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

اس ڈیک کے کارڈز

1. کارڈ 1

   سوال

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

   جواب

   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. کارڈ 2

   سوال

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

   جواب

   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. کارڈ 3

   سوال

   When does direct substitution evaluate a limit?

   جواب

   When the function is continuous at the target input. Then

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

4. کارڈ 4

   سوال

   Three conditions for continuity at $x=a$?

   جواب

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

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

5. کارڈ 5

   سوال

   Intermediate Value Theorem: hypotheses and conclusion?

   جواب

   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. کارڈ 6

   سوال

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

   جواب

   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. کارڈ 7

   سوال

   How do you read a finite limit from a graph?

   جواب

   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. کارڈ 8

   سوال

   Limit law for a sum or difference?

   جواب

   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. کارڈ 9

   سوال

   What makes a discontinuity removable?

   جواب

   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. کارڈ 10

    سوال

    Squeeze Theorem: usable form?

    جواب

    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. کارڈ 11

    سوال

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

    جواب

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

12. کارڈ 12

    سوال

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

    جواب

    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. کارڈ 13

    سوال

    Limit law for a product?

    جواب

    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. کارڈ 14

    سوال

    Graph signature of a jump discontinuity?

    جواب

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

15. کارڈ 15

    سوال

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

    جواب

    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. کارڈ 16

    سوال

    Horizontal asymptote from a limit at infinity?

    جواب

    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. کارڈ 17

    سوال

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

    جواب

    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. کارڈ 18

    سوال

    Limit law for a quotient—and its condition?

    جواب

    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. کارڈ 19

    سوال

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

    جواب

    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. کارڈ 20

    سوال

    When is the Squeeze Theorem a natural choice?

    جواب

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

21. کارڈ 21

    سوال

    Vertical asymptote from one-sided behavior?

    جواب

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

22. کارڈ 22

    سوال

    Limit at infinity of equal-degree rational functions?

    جواب

    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. کارڈ 23

    سوال

    When can a limit pass through a continuous outer function?

    جواب

    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. کارڈ 24

    سوال

    What makes a discontinuity infinite?

    جواب

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

25. کارڈ 25

    سوال

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

    جواب

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

26. کارڈ 26

    سوال

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

    جواب

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. کارڈ 27

    سوال

    Continuity of a composition?

    جواب

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

28. کارڈ 28

    سوال

    Limit at infinity when a rational numerator has lower degree?

    جواب

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

29. کارڈ 29

    سوال

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

    جواب

    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. کارڈ 30

    سوال

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

    جواب

    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. کارڈ 31

    سوال

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

    جواب

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

32. کارڈ 32

    سوال

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

    جواب

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

33. کارڈ 33

    سوال

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

    جواب

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

34. کارڈ 34

    سوال

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

    جواب

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

35. کارڈ 35

    سوال

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

    جواب

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

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

36. کارڈ 36

    سوال

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

    جواب

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

    The derivative exists only if this finite limit exists.

37. کارڈ 37

    سوال

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

    جواب

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

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

38. کارڈ 38

    سوال

    What does differentiability imply about continuity?

    جواب

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

39. کارڈ 39

    سوال

    Power rule for derivatives?

    جواب

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

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

40. کارڈ 40

    سوال

    Units of $f'(x)$?

    جواب

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

41. کارڈ 41

    سوال

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

    جواب

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

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

42. کارڈ 42

    سوال

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

    جواب

    $f'(x)>0$ 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. کارڈ 43

    سوال

    Derivative of a constant?

    جواب

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

    A constant function has zero rate of change.

44. کارڈ 44

    سوال

    Derivative of $\sin x$?

    جواب

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

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

45. کارڈ 45

    سوال

    Product rule?

    جواب

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

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

46. کارڈ 46

    سوال

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

    جواب

    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. کارڈ 47

    سوال

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

    جواب

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

48. کارڈ 48

    سوال

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

    جواب

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

49. کارڈ 49

    سوال

    Derivative of a sum or difference?

    جواب

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

50. کارڈ 50

    سوال

    Derivative of $\cos x$?

    جواب

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

    The standard formula assumes radians.

51. کارڈ 51

    سوال

    Quotient rule?

    جواب

    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. کارڈ 52

    سوال

    Common notations for the first derivative?

    جواب

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

53. کارڈ 53

    سوال

    Derivative of $e^x$?

    جواب

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

54. کارڈ 54

    سوال

    What graph features can make $f$ nondifferentiable?

    جواب

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

55. کارڈ 55

    سوال

    Derivative of $\tan x$?

    جواب

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. کارڈ 56

    سوال

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

    جواب

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

57. کارڈ 57

    سوال

    Derivative of $\ln x$?

    جواب

    For $x>0$,

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

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

58. کارڈ 58

    سوال

    How does the power rule handle roots or negative powers?

    جواب

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

59. کارڈ 59

    سوال

    Derivative of $\csc x$?

    جواب

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. کارڈ 60

    سوال

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

    جواب

    $f$ is increasing on that interval.

61. کارڈ 61

    سوال

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

    جواب

    For $a>0$,

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

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

62. کارڈ 62

    سوال

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

    جواب

    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. کارڈ 63

    سوال

    Derivative of $\sec x$?

    جواب

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. کارڈ 64

    سوال

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

    جواب

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

65. کارڈ 65

    سوال

    Derivative of $\log_a x$?

    جواب

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

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

66. کارڈ 66

    سوال

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

    جواب

    For $h=fg$,

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

    Use the four table entries at the same input.

67. کارڈ 67

    سوال

    Derivative of $\cot x$?

    جواب

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. کارڈ 68

    سوال

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

    جواب

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

69. کارڈ 69

    سوال

    Constant-multiple rule?

    جواب

    For a constant $c$,

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

70. کارڈ 70

    سوال

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

    جواب

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

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

71. کارڈ 71

    سوال

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

    جواب

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

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

72. کارڈ 72

    سوال

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

    جواب

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

73. کارڈ 73

    سوال

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

    جواب

    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. کارڈ 74

    سوال

    Derivative of an inverse function at $x$?

    جواب

    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. کارڈ 75

    سوال

    Derivative of $\arcsin x$?

    جواب

    For $|x|<1$,

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

76. کارڈ 76

    سوال

    Notation for the third derivative of $f$?

    جواب

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

77. کارڈ 77

    سوال

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

    جواب

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

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

78. کارڈ 78

    سوال

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

    جواب

    Where $y\ne0$,

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

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

79. کارڈ 79

    سوال

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

    جواب

    Provided $f'(a)\ne0$,

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

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

80. کارڈ 80

    سوال

    Derivative of $\arctan x$?

    جواب

    For every real $x$,

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

81. کارڈ 81

    سوال

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

    جواب

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

    The extra factor is the chain rule.

82. کارڈ 82

    سوال

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

    جواب

    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. کارڈ 83

    سوال

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

    جواب

    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. کارڈ 84

    سوال

    Derivative of $\arccos x$?

    جواب

    For $|x|<1$,

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

85. کارڈ 85

    سوال

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

    جواب

    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. کارڈ 86

    سوال

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

    جواب

    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. کارڈ 87

    سوال

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

    جواب

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

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

88. کارڈ 88

    سوال

    How are tangent slopes of inverse graphs related?

    جواب

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

89. کارڈ 89

    سوال

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

    جواب

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

90. کارڈ 90

    سوال

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

    جواب

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

91. کارڈ 91

    سوال

    Horizontal tangent on an implicit curve: derivative condition?

    جواب

    $dy/dx=0$ 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. کارڈ 92

    سوال

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

    جواب

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

93. کارڈ 93

    سوال

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

    جواب

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

94. کارڈ 94

    سوال

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

    جواب

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

    This combines the power rule with the chain rule.

95. کارڈ 95

    سوال

    Vertical tangent on an implicit curve: derivative clue?

    جواب

    $dy/dx$ 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. کارڈ 96

    سوال

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

    جواب

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

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

97. کارڈ 97

    سوال

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

    جواب

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

98. کارڈ 98

    سوال

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

    جواب

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

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

99. کارڈ 99

    سوال

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

    جواب

    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. کارڈ 100

     سوال

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

     جواب

     When the functions are differentiable, the chain rule gives

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

101. کارڈ 101

     سوال

     What local property lets a function have an inverse derivative?

     جواب

     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. کارڈ 102

     سوال

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

     جواب

     For $a>0$,

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

103. کارڈ 103

     سوال

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

     جواب

     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. کارڈ 104

     سوال

     Position, velocity, and acceleration relationships?

     جواب

     For position $s(t)$,

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

105. کارڈ 105

     سوال

     Central idea of a related-rates problem?

     جواب

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

106. کارڈ 106

     سوال

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

     جواب

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

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

107. کارڈ 107

     سوال

     L’Hospital’s Rule: basic conditions?

     جواب

     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. کارڈ 108

     سوال

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

     جواب

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

109. کارڈ 109

     سوال

     Speed in terms of velocity?

     جواب

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

     Velocity includes direction; speed is nonnegative magnitude.

110. کارڈ 110

     سوال

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

     جواب

     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. کارڈ 111

     سوال

     Differential approximation connecting $dx$ and $dy$?

     جواب

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

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

112. کارڈ 112

     سوال

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

     جواب

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

113. کارڈ 113

     سوال

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

     جواب

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

114. کارڈ 114

     سوال

     What does positive acceleration say about velocity?

     جواب

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

115. کارڈ 115

     سوال

     Related rates: when should numerical values be substituted?

     جواب

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

116. کارڈ 116

     سوال

     How does concavity predict linearization error?

     جواب

     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. کارڈ 117

     سوال

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

     جواب

     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. کارڈ 118

     سوال

     When is a particle moving in the positive direction?

     جواب

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

119. کارڈ 119

     سوال

     How can velocity show a change of direction?

     جواب

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

120. کارڈ 120

     سوال

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

     جواب

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

121. کارڈ 121

     سوال

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

     جواب

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

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

122. کارڈ 122

     سوال

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

     جواب

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

123. کارڈ 123

     سوال

     What must a contextual derivative sentence include?

     جواب

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

124. کارڈ 124

     سوال

     Velocity negative and acceleration positive: what happens?

     جواب

     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. کارڈ 125

     سوال

     How should a negative related rate be interpreted?

     جواب

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

126. کارڈ 126

     سوال

     When is local linearity a sound approximation tool?

     جواب

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

127. کارڈ 127

     سوال

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

     جواب

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

128. کارڈ 128

     سوال

     When is speed increasing?

     جواب

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

129. کارڈ 129

     سوال

     Volume changes with time: notation for its rate?

     جواب

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

130. کارڈ 130

     سوال

     Why are similar triangles useful in related rates?

     جواب

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

131. کارڈ 131

     سوال

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

     جواب

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

132. کارڈ 132

     سوال

     What conclusion does L’Hospital’s Rule permit?

     جواب

     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. کارڈ 133

     سوال

     When is speed decreasing?

     جواب

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

134. کارڈ 134

     سوال

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

     جواب

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

135. کارڈ 135

     سوال

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

     جواب

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

136. کارڈ 136

     سوال

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

     جواب

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

137. کارڈ 137

     سوال

     Extreme Value Theorem: hypothesis and conclusion?

     جواب

     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. کارڈ 138

     سوال

     What is a critical number of $f$?

     جواب

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

139. کارڈ 139

     سوال

     First derivative test for a local maximum?

     جواب

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

140. کارڈ 140

     سوال

     Second-derivative sign for concave up?

     جواب

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

141. کارڈ 141

     سوال

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

     جواب

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

142. کارڈ 142

     سوال

     First step in an optimization model?

     جواب

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

143. کارڈ 143

     سوال

     Mean Value Theorem: hypotheses and conclusion?

     جواب

     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. کارڈ 144

     سوال

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

     جواب

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

     

     288 کارڈز

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

     یہ ڈیک مفت پڑھیں

     مطالعہ شروع کرنے کے لیے Nibomo کھلے گا۔

145. کارڈ 145

     سوال

     First derivative test for a local minimum?

     جواب

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

146. کارڈ 146

     سوال

     What must happen at an inflection point?

     جواب

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

147. کارڈ 147

     سوال

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

     جواب

     $f''$ 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. کارڈ 148

     سوال

     How do you confirm an optimization answer is absolute?

     جواب

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

149. کارڈ 149

     سوال

     Rolle’s Theorem: hypotheses and conclusion?

     جواب

     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. کارڈ 150

     سوال

     Difference between absolute and relative extrema?

     جواب

     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. کارڈ 151

     سوال

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

     جواب

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

152. کارڈ 152

     سوال

     Second derivative test for a local minimum?

     جواب

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

153. کارڈ 153

     سوال

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

     جواب

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

154. کارڈ 154

     سوال

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

     جواب

     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. کارڈ 155

     سوال

     Which theorem links an average slope to an instantaneous slope?

     جواب

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

156. کارڈ 156

     سوال

     How can an implicit derivative locate a horizontal tangent?

     جواب

     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. کارڈ 157

     سوال

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

     جواب

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

158. کارڈ 158

     سوال

     Second derivative test for a local maximum?

     جواب

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

159. کارڈ 159

     سوال

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

     جواب

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

160. کارڈ 160

     سوال

     Why must an optimization domain be stated?

     جواب

     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. کارڈ 161

     سوال

     Which theorem guarantees absolute extrema, not where they occur?

     جواب

     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. کارڈ 162

     سوال

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

     جواب

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

163. کارڈ 163

     سوال

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

     جواب

     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. کارڈ 164

     سوال

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

     جواب

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

165. کارڈ 165

     سوال

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

     جواب

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

166. کارڈ 166

     سوال

     How can an implicit derivative locate a vertical tangent?

     جواب

     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. کارڈ 167

     سوال

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

     جواب

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

168. کارڈ 168

     سوال

     Why are endpoints included in the candidates test?

     جواب

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

169. کارڈ 169

     سوال

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

     جواب

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

170. کارڈ 170

     سوال

     Second-derivative sign for concave down?

     جواب

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

171. کارڈ 171

     سوال

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

     جواب

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

172. کارڈ 172

     سوال

     What should the final line of an optimization solution state?

     جواب

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

173. کارڈ 173

     سوال

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

     جواب

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

174. کارڈ 174

     سوال

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

     جواب

     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. کارڈ 175

     سوال

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

     جواب

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

176. کارڈ 176

     سوال

     Left Riemann sum on equal subintervals?

     جواب

     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. کارڈ 177

     سوال

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

     جواب

     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. کارڈ 178

     سوال

     Fundamental Theorem of Calculus: evaluate a definite integral?

     جواب

     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. کارڈ 179

     سوال

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

     جواب

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. کارڈ 180

     سوال

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

     جواب

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

181. کارڈ 181

     سوال

     Right Riemann sum on equal subintervals?

     جواب

     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. کارڈ 182

     سوال

     How does reversing integral bounds change the value?

     جواب

     It changes the sign:

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

183. کارڈ 183

     سوال

     Net Change Theorem?

     جواب

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

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

184. کارڈ 184

     سوال

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

     جواب

     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. کارڈ 185

     سوال

     Power rule for antiderivatives?

     جواب

     For $n\ne-1$,

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

186. کارڈ 186

     سوال

     Midpoint Riemann sum on equal subintervals?

     جواب

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

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

187. کارڈ 187

     سوال

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

     جواب

     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. کارڈ 188

     سوال

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

     جواب

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. کارڈ 189

     سوال

     Antiderivative of $1/x$?

     جواب

     On any interval not crossing zero,

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

190. کارڈ 190

     سوال

     Trapezoidal approximation on equal subintervals?

     جواب

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

191. کارڈ 191

     سوال

     How do geometric regions help evaluate a definite integral?

     جواب

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

192. کارڈ 192

     سوال

     Basic antiderivatives of sine and cosine?

     جواب

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

193. کارڈ 193

     سوال

     Definite integral as a limit of Riemann sums?

     جواب

     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. کارڈ 194

     سوال

     Constant-multiple rule for integrals?

     جواب

     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. کارڈ 195

     سوال

     What pattern suggests $u$-substitution?

     جواب

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

196. کارڈ 196

     سوال

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

     جواب

     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. کارڈ 197

     سوال

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

     جواب

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

198. کارڈ 198

     سوال

     Sum-and-difference rule for definite integrals?

     جواب

     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. کارڈ 199

     سوال

     Basic antiderivative of $e^x$?

     جواب

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

200. کارڈ 200

     سوال

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

     جواب

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

201. کارڈ 201

     سوال

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

     جواب

     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. کارڈ 202

     سوال

     How does concavity predict trapezoidal and midpoint error?

     جواب

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

203. کارڈ 203

     سوال

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

     جواب

     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. کارڈ 204

     سوال

     What denominator pattern suggests an arctangent antiderivative?

     جواب

     After completing the square and scaling, a form like

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

205. کارڈ 205

     سوال

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

     جواب

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

206. کارڈ 206

     سوال

     How does an initial condition determine an antiderivative?

     جواب

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

207. کارڈ 207

     سوال

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

     جواب

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

208. کارڈ 208

     سوال

     Why does an indefinite integral include $+C$?

     جواب

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

209. کارڈ 209

     سوال

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

     جواب

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

210. کارڈ 210

     سوال

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

     جواب

     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. کارڈ 211

     سوال

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

     جواب

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

212. کارڈ 212

     سوال

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

     جواب

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

213. کارڈ 213

     سوال

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

     جواب

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

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

214. کارڈ 214

     سوال

     Riemann sum for unequal subinterval widths?

     جواب

     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. کارڈ 215

     سوال

     Does continuity guarantee integrability on a closed interval?

     جواب

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

216. کارڈ 216

     سوال

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

     جواب

     Where $g(x)\ne0$,

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

217. کارڈ 217

     سوال

     What algebraic rewrites often reveal a basic antiderivative?

     جواب

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

218. کارڈ 218

     سوال

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

     جواب

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

219. کارڈ 219

     سوال

     What is a differential equation?

     جواب

     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. کارڈ 220

     سوال

     How does a verbal rate statement become a differential equation?

     جواب

     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. کارڈ 221

     سوال

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

     جواب

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

222. کارڈ 222

     سوال

     General solution versus particular solution?

     جواب

     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. کارڈ 223

     سوال

     What does one segment in a slope field show?

     جواب

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

224. کارڈ 224

     سوال

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

     جواب

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

225. کارڈ 225

     سوال

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

     جواب

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

226. کارڈ 226

     سوال

     What makes a first-order differential equation separable?

     جواب

     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. کارڈ 227

     سوال

     What is an initial value problem?

     جواب

     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. کارڈ 228

     سوال

     What is an isocline in a slope field?

     جواب

     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. کارڈ 229

     سوال

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

     جواب

     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. کارڈ 230

     سوال

     General solution of $dy/dt=ky$?

     جواب

     $$y=Ce^{kt}$$

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

231. کارڈ 231

     سوال

     Core method for solving a separable differential equation?

     جواب

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

232. کارڈ 232

     سوال

     How should a solution curve follow a slope field?

     جواب

     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. کارڈ 233

     سوال

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

     جواب

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

234. کارڈ 234

     سوال

     Why is one integration constant enough after integrating both sides?

     جواب

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

235. کارڈ 235

     سوال

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

     جواب

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

236. کارڈ 236

     سوال

     Can one differential equation have infinitely many solutions?

     جواب

     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. کارڈ 237

     سوال

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

     جواب

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

238. کارڈ 238

     سوال

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

     جواب

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

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

239. کارڈ 239

     سوال

     What can be lost when dividing to separate variables?

     جواب

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

240. کارڈ 240

     سوال

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

     جواب

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

241. کارڈ 241

     سوال

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

     جواب

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

242. کارڈ 242

     سوال

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

     جواب

     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. کارڈ 243

     سوال

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

     جواب

     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. کارڈ 244

     سوال

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

     جواب

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

245. کارڈ 245

     سوال

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

     جواب

     For $k>0$,

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

     It is independent of the initial amount.

246. کارڈ 246

     سوال

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

     جواب

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

247. کارڈ 247

     سوال

     How is an initial condition used after separation?

     جواب

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

248. کارڈ 248

     سوال

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

     جواب

     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. کارڈ 249

     سوال

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

     جواب

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

250. کارڈ 250

     سوال

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

     جواب

     For $k<0$,

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

251. کارڈ 251

     سوال

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

     جواب

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

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

252. کارڈ 252

     سوال

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

     جواب

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

     Velocity below zero contributes negative displacement.

253. کارڈ 253

     سوال

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

     جواب

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

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

     Think top minus bottom.

254. کارڈ 254

     سوال

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

     جواب

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

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

255. کارڈ 255

     سوال

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     جواب

     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. کارڈ 256

     سوال

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

     جواب

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

257. کارڈ 257

     سوال

     Cross-sectional area when each slice is a square?

     جواب

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

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

258. کارڈ 258

     سوال

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

     جواب

     Integrate the net rate:

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

259. کارڈ 259

     سوال

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

     جواب

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

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

     Think right minus left.

260. کارڈ 260

     سوال

     Disc-method volume formula?

     جواب

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

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

261. کارڈ 261

     سوال

     What units does average value have?

     جواب

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

262. کارڈ 262

     سوال

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

     جواب

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

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

263. کارڈ 263

     سوال

     Cross-sectional area when each slice is a rectangle?

     جواب

     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. کارڈ 264

     سوال

     How do you determine bounds for area between curves?

     جواب

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

265. کارڈ 265

     سوال

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

     جواب

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

266. کارڈ 266

     سوال

     When is a particle moving to the right or left?

     جواب

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

267. کارڈ 267

     سوال

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

     جواب

     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. کارڈ 268

     سوال

     Why must an area integral be split where curves intersect?

     جواب

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

269. کارڈ 269

     سوال

     How can a velocity table approximate displacement?

     جواب

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

270. کارڈ 270

     سوال

     Washer-method volume formula?

     جواب

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

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

271. کارڈ 271

     سوال

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

     جواب

     If $s(a)$ is known,

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

272. کارڈ 272

     سوال

     How do you choose between vertical and horizontal area slices?

     جواب

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

273. کارڈ 273

     سوال

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

     جواب

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

274. کارڈ 274

     سوال

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

     جواب

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

275. کارڈ 275

     سوال

     Single expression for area between two curves?

     جواب

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. کارڈ 276

     سوال

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

     جواب

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

277. کارڈ 277

     سوال

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

     جواب

     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. کارڈ 278

     سوال

     When should a volume integral use $dy$?

     جواب

     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. کارڈ 279

     سوال

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

     جواب

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

280. کارڈ 280

     سوال

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

     جواب

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

281. کارڈ 281

     سوال

     Why must total distance split at velocity sign changes?

     جواب

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

282. کارڈ 282

     سوال

     When does an accumulated quantity reach a local maximum?

     جواب

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

283. کارڈ 283

     سوال

     What distinguishes area from a definite integral?

     جواب

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

284. کارڈ 284

     سوال

     How do position, velocity, and acceleration graphs correspond?

     جواب

     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. کارڈ 285

     سوال

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

     جواب

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

286. کارڈ 286

     سوال

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

     جواب

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

287. کارڈ 287

     سوال

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

     جواب

     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. کارڈ 288

     سوال

     Why should a contextual integral answer include a sentence?

     جواب

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