?

    Jawaban

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

29. Kartu 29

    Pertanyaan

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

    Jawaban

    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. Kartu 30

    Pertanyaan

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

    Jawaban

    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. Kartu 31

    Pertanyaan

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

    Jawaban

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

32. Kartu 32

    Pertanyaan

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

    Jawaban

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

33. Kartu 33

    Pertanyaan

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

    Jawaban

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

34. Kartu 34

    Pertanyaan

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

    Jawaban

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

35. Kartu 35

    Pertanyaan

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

    Jawaban

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

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

36. Kartu 36

    Pertanyaan

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

    Jawaban

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

    The derivative exists only if this finite limit exists.

37. Kartu 37

    Pertanyaan

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

    Jawaban

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

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

38. Kartu 38

    Pertanyaan

    What does differentiability imply about continuity?

    Jawaban

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

39. Kartu 39

    Pertanyaan

    Power rule for derivatives?

    Jawaban

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

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

40. Kartu 40

    Pertanyaan

    Units of $f'(x)$?

    Jawaban

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

41. Kartu 41

    Pertanyaan

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

    Jawaban

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

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

42. Kartu 42

    Pertanyaan

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

    Jawaban

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

    Pertanyaan

    Derivative of a constant?

    Jawaban

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

    A constant function has zero rate of change.

44. Kartu 44

    Pertanyaan

    Derivative of $\sin x$?

    Jawaban

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

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

45. Kartu 45

    Pertanyaan

    Product rule?

    Jawaban

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

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

46. Kartu 46

    Pertanyaan

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

    Jawaban

    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. Kartu 47

    Pertanyaan

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

    Jawaban

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

48. Kartu 48

    Pertanyaan

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

    Jawaban

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

49. Kartu 49

    Pertanyaan

    Derivative of a sum or difference?

    Jawaban

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

50. Kartu 50

    Pertanyaan

    Derivative of $\cos x$?

    Jawaban

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

    The standard formula assumes radians.

51. Kartu 51

    Pertanyaan

    Quotient rule?

    Jawaban

    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. Kartu 52

    Pertanyaan

    Common notations for the first derivative?

    Jawaban

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

53. Kartu 53

    Pertanyaan

    Derivative of $e^x$?

    Jawaban

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

54. Kartu 54

    Pertanyaan

    What graph features can make $f$ nondifferentiable?

    Jawaban

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

55. Kartu 55

    Pertanyaan

    Derivative of $\tan x$?

    Jawaban

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kartu 56

    Pertanyaan

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

    Jawaban

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

57. Kartu 57

    Pertanyaan

    Derivative of $\ln x$?

    Jawaban

    For $x>0$,

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

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

58. Kartu 58

    Pertanyaan

    How does the power rule handle roots or negative powers?

    Jawaban

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

59. Kartu 59

    Pertanyaan

    Derivative of $\csc x$?

    Jawaban

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kartu 60

    Pertanyaan

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

    Jawaban

    $f$ is increasing on that interval.

61. Kartu 61

    Pertanyaan

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

    Jawaban

    For $a>0$,

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

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

62. Kartu 62

    Pertanyaan

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

    Jawaban

    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. Kartu 63

    Pertanyaan

    Derivative of $\sec x$?

    Jawaban

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kartu 64

    Pertanyaan

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

    Jawaban

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

65. Kartu 65

    Pertanyaan

    Derivative of $\log_a x$?

    Jawaban

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

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

66. Kartu 66

    Pertanyaan

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

    Jawaban

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kartu 67

    Pertanyaan

    Derivative of $\cot x$?

    Jawaban

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kartu 68

    Pertanyaan

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

    Jawaban

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

69. Kartu 69

    Pertanyaan

    Constant-multiple rule?

    Jawaban

    For a constant $c$,

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

70. Kartu 70

    Pertanyaan

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

    Jawaban

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

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

71. Kartu 71

    Pertanyaan

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

    Jawaban

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

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

72. Kartu 72

    Pertanyaan

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

    Jawaban

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

73. Kartu 73

    Pertanyaan

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

    Jawaban

    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. Kartu 74

    Pertanyaan

    Derivative of an inverse function at $x$?

    Jawaban

    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. Kartu 75

    Pertanyaan

    Derivative of $\arcsin x$?

    Jawaban

    For $|x|<1$,

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

76. Kartu 76

    Pertanyaan

    Notation for the third derivative of $f$?

    Jawaban

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

77. Kartu 77

    Pertanyaan

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

    Jawaban

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

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

78. Kartu 78

    Pertanyaan

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

    Jawaban

    Where $y\ne0$,

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

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

79. Kartu 79

    Pertanyaan

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

    Jawaban

    Provided $f'(a)\ne0$,

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

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

80. Kartu 80

    Pertanyaan

    Derivative of $\arctan x$?

    Jawaban

    For every real $x$,

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

81. Kartu 81

    Pertanyaan

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

    Jawaban

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

    The extra factor is the chain rule.

82. Kartu 82

    Pertanyaan

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

    Jawaban

    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. Kartu 83

    Pertanyaan

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

    Jawaban

    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. Kartu 84

    Pertanyaan

    Derivative of $\arccos x$?

    Jawaban

    For $|x|<1$,

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

85. Kartu 85

    Pertanyaan

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

    Jawaban

    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. Kartu 86

    Pertanyaan

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

    Jawaban

    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. Kartu 87

    Pertanyaan

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

    Jawaban

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

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

88. Kartu 88

    Pertanyaan

    How are tangent slopes of inverse graphs related?

    Jawaban

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

89. Kartu 89

    Pertanyaan

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

    Jawaban

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

90. Kartu 90

    Pertanyaan

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

    Jawaban

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

91. Kartu 91

    Pertanyaan

    Horizontal tangent on an implicit curve: derivative condition?

    Jawaban

    $dy/dx=0$ 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. Kartu 92

    Pertanyaan

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

    Jawaban

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

93. Kartu 93

    Pertanyaan

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

    Jawaban

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

94. Kartu 94

    Pertanyaan

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

    Jawaban

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

    This combines the power rule with the chain rule.

95. Kartu 95

    Pertanyaan

    Vertical tangent on an implicit curve: derivative clue?

    Jawaban

    $dy/dx$ 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. Kartu 96

    Pertanyaan

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

    Jawaban

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

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

97. Kartu 97

    Pertanyaan

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

    Jawaban

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

98. Kartu 98

    Pertanyaan

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

    Jawaban

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

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

99. Kartu 99

    Pertanyaan

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

    Jawaban

    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. Kartu 100

     Pertanyaan

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

     Jawaban

     When the functions are differentiable, the chain rule gives

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

101. Kartu 101

     Pertanyaan

     What local property lets a function have an inverse derivative?

     Jawaban

     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. Kartu 102

     Pertanyaan

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

     Jawaban

     For $a>0$,

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

103. Kartu 103

     Pertanyaan

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

     Jawaban

     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. Kartu 104

     Pertanyaan

     Position, velocity, and acceleration relationships?

     Jawaban

     For position $s(t)$,

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

105. Kartu 105

     Pertanyaan

     Central idea of a related-rates problem?

     Jawaban

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

106. Kartu 106

     Pertanyaan

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

     Jawaban

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

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

107. Kartu 107

     Pertanyaan

     L’Hospital’s Rule: basic conditions?

     Jawaban

     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. Kartu 108

     Pertanyaan

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

     Jawaban

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

109. Kartu 109

     Pertanyaan

     Speed in terms of velocity?

     Jawaban

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kartu 110

     Pertanyaan

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

     Jawaban

     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. Kartu 111

     Pertanyaan

     Differential approximation connecting $dx$ and $dy$?

     Jawaban

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

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

112. Kartu 112

     Pertanyaan

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

     Jawaban

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

113. Kartu 113

     Pertanyaan

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

     Jawaban

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

114. Kartu 114

     Pertanyaan

     What does positive acceleration say about velocity?

     Jawaban

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

115. Kartu 115

     Pertanyaan

     Related rates: when should numerical values be substituted?

     Jawaban

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

116. Kartu 116

     Pertanyaan

     How does concavity predict linearization error?

     Jawaban

     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. Kartu 117

     Pertanyaan

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

     Jawaban

     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. Kartu 118

     Pertanyaan

     When is a particle moving in the positive direction?

     Jawaban

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

119. Kartu 119

     Pertanyaan

     How can velocity show a change of direction?

     Jawaban

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

120. Kartu 120

     Pertanyaan

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

     Jawaban

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

121. Kartu 121

     Pertanyaan

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

     Jawaban

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

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

122. Kartu 122

     Pertanyaan

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

     Jawaban

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

123. Kartu 123

     Pertanyaan

     What must a contextual derivative sentence include?

     Jawaban

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

124. Kartu 124

     Pertanyaan

     Velocity negative and acceleration positive: what happens?

     Jawaban

     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. Kartu 125

     Pertanyaan

     How should a negative related rate be interpreted?

     Jawaban

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

126. Kartu 126

     Pertanyaan

     When is local linearity a sound approximation tool?

     Jawaban

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

127. Kartu 127

     Pertanyaan

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

     Jawaban

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

128. Kartu 128

     Pertanyaan

     When is speed increasing?

     Jawaban

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

129. Kartu 129

     Pertanyaan

     Volume changes with time: notation for its rate?

     Jawaban

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

130. Kartu 130

     Pertanyaan

     Why are similar triangles useful in related rates?

     Jawaban

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

131. Kartu 131

     Pertanyaan

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

     Jawaban

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

132. Kartu 132

     Pertanyaan

     What conclusion does L’Hospital’s Rule permit?

     Jawaban

     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. Kartu 133

     Pertanyaan

     When is speed decreasing?

     Jawaban

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

134. Kartu 134

     Pertanyaan

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

     Jawaban

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

135. Kartu 135

     Pertanyaan

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

     Jawaban

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

136. Kartu 136

     Pertanyaan

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

     Jawaban

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

137. Kartu 137

     Pertanyaan

     Extreme Value Theorem: hypothesis and conclusion?

     Jawaban

     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. Kartu 138

     Pertanyaan

     What is a critical number of $f$?

     Jawaban

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

139. Kartu 139

     Pertanyaan

     First derivative test for a local maximum?

     Jawaban

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

140. Kartu 140

     Pertanyaan

     Second-derivative sign for concave up?

     Jawaban

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

141. Kartu 141

     Pertanyaan

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

     Jawaban

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

142. Kartu 142

     Pertanyaan

     First step in an optimization model?

     Jawaban

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

143. Kartu 143

     Pertanyaan

     Mean Value Theorem: hypotheses and conclusion?

     Jawaban

     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. Kartu 144

     Pertanyaan

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

     Jawaban

     Assuming $f$ is continuous on $[a,b]$, evaluate $f$ at every critical number in $(a,b)$ and at both endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.

     

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145. Kartu 145

     Pertanyaan

     First derivative test for a local minimum?

     Jawaban

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

146. Kartu 146

     Pertanyaan

     What must happen at an inflection point?

     Jawaban

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

147. Kartu 147

     Pertanyaan

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

     Jawaban

     $f''$ 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. Kartu 148

     Pertanyaan

     How do you confirm an optimization answer is absolute?

     Jawaban

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

149. Kartu 149

     Pertanyaan

     Rolle’s Theorem: hypotheses and conclusion?

     Jawaban

     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. Kartu 150

     Pertanyaan

     Difference between absolute and relative extrema?

     Jawaban

     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. Kartu 151

     Pertanyaan

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

     Jawaban

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

152. Kartu 152

     Pertanyaan

     Second derivative test for a local minimum?

     Jawaban

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

153. Kartu 153

     Pertanyaan

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

     Jawaban

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

154. Kartu 154

     Pertanyaan

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

     Jawaban

     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. Kartu 155

     Pertanyaan

     Which theorem links an average slope to an instantaneous slope?

     Jawaban

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

156. Kartu 156

     Pertanyaan

     How can an implicit derivative locate a horizontal tangent?

     Jawaban

     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. Kartu 157

     Pertanyaan

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

     Jawaban

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

158. Kartu 158

     Pertanyaan

     Second derivative test for a local maximum?

     Jawaban

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

159. Kartu 159

     Pertanyaan

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

     Jawaban

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

160. Kartu 160

     Pertanyaan

     Why must an optimization domain be stated?

     Jawaban

     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. Kartu 161

     Pertanyaan

     Which theorem guarantees absolute extrema, not where they occur?

     Jawaban

     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. Kartu 162

     Pertanyaan

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

     Jawaban

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

163. Kartu 163

     Pertanyaan

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

     Jawaban

     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. Kartu 164

     Pertanyaan

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

     Jawaban

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

165. Kartu 165

     Pertanyaan

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

     Jawaban

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

166. Kartu 166

     Pertanyaan

     How can an implicit derivative locate a vertical tangent?

     Jawaban

     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. Kartu 167

     Pertanyaan

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

     Jawaban

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

168. Kartu 168

     Pertanyaan

     Why are endpoints included in the candidates test?

     Jawaban

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

169. Kartu 169

     Pertanyaan

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

     Jawaban

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

170. Kartu 170

     Pertanyaan

     Second-derivative sign for concave down?

     Jawaban

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

171. Kartu 171

     Pertanyaan

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

     Jawaban

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

172. Kartu 172

     Pertanyaan

     What should the final line of an optimization solution state?

     Jawaban

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

173. Kartu 173

     Pertanyaan

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

     Jawaban

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

174. Kartu 174

     Pertanyaan

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

     Jawaban

     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. Kartu 175

     Pertanyaan

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

     Jawaban

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

176. Kartu 176

     Pertanyaan

     Left Riemann sum on equal subintervals?

     Jawaban

     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. Kartu 177

     Pertanyaan

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

     Jawaban

     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. Kartu 178

     Pertanyaan

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Jawaban

     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. Kartu 179

     Pertanyaan

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

     Jawaban

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kartu 180

     Pertanyaan

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

     Jawaban

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

181. Kartu 181

     Pertanyaan

     Right Riemann sum on equal subintervals?

     Jawaban

     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. Kartu 182

     Pertanyaan

     How does reversing integral bounds change the value?

     Jawaban

     It changes the sign:

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

183. Kartu 183

     Pertanyaan

     Net Change Theorem?

     Jawaban

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

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

184. Kartu 184

     Pertanyaan

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

     Jawaban

     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. Kartu 185

     Pertanyaan

     Power rule for antiderivatives?

     Jawaban

     For $n\ne-1$,

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

186. Kartu 186

     Pertanyaan

     Midpoint Riemann sum on equal subintervals?

     Jawaban

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

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

187. Kartu 187

     Pertanyaan

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

     Jawaban

     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. Kartu 188

     Pertanyaan

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

     Jawaban

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kartu 189

     Pertanyaan

     Antiderivative of $1/x$?

     Jawaban

     On any interval not crossing zero,

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

190. Kartu 190

     Pertanyaan

     Trapezoidal approximation on equal subintervals?

     Jawaban

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

191. Kartu 191

     Pertanyaan

     How do geometric regions help evaluate a definite integral?

     Jawaban

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

192. Kartu 192

     Pertanyaan

     Basic antiderivatives of sine and cosine?

     Jawaban

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

193. Kartu 193

     Pertanyaan

     Definite integral as a limit of Riemann sums?

     Jawaban

     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. Kartu 194

     Pertanyaan

     Constant-multiple rule for integrals?

     Jawaban

     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. Kartu 195

     Pertanyaan

     What pattern suggests $u$-substitution?

     Jawaban

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

196. Kartu 196

     Pertanyaan

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

     Jawaban

     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. Kartu 197

     Pertanyaan

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

     Jawaban

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

198. Kartu 198

     Pertanyaan

     Sum-and-difference rule for definite integrals?

     Jawaban

     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. Kartu 199

     Pertanyaan

     Basic antiderivative of $e^x$?

     Jawaban

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

200. Kartu 200

     Pertanyaan

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

     Jawaban

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

201. Kartu 201

     Pertanyaan

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

     Jawaban

     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. Kartu 202

     Pertanyaan

     How does concavity predict trapezoidal and midpoint error?

     Jawaban

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

203. Kartu 203

     Pertanyaan

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

     Jawaban

     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. Kartu 204

     Pertanyaan

     What denominator pattern suggests an arctangent antiderivative?

     Jawaban

     After completing the square and scaling, a form like

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

205. Kartu 205

     Pertanyaan

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

     Jawaban

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

206. Kartu 206

     Pertanyaan

     How does an initial condition determine an antiderivative?

     Jawaban

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

207. Kartu 207

     Pertanyaan

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

     Jawaban

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

208. Kartu 208

     Pertanyaan

     Why does an indefinite integral include $+C$?

     Jawaban

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

209. Kartu 209

     Pertanyaan

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

     Jawaban

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

210. Kartu 210

     Pertanyaan

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

     Jawaban

     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. Kartu 211

     Pertanyaan

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

     Jawaban

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

212. Kartu 212

     Pertanyaan

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

     Jawaban

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

213. Kartu 213

     Pertanyaan

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

     Jawaban

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

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

214. Kartu 214

     Pertanyaan

     Riemann sum for unequal subinterval widths?

     Jawaban

     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. Kartu 215

     Pertanyaan

     Does continuity guarantee integrability on a closed interval?

     Jawaban

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

216. Kartu 216

     Pertanyaan

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

     Jawaban

     Where $g(x)\ne0$,

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

217. Kartu 217

     Pertanyaan

     What algebraic rewrites often reveal a basic antiderivative?

     Jawaban

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

218. Kartu 218

     Pertanyaan

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

     Jawaban

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

219. Kartu 219

     Pertanyaan

     What is a differential equation?

     Jawaban

     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. Kartu 220

     Pertanyaan

     How does a verbal rate statement become a differential equation?

     Jawaban

     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. Kartu 221

     Pertanyaan

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

     Jawaban

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

222. Kartu 222

     Pertanyaan

     General solution versus particular solution?

     Jawaban

     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. Kartu 223

     Pertanyaan

     What does one segment in a slope field show?

     Jawaban

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

224. Kartu 224

     Pertanyaan

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

     Jawaban

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

225. Kartu 225

     Pertanyaan

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

     Jawaban

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

226. Kartu 226

     Pertanyaan

     What makes a first-order differential equation separable?

     Jawaban

     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. Kartu 227

     Pertanyaan

     What is an initial value problem?

     Jawaban

     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. Kartu 228

     Pertanyaan

     What is an isocline in a slope field?

     Jawaban

     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. Kartu 229

     Pertanyaan

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

     Jawaban

     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. Kartu 230

     Pertanyaan

     General solution of $dy/dt=ky$?

     Jawaban

     $$y=Ce^{kt}$$

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

231. Kartu 231

     Pertanyaan

     Core method for solving a separable differential equation?

     Jawaban

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

232. Kartu 232

     Pertanyaan

     How should a solution curve follow a slope field?

     Jawaban

     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. Kartu 233

     Pertanyaan

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

     Jawaban

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

234. Kartu 234

     Pertanyaan

     Why is one integration constant enough after integrating both sides?

     Jawaban

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

235. Kartu 235

     Pertanyaan

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

     Jawaban

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

236. Kartu 236

     Pertanyaan

     Can one differential equation have infinitely many solutions?

     Jawaban

     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. Kartu 237

     Pertanyaan

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

     Jawaban

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

238. Kartu 238

     Pertanyaan

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

     Jawaban

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

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

239. Kartu 239

     Pertanyaan

     What can be lost when dividing to separate variables?

     Jawaban

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

240. Kartu 240

     Pertanyaan

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

     Jawaban

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

241. Kartu 241

     Pertanyaan

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

     Jawaban

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

242. Kartu 242

     Pertanyaan

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

     Jawaban

     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. Kartu 243

     Pertanyaan

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

     Jawaban

     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. Kartu 244

     Pertanyaan

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

     Jawaban

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

245. Kartu 245

     Pertanyaan

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

     Jawaban

     For $k>0$,

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

     It is independent of the initial amount.

246. Kartu 246

     Pertanyaan

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

     Jawaban

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

247. Kartu 247

     Pertanyaan

     How is an initial condition used after separation?

     Jawaban

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

248. Kartu 248

     Pertanyaan

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

     Jawaban

     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. Kartu 249

     Pertanyaan

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

     Jawaban

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

250. Kartu 250

     Pertanyaan

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

     Jawaban

     For $k<0$,

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

251. Kartu 251

     Pertanyaan

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

     Jawaban

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

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

252. Kartu 252

     Pertanyaan

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

     Jawaban

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

     Velocity below zero contributes negative displacement.

253. Kartu 253

     Pertanyaan

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

     Jawaban

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

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

     Think top minus bottom.

254. Kartu 254

     Pertanyaan

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

     Jawaban

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

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

255. Kartu 255

     Pertanyaan

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Jawaban

     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. Kartu 256

     Pertanyaan

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

     Jawaban

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

257. Kartu 257

     Pertanyaan

     Cross-sectional area when each slice is a square?

     Jawaban

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

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

258. Kartu 258

     Pertanyaan

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

     Jawaban

     Integrate the net rate:

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

259. Kartu 259

     Pertanyaan

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

     Jawaban

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

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

     Think right minus left.

260. Kartu 260

     Pertanyaan

     Disc-method volume formula?

     Jawaban

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

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

261. Kartu 261

     Pertanyaan

     What units does average value have?

     Jawaban

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

262. Kartu 262

     Pertanyaan

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

     Jawaban

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

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

263. Kartu 263

     Pertanyaan

     Cross-sectional area when each slice is a rectangle?

     Jawaban

     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. Kartu 264

     Pertanyaan

     How do you determine bounds for area between curves?

     Jawaban

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

265. Kartu 265

     Pertanyaan

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

     Jawaban

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

266. Kartu 266

     Pertanyaan

     When is a particle moving to the right or left?

     Jawaban

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

267. Kartu 267

     Pertanyaan

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

     Jawaban

     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. Kartu 268

     Pertanyaan

     Why must an area integral be split where curves intersect?

     Jawaban

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

269. Kartu 269

     Pertanyaan

     How can a velocity table approximate displacement?

     Jawaban

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

270. Kartu 270

     Pertanyaan

     Washer-method volume formula?

     Jawaban

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

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

271. Kartu 271

     Pertanyaan

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

     Jawaban

     If $s(a)$ is known,

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

272. Kartu 272

     Pertanyaan

     How do you choose between vertical and horizontal area slices?

     Jawaban

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

273. Kartu 273

     Pertanyaan

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

     Jawaban

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

274. Kartu 274

     Pertanyaan

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

     Jawaban

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

275. Kartu 275

     Pertanyaan

     Single expression for area between two curves?

     Jawaban

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kartu 276

     Pertanyaan

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

     Jawaban

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

277. Kartu 277

     Pertanyaan

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

     Jawaban

     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. Kartu 278

     Pertanyaan

     When should a volume integral use $dy$?

     Jawaban

     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. Kartu 279

     Pertanyaan

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

     Jawaban

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

280. Kartu 280

     Pertanyaan

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

     Jawaban

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

281. Kartu 281

     Pertanyaan

     Why must total distance split at velocity sign changes?

     Jawaban

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

282. Kartu 282

     Pertanyaan

     When does an accumulated quantity reach a local maximum?

     Jawaban

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

283. Kartu 283

     Pertanyaan

     What distinguishes area from a definite integral?

     Jawaban

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

284. Kartu 284

     Pertanyaan

     How do position, velocity, and acceleration graphs correspond?

     Jawaban

     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. Kartu 285

     Pertanyaan

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

     Jawaban

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

286. Kartu 286

     Pertanyaan

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

     Jawaban

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

287. Kartu 287

     Pertanyaan

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

     Jawaban

     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. Kartu 288

     Pertanyaan

     Why should a contextual integral answer include a sentence?

     Jawaban

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