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

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1. Kad 1

   Soalan

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

   Jawapan

   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. Kad 2

   Soalan

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

   Jawapan

   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. Kad 3

   Soalan

   When does direct substitution evaluate a limit?

   Jawapan

   When the function is continuous at the target input. Then

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

4. Kad 4

   Soalan

   Three conditions for continuity at $x=a$?

   Jawapan

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

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

5. Kad 5

   Soalan

   Intermediate Value Theorem: hypotheses and conclusion?

   Jawapan

   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. Kad 6

   Soalan

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

   Jawapan

   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. Kad 7

   Soalan

   How do you read a finite limit from a graph?

   Jawapan

   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. Kad 8

   Soalan

   Limit law for a sum or difference?

   Jawapan

   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. Kad 9

   Soalan

   What makes a discontinuity removable?

   Jawapan

   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. Kad 10

    Soalan

    Squeeze Theorem: usable form?

    Jawapan

    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. Kad 11

    Soalan

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

    Jawapan

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

12. Kad 12

    Soalan

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

    Jawapan

    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. Kad 13

    Soalan

    Limit law for a product?

    Jawapan

    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. Kad 14

    Soalan

    Graph signature of a jump discontinuity?

    Jawapan

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

15. Kad 15

    Soalan

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

    Jawapan

    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. Kad 16

    Soalan

    Horizontal asymptote from a limit at infinity?

    Jawapan

    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. Kad 17

    Soalan

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

    Jawapan

    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. Kad 18

    Soalan

    Limit law for a quotient—and its condition?

    Jawapan

    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. Kad 19

    Soalan

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

    Jawapan

    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. Kad 20

    Soalan

    When is the Squeeze Theorem a natural choice?

    Jawapan

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

21. Kad 21

    Soalan

    Vertical asymptote from one-sided behavior?

    Jawapan

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

22. Kad 22

    Soalan

    Limit at infinity of equal-degree rational functions?

    Jawapan

    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. Kad 23

    Soalan

    When can a limit pass through a continuous outer function?

    Jawapan

    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. Kad 24

    Soalan

    What makes a discontinuity infinite?

    Jawapan

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

25. Kad 25

    Soalan

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

    Jawapan

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

26. Kad 26

    Soalan

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

    Jawapan

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kad 27

    Soalan

    Continuity of a composition?

    Jawapan

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

28. Kad 28

    Soalan

    Limit at infinity when a rational numerator has lower degree?

    Jawapan

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

29. Kad 29

    Soalan

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

    Jawapan

    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. Kad 30

    Soalan

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

    Jawapan

    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. Kad 31

    Soalan

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

    Jawapan

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

32. Kad 32

    Soalan

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

    Jawapan

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

33. Kad 33

    Soalan

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

    Jawapan

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

34. Kad 34

    Soalan

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

    Jawapan

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

35. Kad 35

    Soalan

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

    Jawapan

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

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

36. Kad 36

    Soalan

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

    Jawapan

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

    The derivative exists only if this finite limit exists.

37. Kad 37

    Soalan

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

    Jawapan

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

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

38. Kad 38

    Soalan

    What does differentiability imply about continuity?

    Jawapan

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

39. Kad 39

    Soalan

    Power rule for derivatives?

    Jawapan

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

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

40. Kad 40

    Soalan

    Units of $f'(x)$?

    Jawapan

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

41. Kad 41

    Soalan

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

    Jawapan

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

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

42. Kad 42

    Soalan

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

    Jawapan

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

    Soalan

    Derivative of a constant?

    Jawapan

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

    A constant function has zero rate of change.

44. Kad 44

    Soalan

    Derivative of $\sin x$?

    Jawapan

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

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

45. Kad 45

    Soalan

    Product rule?

    Jawapan

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

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

46. Kad 46

    Soalan

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

    Jawapan

    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. Kad 47

    Soalan

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

    Jawapan

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

48. Kad 48

    Soalan

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

    Jawapan

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

49. Kad 49

    Soalan

    Derivative of a sum or difference?

    Jawapan

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

50. Kad 50

    Soalan

    Derivative of $\cos x$?

    Jawapan

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

    The standard formula assumes radians.

51. Kad 51

    Soalan

    Quotient rule?

    Jawapan

    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. Kad 52

    Soalan

    Common notations for the first derivative?

    Jawapan

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

53. Kad 53

    Soalan

    Derivative of $e^x$?

    Jawapan

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

54. Kad 54

    Soalan

    What graph features can make $f$ nondifferentiable?

    Jawapan

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

55. Kad 55

    Soalan

    Derivative of $\tan x$?

    Jawapan

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kad 56

    Soalan

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

    Jawapan

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

57. Kad 57

    Soalan

    Derivative of $\ln x$?

    Jawapan

    For $x>0$,

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

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

58. Kad 58

    Soalan

    How does the power rule handle roots or negative powers?

    Jawapan

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

59. Kad 59

    Soalan

    Derivative of $\csc x$?

    Jawapan

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kad 60

    Soalan

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

    Jawapan

    $f$ is increasing on that interval.

61. Kad 61

    Soalan

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

    Jawapan

    For $a>0$,

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

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

62. Kad 62

    Soalan

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

    Jawapan

    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. Kad 63

    Soalan

    Derivative of $\sec x$?

    Jawapan

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kad 64

    Soalan

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

    Jawapan

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

65. Kad 65

    Soalan

    Derivative of $\log_a x$?

    Jawapan

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

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

66. Kad 66

    Soalan

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

    Jawapan

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kad 67

    Soalan

    Derivative of $\cot x$?

    Jawapan

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kad 68

    Soalan

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

    Jawapan

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

69. Kad 69

    Soalan

    Constant-multiple rule?

    Jawapan

    For a constant $c$,

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

70. Kad 70

    Soalan

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

    Jawapan

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

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

71. Kad 71

    Soalan

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

    Jawapan

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

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

72. Kad 72

    Soalan

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

    Jawapan

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

73. Kad 73

    Soalan

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

    Jawapan

    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. Kad 74

    Soalan

    Derivative of an inverse function at $x$?

    Jawapan

    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. Kad 75

    Soalan

    Derivative of $\arcsin x$?

    Jawapan

    For $|x|<1$,

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

76. Kad 76

    Soalan

    Notation for the third derivative of $f$?

    Jawapan

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

77. Kad 77

    Soalan

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

    Jawapan

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

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

78. Kad 78

    Soalan

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

    Jawapan

    Where $y\ne0$,

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

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

79. Kad 79

    Soalan

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

    Jawapan

    Provided $f'(a)\ne0$,

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

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

80. Kad 80

    Soalan

    Derivative of $\arctan x$?

    Jawapan

    For every real $x$,

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

81. Kad 81

    Soalan

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

    Jawapan

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

    The extra factor is the chain rule.

82. Kad 82

    Soalan

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

    Jawapan

    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. Kad 83

    Soalan

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

    Jawapan

    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. Kad 84

    Soalan

    Derivative of $\arccos x$?

    Jawapan

    For $|x|<1$,

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

85. Kad 85

    Soalan

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

    Jawapan

    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. Kad 86

    Soalan

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

    Jawapan

    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. Kad 87

    Soalan

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

    Jawapan

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

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

88. Kad 88

    Soalan

    How are tangent slopes of inverse graphs related?

    Jawapan

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

89. Kad 89

    Soalan

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

    Jawapan

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

90. Kad 90

    Soalan

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

    Jawapan

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

91. Kad 91

    Soalan

    Horizontal tangent on an implicit curve: derivative condition?

    Jawapan

    $dy/dx=0$ 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. Kad 92

    Soalan

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

    Jawapan

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

93. Kad 93

    Soalan

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

    Jawapan

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

94. Kad 94

    Soalan

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

    Jawapan

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

    This combines the power rule with the chain rule.

95. Kad 95

    Soalan

    Vertical tangent on an implicit curve: derivative clue?

    Jawapan

    $dy/dx$ 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. Kad 96

    Soalan

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

    Jawapan

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

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

97. Kad 97

    Soalan

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

    Jawapan

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

98. Kad 98

    Soalan

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

    Jawapan

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

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

99. Kad 99

    Soalan

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

    Jawapan

    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. Kad 100

     Soalan

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

     Jawapan

     When the functions are differentiable, the chain rule gives

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

101. Kad 101

     Soalan

     What local property lets a function have an inverse derivative?

     Jawapan

     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. Kad 102

     Soalan

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

     Jawapan

     For $a>0$,

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

103. Kad 103

     Soalan

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

     Jawapan

     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. Kad 104

     Soalan

     Position, velocity, and acceleration relationships?

     Jawapan

     For position $s(t)$,

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

105. Kad 105

     Soalan

     Central idea of a related-rates problem?

     Jawapan

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

106. Kad 106

     Soalan

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

     Jawapan

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

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

107. Kad 107

     Soalan

     L’Hospital’s Rule: basic conditions?

     Jawapan

     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. Kad 108

     Soalan

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

     Jawapan

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

109. Kad 109

     Soalan

     Speed in terms of velocity?

     Jawapan

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kad 110

     Soalan

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

     Jawapan

     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. Kad 111

     Soalan

     Differential approximation connecting $dx$ and $dy$?

     Jawapan

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

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

112. Kad 112

     Soalan

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

     Jawapan

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

113. Kad 113

     Soalan

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

     Jawapan

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

114. Kad 114

     Soalan

     What does positive acceleration say about velocity?

     Jawapan

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

115. Kad 115

     Soalan

     Related rates: when should numerical values be substituted?

     Jawapan

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

116. Kad 116

     Soalan

     How does concavity predict linearization error?

     Jawapan

     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. Kad 117

     Soalan

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

     Jawapan

     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. Kad 118

     Soalan

     When is a particle moving in the positive direction?

     Jawapan

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

119. Kad 119

     Soalan

     How can velocity show a change of direction?

     Jawapan

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

120. Kad 120

     Soalan

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

     Jawapan

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

121. Kad 121

     Soalan

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

     Jawapan

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

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

122. Kad 122

     Soalan

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

     Jawapan

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

123. Kad 123

     Soalan

     What must a contextual derivative sentence include?

     Jawapan

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

124. Kad 124

     Soalan

     Velocity negative and acceleration positive: what happens?

     Jawapan

     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. Kad 125

     Soalan

     How should a negative related rate be interpreted?

     Jawapan

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

126. Kad 126

     Soalan

     When is local linearity a sound approximation tool?

     Jawapan

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

127. Kad 127

     Soalan

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

     Jawapan

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

128. Kad 128

     Soalan

     When is speed increasing?

     Jawapan

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

129. Kad 129

     Soalan

     Volume changes with time: notation for its rate?

     Jawapan

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

130. Kad 130

     Soalan

     Why are similar triangles useful in related rates?

     Jawapan

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

131. Kad 131

     Soalan

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

     Jawapan

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

132. Kad 132

     Soalan

     What conclusion does L’Hospital’s Rule permit?

     Jawapan

     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. Kad 133

     Soalan

     When is speed decreasing?

     Jawapan

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

134. Kad 134

     Soalan

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

     Jawapan

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

135. Kad 135

     Soalan

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

     Jawapan

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

136. Kad 136

     Soalan

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

     Jawapan

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

137. Kad 137

     Soalan

     Extreme Value Theorem: hypothesis and conclusion?

     Jawapan

     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. Kad 138

     Soalan

     What is a critical number of $f$?

     Jawapan

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

139. Kad 139

     Soalan

     First derivative test for a local maximum?

     Jawapan

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

140. Kad 140

     Soalan

     Second-derivative sign for concave up?

     Jawapan

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

141. Kad 141

     Soalan

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

     Jawapan

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

142. Kad 142

     Soalan

     First step in an optimization model?

     Jawapan

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

143. Kad 143

     Soalan

     Mean Value Theorem: hypotheses and conclusion?

     Jawapan

     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. Kad 144

     Soalan

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

     Jawapan

     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. Kad 145

     Soalan

     First derivative test for a local minimum?

     Jawapan

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

146. Kad 146

     Soalan

     What must happen at an inflection point?

     Jawapan

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

147. Kad 147

     Soalan

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

     Jawapan

     $f''$ 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. Kad 148

     Soalan

     How do you confirm an optimization answer is absolute?

     Jawapan

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

149. Kad 149

     Soalan

     Rolle’s Theorem: hypotheses and conclusion?

     Jawapan

     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. Kad 150

     Soalan

     Difference between absolute and relative extrema?

     Jawapan

     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. Kad 151

     Soalan

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

     Jawapan

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

152. Kad 152

     Soalan

     Second derivative test for a local minimum?

     Jawapan

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

153. Kad 153

     Soalan

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

     Jawapan

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

154. Kad 154

     Soalan

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

     Jawapan

     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. Kad 155

     Soalan

     Which theorem links an average slope to an instantaneous slope?

     Jawapan

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

156. Kad 156

     Soalan

     How can an implicit derivative locate a horizontal tangent?

     Jawapan

     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. Kad 157

     Soalan

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

     Jawapan

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

158. Kad 158

     Soalan

     Second derivative test for a local maximum?

     Jawapan

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

159. Kad 159

     Soalan

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

     Jawapan

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

160. Kad 160

     Soalan

     Why must an optimization domain be stated?

     Jawapan

     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. Kad 161

     Soalan

     Which theorem guarantees absolute extrema, not where they occur?

     Jawapan

     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. Kad 162

     Soalan

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

     Jawapan

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

163. Kad 163

     Soalan

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

     Jawapan

     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. Kad 164

     Soalan

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

     Jawapan

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

165. Kad 165

     Soalan

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

     Jawapan

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

166. Kad 166

     Soalan

     How can an implicit derivative locate a vertical tangent?

     Jawapan

     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. Kad 167

     Soalan

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

     Jawapan

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

168. Kad 168

     Soalan

     Why are endpoints included in the candidates test?

     Jawapan

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

169. Kad 169

     Soalan

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

     Jawapan

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

170. Kad 170

     Soalan

     Second-derivative sign for concave down?

     Jawapan

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

171. Kad 171

     Soalan

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

     Jawapan

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

172. Kad 172

     Soalan

     What should the final line of an optimization solution state?

     Jawapan

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

173. Kad 173

     Soalan

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

     Jawapan

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

174. Kad 174

     Soalan

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

     Jawapan

     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. Kad 175

     Soalan

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

     Jawapan

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

176. Kad 176

     Soalan

     Left Riemann sum on equal subintervals?

     Jawapan

     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. Kad 177

     Soalan

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

     Jawapan

     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. Kad 178

     Soalan

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Jawapan

     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. Kad 179

     Soalan

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

     Jawapan

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kad 180

     Soalan

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

     Jawapan

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

181. Kad 181

     Soalan

     Right Riemann sum on equal subintervals?

     Jawapan

     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. Kad 182

     Soalan

     How does reversing integral bounds change the value?

     Jawapan

     It changes the sign:

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

183. Kad 183

     Soalan

     Net Change Theorem?

     Jawapan

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

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

184. Kad 184

     Soalan

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

     Jawapan

     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. Kad 185

     Soalan

     Power rule for antiderivatives?

     Jawapan

     For $n\ne-1$,

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

186. Kad 186

     Soalan

     Midpoint Riemann sum on equal subintervals?

     Jawapan

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

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

187. Kad 187

     Soalan

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

     Jawapan

     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. Kad 188

     Soalan

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

     Jawapan

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kad 189

     Soalan

     Antiderivative of $1/x$?

     Jawapan

     On any interval not crossing zero,

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

190. Kad 190

     Soalan

     Trapezoidal approximation on equal subintervals?

     Jawapan

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

191. Kad 191

     Soalan

     How do geometric regions help evaluate a definite integral?

     Jawapan

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

192. Kad 192

     Soalan

     Basic antiderivatives of sine and cosine?

     Jawapan

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

193. Kad 193

     Soalan

     Definite integral as a limit of Riemann sums?

     Jawapan

     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. Kad 194

     Soalan

     Constant-multiple rule for integrals?

     Jawapan

     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. Kad 195

     Soalan

     What pattern suggests $u$-substitution?

     Jawapan

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

196. Kad 196

     Soalan

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

     Jawapan

     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. Kad 197

     Soalan

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

     Jawapan

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

198. Kad 198

     Soalan

     Sum-and-difference rule for definite integrals?

     Jawapan

     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. Kad 199

     Soalan

     Basic antiderivative of $e^x$?

     Jawapan

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

200. Kad 200

     Soalan

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

     Jawapan

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

201. Kad 201

     Soalan

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

     Jawapan

     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. Kad 202

     Soalan

     How does concavity predict trapezoidal and midpoint error?

     Jawapan

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

203. Kad 203

     Soalan

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

     Jawapan

     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. Kad 204

     Soalan

     What denominator pattern suggests an arctangent antiderivative?

     Jawapan

     After completing the square and scaling, a form like

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

205. Kad 205

     Soalan

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

     Jawapan

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

206. Kad 206

     Soalan

     How does an initial condition determine an antiderivative?

     Jawapan

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

207. Kad 207

     Soalan

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

     Jawapan

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

208. Kad 208

     Soalan

     Why does an indefinite integral include $+C$?

     Jawapan

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

209. Kad 209

     Soalan

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

     Jawapan

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

210. Kad 210

     Soalan

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

     Jawapan

     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. Kad 211

     Soalan

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

     Jawapan

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

212. Kad 212

     Soalan

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

     Jawapan

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

213. Kad 213

     Soalan

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

     Jawapan

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

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

214. Kad 214

     Soalan

     Riemann sum for unequal subinterval widths?

     Jawapan

     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. Kad 215

     Soalan

     Does continuity guarantee integrability on a closed interval?

     Jawapan

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

216. Kad 216

     Soalan

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

     Jawapan

     Where $g(x)\ne0$,

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

217. Kad 217

     Soalan

     What algebraic rewrites often reveal a basic antiderivative?

     Jawapan

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

218. Kad 218

     Soalan

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

     Jawapan

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

219. Kad 219

     Soalan

     What is a differential equation?

     Jawapan

     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. Kad 220

     Soalan

     How does a verbal rate statement become a differential equation?

     Jawapan

     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. Kad 221

     Soalan

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

     Jawapan

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

222. Kad 222

     Soalan

     General solution versus particular solution?

     Jawapan

     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. Kad 223

     Soalan

     What does one segment in a slope field show?

     Jawapan

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

224. Kad 224

     Soalan

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

     Jawapan

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

225. Kad 225

     Soalan

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

     Jawapan

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

226. Kad 226

     Soalan

     What makes a first-order differential equation separable?

     Jawapan

     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. Kad 227

     Soalan

     What is an initial value problem?

     Jawapan

     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. Kad 228

     Soalan

     What is an isocline in a slope field?

     Jawapan

     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. Kad 229

     Soalan

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

     Jawapan

     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. Kad 230

     Soalan

     General solution of $dy/dt=ky$?

     Jawapan

     $$y=Ce^{kt}$$

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

231. Kad 231

     Soalan

     Core method for solving a separable differential equation?

     Jawapan

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

232. Kad 232

     Soalan

     How should a solution curve follow a slope field?

     Jawapan

     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. Kad 233

     Soalan

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

     Jawapan

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

234. Kad 234

     Soalan

     Why is one integration constant enough after integrating both sides?

     Jawapan

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

235. Kad 235

     Soalan

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

     Jawapan

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

236. Kad 236

     Soalan

     Can one differential equation have infinitely many solutions?

     Jawapan

     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. Kad 237

     Soalan

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

     Jawapan

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

238. Kad 238

     Soalan

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

     Jawapan

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

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

239. Kad 239

     Soalan

     What can be lost when dividing to separate variables?

     Jawapan

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

240. Kad 240

     Soalan

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

     Jawapan

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

241. Kad 241

     Soalan

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

     Jawapan

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

242. Kad 242

     Soalan

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

     Jawapan

     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. Kad 243

     Soalan

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

     Jawapan

     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. Kad 244

     Soalan

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

     Jawapan

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

245. Kad 245

     Soalan

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

     Jawapan

     For $k>0$,

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

     It is independent of the initial amount.

246. Kad 246

     Soalan

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

     Jawapan

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

247. Kad 247

     Soalan

     How is an initial condition used after separation?

     Jawapan

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

248. Kad 248

     Soalan

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

     Jawapan

     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. Kad 249

     Soalan

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

     Jawapan

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

250. Kad 250

     Soalan

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

     Jawapan

     For $k<0$,

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

251. Kad 251

     Soalan

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

     Jawapan

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

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

252. Kad 252

     Soalan

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

     Jawapan

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

     Velocity below zero contributes negative displacement.

253. Kad 253

     Soalan

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

     Jawapan

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

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

     Think top minus bottom.

254. Kad 254

     Soalan

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

     Jawapan

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

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

255. Kad 255

     Soalan

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Jawapan

     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. Kad 256

     Soalan

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

     Jawapan

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

257. Kad 257

     Soalan

     Cross-sectional area when each slice is a square?

     Jawapan

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

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

258. Kad 258

     Soalan

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

     Jawapan

     Integrate the net rate:

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

259. Kad 259

     Soalan

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

     Jawapan

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

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

     Think right minus left.

260. Kad 260

     Soalan

     Disc-method volume formula?

     Jawapan

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

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

261. Kad 261

     Soalan

     What units does average value have?

     Jawapan

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

262. Kad 262

     Soalan

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

     Jawapan

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

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

263. Kad 263

     Soalan

     Cross-sectional area when each slice is a rectangle?

     Jawapan

     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. Kad 264

     Soalan

     How do you determine bounds for area between curves?

     Jawapan

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

265. Kad 265

     Soalan

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

     Jawapan

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

266. Kad 266

     Soalan

     When is a particle moving to the right or left?

     Jawapan

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

267. Kad 267

     Soalan

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

     Jawapan

     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. Kad 268

     Soalan

     Why must an area integral be split where curves intersect?

     Jawapan

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

269. Kad 269

     Soalan

     How can a velocity table approximate displacement?

     Jawapan

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

270. Kad 270

     Soalan

     Washer-method volume formula?

     Jawapan

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

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

271. Kad 271

     Soalan

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

     Jawapan

     If $s(a)$ is known,

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

272. Kad 272

     Soalan

     How do you choose between vertical and horizontal area slices?

     Jawapan

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

273. Kad 273

     Soalan

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

     Jawapan

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

274. Kad 274

     Soalan

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

     Jawapan

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

275. Kad 275

     Soalan

     Single expression for area between two curves?

     Jawapan

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kad 276

     Soalan

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

     Jawapan

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

277. Kad 277

     Soalan

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

     Jawapan

     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. Kad 278

     Soalan

     When should a volume integral use $dy$?

     Jawapan

     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. Kad 279

     Soalan

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

     Jawapan

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

280. Kad 280

     Soalan

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

     Jawapan

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

281. Kad 281

     Soalan

     Why must total distance split at velocity sign changes?

     Jawapan

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

282. Kad 282

     Soalan

     When does an accumulated quantity reach a local maximum?

     Jawapan

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

283. Kad 283

     Soalan

     What distinguishes area from a definite integral?

     Jawapan

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

284. Kad 284

     Soalan

     How do position, velocity, and acceleration graphs correspond?

     Jawapan

     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. Kad 285

     Soalan

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

     Jawapan

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

286. Kad 286

     Soalan

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

     Jawapan

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

287. Kad 287

     Soalan

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

     Jawapan

     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. Kad 288

     Soalan

     Why should a contextual integral answer include a sentence?

     Jawapan

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