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

کارت‌های این دسته

1. کارت ۱

   پرسش

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

   پاسخ

   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. کارت ۲

   پرسش

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

   پاسخ

   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. کارت ۳

   پرسش

   When does direct substitution evaluate a limit?

   پاسخ

   When the function is continuous at the target input. Then

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

4. کارت ۴

   پرسش

   Three conditions for continuity at $x=a$?

   پاسخ

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

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

5. کارت ۵

   پرسش

   Intermediate Value Theorem: hypotheses and conclusion?

   پاسخ

   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. کارت ۶

   پرسش

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

   پاسخ

   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. کارت ۷

   پرسش

   How do you read a finite limit from a graph?

   پاسخ

   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. کارت ۸

   پرسش

   Limit law for a sum or difference?

   پاسخ

   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. کارت ۹

   پرسش

   What makes a discontinuity removable?

   پاسخ

   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. کارت ۱۰

    پرسش

    Squeeze Theorem: usable form?

    پاسخ

    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. کارت ۱۱

    پرسش

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

    پاسخ

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

12. کارت ۱۲

    پرسش

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

    پاسخ

    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. کارت ۱۳

    پرسش

    Limit law for a product?

    پاسخ

    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. کارت ۱۴

    پرسش

    Graph signature of a jump discontinuity?

    پاسخ

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

15. کارت ۱۵

    پرسش

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

    پاسخ

    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. کارت ۱۶

    پرسش

    Horizontal asymptote from a limit at infinity?

    پاسخ

    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. کارت ۱۷

    پرسش

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

    پاسخ

    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. کارت ۱۸

    پرسش

    Limit law for a quotient—and its condition?

    پاسخ

    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. کارت ۱۹

    پرسش

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

    پاسخ

    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. کارت ۲۰

    پرسش

    When is the Squeeze Theorem a natural choice?

    پاسخ

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

21. کارت ۲۱

    پرسش

    Vertical asymptote from one-sided behavior?

    پاسخ

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

22. کارت ۲۲

    پرسش

    Limit at infinity of equal-degree rational functions?

    پاسخ

    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. کارت ۲۳

    پرسش

    When can a limit pass through a continuous outer function?

    پاسخ

    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. کارت ۲۴

    پرسش

    What makes a discontinuity infinite?

    پاسخ

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

25. کارت ۲۵

    پرسش

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

    پاسخ

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

26. کارت ۲۶

    پرسش

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

    پاسخ

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. کارت ۲۷

    پرسش

    Continuity of a composition?

    پاسخ

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

28. کارت ۲۸

    پرسش

    Limit at infinity when a rational numerator has lower degree?

    پاسخ

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

29. کارت ۲۹

    پرسش

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

    پاسخ

    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. کارت ۳۰

    پرسش

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

    پاسخ

    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. کارت ۳۱

    پرسش

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

    پاسخ

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

32. کارت ۳۲

    پرسش

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

    پاسخ

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

33. کارت ۳۳

    پرسش

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

    پاسخ

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

34. کارت ۳۴

    پرسش

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

    پاسخ

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

35. کارت ۳۵

    پرسش

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

    پاسخ

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

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

36. کارت ۳۶

    پرسش

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

    پاسخ

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

    The derivative exists only if this finite limit exists.

37. کارت ۳۷

    پرسش

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

    پاسخ

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

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

38. کارت ۳۸

    پرسش

    What does differentiability imply about continuity?

    پاسخ

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

39. کارت ۳۹

    پرسش

    Power rule for derivatives?

    پاسخ

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

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

40. کارت ۴۰

    پرسش

    Units of $f'(x)$?

    پاسخ

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

41. کارت ۴۱

    پرسش

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

    پاسخ

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

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

42. کارت ۴۲

    پرسش

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

    پاسخ

    $f'(x)>0$ 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. کارت ۴۳

    پرسش

    Derivative of a constant?

    پاسخ

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

    A constant function has zero rate of change.

44. کارت ۴۴

    پرسش

    Derivative of $\sin x$?

    پاسخ

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

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

45. کارت ۴۵

    پرسش

    Product rule?

    پاسخ

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

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

46. کارت ۴۶

    پرسش

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

    پاسخ

    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. کارت ۴۷

    پرسش

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

    پاسخ

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

48. کارت ۴۸

    پرسش

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

    پاسخ

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

49. کارت ۴۹

    پرسش

    Derivative of a sum or difference?

    پاسخ

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

50. کارت ۵۰

    پرسش

    Derivative of $\cos x$?

    پاسخ

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

    The standard formula assumes radians.

51. کارت ۵۱

    پرسش

    Quotient rule?

    پاسخ

    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. کارت ۵۲

    پرسش

    Common notations for the first derivative?

    پاسخ

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

53. کارت ۵۳

    پرسش

    Derivative of $e^x$?

    پاسخ

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

54. کارت ۵۴

    پرسش

    What graph features can make $f$ nondifferentiable?

    پاسخ

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

55. کارت ۵۵

    پرسش

    Derivative of $\tan x$?

    پاسخ

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. کارت ۵۶

    پرسش

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

    پاسخ

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

57. کارت ۵۷

    پرسش

    Derivative of $\ln x$?

    پاسخ

    For $x>0$,

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

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

58. کارت ۵۸

    پرسش

    How does the power rule handle roots or negative powers?

    پاسخ

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

59. کارت ۵۹

    پرسش

    Derivative of $\csc x$?

    پاسخ

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. کارت ۶۰

    پرسش

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

    پاسخ

    $f$ is increasing on that interval.

61. کارت ۶۱

    پرسش

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

    پاسخ

    For $a>0$,

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

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

62. کارت ۶۲

    پرسش

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

    پاسخ

    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. کارت ۶۳

    پرسش

    Derivative of $\sec x$?

    پاسخ

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. کارت ۶۴

    پرسش

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

    پاسخ

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

65. کارت ۶۵

    پرسش

    Derivative of $\log_a x$?

    پاسخ

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

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

66. کارت ۶۶

    پرسش

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

    پاسخ

    For $h=fg$,

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

    Use the four table entries at the same input.

67. کارت ۶۷

    پرسش

    Derivative of $\cot x$?

    پاسخ

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. کارت ۶۸

    پرسش

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

    پاسخ

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

69. کارت ۶۹

    پرسش

    Constant-multiple rule?

    پاسخ

    For a constant $c$,

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

70. کارت ۷۰

    پرسش

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

    پاسخ

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

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

71. کارت ۷۱

    پرسش

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

    پاسخ

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

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

72. کارت ۷۲

    پرسش

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

    پاسخ

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

73. کارت ۷۳

    پرسش

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

    پاسخ

    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. کارت ۷۴

    پرسش

    Derivative of an inverse function at $x$?

    پاسخ

    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. کارت ۷۵

    پرسش

    Derivative of $\arcsin x$?

    پاسخ

    For $|x|<1$,

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

76. کارت ۷۶

    پرسش

    Notation for the third derivative of $f$?

    پاسخ

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

77. کارت ۷۷

    پرسش

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

    پاسخ

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

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

78. کارت ۷۸

    پرسش

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

    پاسخ

    Where $y\ne0$,

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

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

79. کارت ۷۹

    پرسش

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

    پاسخ

    Provided $f'(a)\ne0$,

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

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

80. کارت ۸۰

    پرسش

    Derivative of $\arctan x$?

    پاسخ

    For every real $x$,

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

81. کارت ۸۱

    پرسش

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

    پاسخ

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

    The extra factor is the chain rule.

82. کارت ۸۲

    پرسش

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

    پاسخ

    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. کارت ۸۳

    پرسش

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

    پاسخ

    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. کارت ۸۴

    پرسش

    Derivative of $\arccos x$?

    پاسخ

    For $|x|<1$,

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

85. کارت ۸۵

    پرسش

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

    پاسخ

    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. کارت ۸۶

    پرسش

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

    پاسخ

    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. کارت ۸۷

    پرسش

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

    پاسخ

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

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

88. کارت ۸۸

    پرسش

    How are tangent slopes of inverse graphs related?

    پاسخ

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

89. کارت ۸۹

    پرسش

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

    پاسخ

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

90. کارت ۹۰

    پرسش

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

    پاسخ

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

91. کارت ۹۱

    پرسش

    Horizontal tangent on an implicit curve: derivative condition?

    پاسخ

    $dy/dx=0$ 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. کارت ۹۲

    پرسش

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

    پاسخ

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

93. کارت ۹۳

    پرسش

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

    پاسخ

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

94. کارت ۹۴

    پرسش

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

    پاسخ

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

    This combines the power rule with the chain rule.

95. کارت ۹۵

    پرسش

    Vertical tangent on an implicit curve: derivative clue?

    پاسخ

    $dy/dx$ 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. کارت ۹۶

    پرسش

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

    پاسخ

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

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

97. کارت ۹۷

    پرسش

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

    پاسخ

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

98. کارت ۹۸

    پرسش

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

    پاسخ

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

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

99. کارت ۹۹

    پرسش

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

    پاسخ

    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. کارت ۱۰۰

     پرسش

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

     پاسخ

     When the functions are differentiable, the chain rule gives

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

101. کارت ۱۰۱

     پرسش

     What local property lets a function have an inverse derivative?

     پاسخ

     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. کارت ۱۰۲

     پرسش

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

     پاسخ

     For $a>0$,

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

103. کارت ۱۰۳

     پرسش

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

     پاسخ

     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. کارت ۱۰۴

     پرسش

     Position, velocity, and acceleration relationships?

     پاسخ

     For position $s(t)$,

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

105. کارت ۱۰۵

     پرسش

     Central idea of a related-rates problem?

     پاسخ

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

106. کارت ۱۰۶

     پرسش

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

     پاسخ

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

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

107. کارت ۱۰۷

     پرسش

     L’Hospital’s Rule: basic conditions?

     پاسخ

     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. کارت ۱۰۸

     پرسش

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

     پاسخ

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

109. کارت ۱۰۹

     پرسش

     Speed in terms of velocity?

     پاسخ

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

     Velocity includes direction; speed is nonnegative magnitude.

110. کارت ۱۱۰

     پرسش

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

     پاسخ

     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. کارت ۱۱۱

     پرسش

     Differential approximation connecting $dx$ and $dy$?

     پاسخ

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

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

112. کارت ۱۱۲

     پرسش

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

     پاسخ

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

113. کارت ۱۱۳

     پرسش

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

     پاسخ

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

114. کارت ۱۱۴

     پرسش

     What does positive acceleration say about velocity?

     پاسخ

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

115. کارت ۱۱۵

     پرسش

     Related rates: when should numerical values be substituted?

     پاسخ

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

116. کارت ۱۱۶

     پرسش

     How does concavity predict linearization error?

     پاسخ

     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. کارت ۱۱۷

     پرسش

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

     پاسخ

     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. کارت ۱۱۸

     پرسش

     When is a particle moving in the positive direction?

     پاسخ

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

119. کارت ۱۱۹

     پرسش

     How can velocity show a change of direction?

     پاسخ

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

120. کارت ۱۲۰

     پرسش

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

     پاسخ

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

121. کارت ۱۲۱

     پرسش

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

     پاسخ

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

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

122. کارت ۱۲۲

     پرسش

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

     پاسخ

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

123. کارت ۱۲۳

     پرسش

     What must a contextual derivative sentence include?

     پاسخ

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

124. کارت ۱۲۴

     پرسش

     Velocity negative and acceleration positive: what happens?

     پاسخ

     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. کارت ۱۲۵

     پرسش

     How should a negative related rate be interpreted?

     پاسخ

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

126. کارت ۱۲۶

     پرسش

     When is local linearity a sound approximation tool?

     پاسخ

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

127. کارت ۱۲۷

     پرسش

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

     پاسخ

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

128. کارت ۱۲۸

     پرسش

     When is speed increasing?

     پاسخ

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

129. کارت ۱۲۹

     پرسش

     Volume changes with time: notation for its rate?

     پاسخ

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

130. کارت ۱۳۰

     پرسش

     Why are similar triangles useful in related rates?

     پاسخ

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

131. کارت ۱۳۱

     پرسش

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

     پاسخ

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

132. کارت ۱۳۲

     پرسش

     What conclusion does L’Hospital’s Rule permit?

     پاسخ

     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. کارت ۱۳۳

     پرسش

     When is speed decreasing?

     پاسخ

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

134. کارت ۱۳۴

     پرسش

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

     پاسخ

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

135. کارت ۱۳۵

     پرسش

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

     پاسخ

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

136. کارت ۱۳۶

     پرسش

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

     پاسخ

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

137. کارت ۱۳۷

     پرسش

     Extreme Value Theorem: hypothesis and conclusion?

     پاسخ

     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. کارت ۱۳۸

     پرسش

     What is a critical number of $f$?

     پاسخ

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

139. کارت ۱۳۹

     پرسش

     First derivative test for a local maximum?

     پاسخ

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

140. کارت ۱۴۰

     پرسش

     Second-derivative sign for concave up?

     پاسخ

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

141. کارت ۱۴۱

     پرسش

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

     پاسخ

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

142. کارت ۱۴۲

     پرسش

     First step in an optimization model?

     پاسخ

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

143. کارت ۱۴۳

     پرسش

     Mean Value Theorem: hypotheses and conclusion?

     پاسخ

     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. کارت ۱۴۴

     پرسش

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

     پاسخ

     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.

     

     ۲۸۸ کارت

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

     این دسته را رایگان مطالعه کنید

     Nibomo باز می‌شود تا بتوانید مطالعه را شروع کنید.

145. کارت ۱۴۵

     پرسش

     First derivative test for a local minimum?

     پاسخ

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

146. کارت ۱۴۶

     پرسش

     What must happen at an inflection point?

     پاسخ

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

147. کارت ۱۴۷

     پرسش

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

     پاسخ

     $f''$ 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. کارت ۱۴۸

     پرسش

     How do you confirm an optimization answer is absolute?

     پاسخ

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

149. کارت ۱۴۹

     پرسش

     Rolle’s Theorem: hypotheses and conclusion?

     پاسخ

     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. کارت ۱۵۰

     پرسش

     Difference between absolute and relative extrema?

     پاسخ

     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. کارت ۱۵۱

     پرسش

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

     پاسخ

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

152. کارت ۱۵۲

     پرسش

     Second derivative test for a local minimum?

     پاسخ

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

153. کارت ۱۵۳

     پرسش

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

     پاسخ

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

154. کارت ۱۵۴

     پرسش

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

     پاسخ

     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. کارت ۱۵۵

     پرسش

     Which theorem links an average slope to an instantaneous slope?

     پاسخ

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

156. کارت ۱۵۶

     پرسش

     How can an implicit derivative locate a horizontal tangent?

     پاسخ

     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. کارت ۱۵۷

     پرسش

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

     پاسخ

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

158. کارت ۱۵۸

     پرسش

     Second derivative test for a local maximum?

     پاسخ

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

159. کارت ۱۵۹

     پرسش

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

     پاسخ

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

160. کارت ۱۶۰

     پرسش

     Why must an optimization domain be stated?

     پاسخ

     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. کارت ۱۶۱

     پرسش

     Which theorem guarantees absolute extrema, not where they occur?

     پاسخ

     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. کارت ۱۶۲

     پرسش

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

     پاسخ

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

163. کارت ۱۶۳

     پرسش

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

     پاسخ

     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. کارت ۱۶۴

     پرسش

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

     پاسخ

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

165. کارت ۱۶۵

     پرسش

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

     پاسخ

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

166. کارت ۱۶۶

     پرسش

     How can an implicit derivative locate a vertical tangent?

     پاسخ

     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. کارت ۱۶۷

     پرسش

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

     پاسخ

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

168. کارت ۱۶۸

     پرسش

     Why are endpoints included in the candidates test?

     پاسخ

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

169. کارت ۱۶۹

     پرسش

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

     پاسخ

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

170. کارت ۱۷۰

     پرسش

     Second-derivative sign for concave down?

     پاسخ

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

171. کارت ۱۷۱

     پرسش

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

     پاسخ

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

172. کارت ۱۷۲

     پرسش

     What should the final line of an optimization solution state?

     پاسخ

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

173. کارت ۱۷۳

     پرسش

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

     پاسخ

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

174. کارت ۱۷۴

     پرسش

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

     پاسخ

     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. کارت ۱۷۵

     پرسش

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

     پاسخ

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

176. کارت ۱۷۶

     پرسش

     Left Riemann sum on equal subintervals?

     پاسخ

     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. کارت ۱۷۷

     پرسش

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

     پاسخ

     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. کارت ۱۷۸

     پرسش

     Fundamental Theorem of Calculus: evaluate a definite integral?

     پاسخ

     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. کارت ۱۷۹

     پرسش

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

     پاسخ

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. کارت ۱۸۰

     پرسش

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

     پاسخ

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

181. کارت ۱۸۱

     پرسش

     Right Riemann sum on equal subintervals?

     پاسخ

     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. کارت ۱۸۲

     پرسش

     How does reversing integral bounds change the value?

     پاسخ

     It changes the sign:

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

183. کارت ۱۸۳

     پرسش

     Net Change Theorem?

     پاسخ

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

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

184. کارت ۱۸۴

     پرسش

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

     پاسخ

     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. کارت ۱۸۵

     پرسش

     Power rule for antiderivatives?

     پاسخ

     For $n\ne-1$,

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

186. کارت ۱۸۶

     پرسش

     Midpoint Riemann sum on equal subintervals?

     پاسخ

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

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

187. کارت ۱۸۷

     پرسش

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

     پاسخ

     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. کارت ۱۸۸

     پرسش

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

     پاسخ

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. کارت ۱۸۹

     پرسش

     Antiderivative of $1/x$?

     پاسخ

     On any interval not crossing zero,

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

190. کارت ۱۹۰

     پرسش

     Trapezoidal approximation on equal subintervals?

     پاسخ

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

191. کارت ۱۹۱

     پرسش

     How do geometric regions help evaluate a definite integral?

     پاسخ

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

192. کارت ۱۹۲

     پرسش

     Basic antiderivatives of sine and cosine?

     پاسخ

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

193. کارت ۱۹۳

     پرسش

     Definite integral as a limit of Riemann sums?

     پاسخ

     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. کارت ۱۹۴

     پرسش

     Constant-multiple rule for integrals?

     پاسخ

     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. کارت ۱۹۵

     پرسش

     What pattern suggests $u$-substitution?

     پاسخ

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

196. کارت ۱۹۶

     پرسش

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

     پاسخ

     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. کارت ۱۹۷

     پرسش

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

     پاسخ

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

198. کارت ۱۹۸

     پرسش

     Sum-and-difference rule for definite integrals?

     پاسخ

     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. کارت ۱۹۹

     پرسش

     Basic antiderivative of $e^x$?

     پاسخ

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

200. کارت ۲۰۰

     پرسش

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

     پاسخ

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

201. کارت ۲۰۱

     پرسش

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

     پاسخ

     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. کارت ۲۰۲

     پرسش

     How does concavity predict trapezoidal and midpoint error?

     پاسخ

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

203. کارت ۲۰۳

     پرسش

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

     پاسخ

     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. کارت ۲۰۴

     پرسش

     What denominator pattern suggests an arctangent antiderivative?

     پاسخ

     After completing the square and scaling, a form like

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

205. کارت ۲۰۵

     پرسش

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

     پاسخ

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

206. کارت ۲۰۶

     پرسش

     How does an initial condition determine an antiderivative?

     پاسخ

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

207. کارت ۲۰۷

     پرسش

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

     پاسخ

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

208. کارت ۲۰۸

     پرسش

     Why does an indefinite integral include $+C$?

     پاسخ

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

209. کارت ۲۰۹

     پرسش

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

     پاسخ

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

210. کارت ۲۱۰

     پرسش

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

     پاسخ

     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. کارت ۲۱۱

     پرسش

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

     پاسخ

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

212. کارت ۲۱۲

     پرسش

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

     پاسخ

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

213. کارت ۲۱۳

     پرسش

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

     پاسخ

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

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

214. کارت ۲۱۴

     پرسش

     Riemann sum for unequal subinterval widths?

     پاسخ

     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. کارت ۲۱۵

     پرسش

     Does continuity guarantee integrability on a closed interval?

     پاسخ

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

216. کارت ۲۱۶

     پرسش

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

     پاسخ

     Where $g(x)\ne0$,

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

217. کارت ۲۱۷

     پرسش

     What algebraic rewrites often reveal a basic antiderivative?

     پاسخ

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

218. کارت ۲۱۸

     پرسش

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

     پاسخ

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

219. کارت ۲۱۹

     پرسش

     What is a differential equation?

     پاسخ

     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. کارت ۲۲۰

     پرسش

     How does a verbal rate statement become a differential equation?

     پاسخ

     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. کارت ۲۲۱

     پرسش

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

     پاسخ

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

222. کارت ۲۲۲

     پرسش

     General solution versus particular solution?

     پاسخ

     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. کارت ۲۲۳

     پرسش

     What does one segment in a slope field show?

     پاسخ

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

224. کارت ۲۲۴

     پرسش

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

     پاسخ

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

225. کارت ۲۲۵

     پرسش

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

     پاسخ

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

226. کارت ۲۲۶

     پرسش

     What makes a first-order differential equation separable?

     پاسخ

     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. کارت ۲۲۷

     پرسش

     What is an initial value problem?

     پاسخ

     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. کارت ۲۲۸

     پرسش

     What is an isocline in a slope field?

     پاسخ

     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. کارت ۲۲۹

     پرسش

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

     پاسخ

     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. کارت ۲۳۰

     پرسش

     General solution of $dy/dt=ky$?

     پاسخ

     $$y=Ce^{kt}$$

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

231. کارت ۲۳۱

     پرسش

     Core method for solving a separable differential equation?

     پاسخ

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

232. کارت ۲۳۲

     پرسش

     How should a solution curve follow a slope field?

     پاسخ

     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. کارت ۲۳۳

     پرسش

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

     پاسخ

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

234. کارت ۲۳۴

     پرسش

     Why is one integration constant enough after integrating both sides?

     پاسخ

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

235. کارت ۲۳۵

     پرسش

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

     پاسخ

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

236. کارت ۲۳۶

     پرسش

     Can one differential equation have infinitely many solutions?

     پاسخ

     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. کارت ۲۳۷

     پرسش

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

     پاسخ

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

238. کارت ۲۳۸

     پرسش

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

     پاسخ

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

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

239. کارت ۲۳۹

     پرسش

     What can be lost when dividing to separate variables?

     پاسخ

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

240. کارت ۲۴۰

     پرسش

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

     پاسخ

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

241. کارت ۲۴۱

     پرسش

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

     پاسخ

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

242. کارت ۲۴۲

     پرسش

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

     پاسخ

     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. کارت ۲۴۳

     پرسش

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

     پاسخ

     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. کارت ۲۴۴

     پرسش

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

     پاسخ

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

245. کارت ۲۴۵

     پرسش

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

     پاسخ

     For $k>0$,

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

     It is independent of the initial amount.

246. کارت ۲۴۶

     پرسش

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

     پاسخ

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

247. کارت ۲۴۷

     پرسش

     How is an initial condition used after separation?

     پاسخ

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

248. کارت ۲۴۸

     پرسش

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

     پاسخ

     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. کارت ۲۴۹

     پرسش

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

     پاسخ

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

250. کارت ۲۵۰

     پرسش

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

     پاسخ

     For $k<0$,

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

251. کارت ۲۵۱

     پرسش

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

     پاسخ

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

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

252. کارت ۲۵۲

     پرسش

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

     پاسخ

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

     Velocity below zero contributes negative displacement.

253. کارت ۲۵۳

     پرسش

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

     پاسخ

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

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

     Think top minus bottom.

254. کارت ۲۵۴

     پرسش

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

     پاسخ

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

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

255. کارت ۲۵۵

     پرسش

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     پاسخ

     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. کارت ۲۵۶

     پرسش

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

     پاسخ

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

257. کارت ۲۵۷

     پرسش

     Cross-sectional area when each slice is a square?

     پاسخ

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

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

258. کارت ۲۵۸

     پرسش

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

     پاسخ

     Integrate the net rate:

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

259. کارت ۲۵۹

     پرسش

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

     پاسخ

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

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

     Think right minus left.

260. کارت ۲۶۰

     پرسش

     Disc-method volume formula?

     پاسخ

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

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

261. کارت ۲۶۱

     پرسش

     What units does average value have?

     پاسخ

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

262. کارت ۲۶۲

     پرسش

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

     پاسخ

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

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

263. کارت ۲۶۳

     پرسش

     Cross-sectional area when each slice is a rectangle?

     پاسخ

     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. کارت ۲۶۴

     پرسش

     How do you determine bounds for area between curves?

     پاسخ

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

265. کارت ۲۶۵

     پرسش

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

     پاسخ

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

266. کارت ۲۶۶

     پرسش

     When is a particle moving to the right or left?

     پاسخ

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

267. کارت ۲۶۷

     پرسش

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

     پاسخ

     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. کارت ۲۶۸

     پرسش

     Why must an area integral be split where curves intersect?

     پاسخ

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

269. کارت ۲۶۹

     پرسش

     How can a velocity table approximate displacement?

     پاسخ

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

270. کارت ۲۷۰

     پرسش

     Washer-method volume formula?

     پاسخ

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

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

271. کارت ۲۷۱

     پرسش

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

     پاسخ

     If $s(a)$ is known,

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

272. کارت ۲۷۲

     پرسش

     How do you choose between vertical and horizontal area slices?

     پاسخ

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

273. کارت ۲۷۳

     پرسش

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

     پاسخ

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

274. کارت ۲۷۴

     پرسش

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

     پاسخ

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

275. کارت ۲۷۵

     پرسش

     Single expression for area between two curves?

     پاسخ

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. کارت ۲۷۶

     پرسش

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

     پاسخ

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

277. کارت ۲۷۷

     پرسش

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

     پاسخ

     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. کارت ۲۷۸

     پرسش

     When should a volume integral use $dy$?

     پاسخ

     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. کارت ۲۷۹

     پرسش

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

     پاسخ

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

280. کارت ۲۸۰

     پرسش

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

     پاسخ

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

281. کارت ۲۸۱

     پرسش

     Why must total distance split at velocity sign changes?

     پاسخ

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

282. کارت ۲۸۲

     پرسش

     When does an accumulated quantity reach a local maximum?

     پاسخ

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

283. کارت ۲۸۳

     پرسش

     What distinguishes area from a definite integral?

     پاسخ

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

284. کارت ۲۸۴

     پرسش

     How do position, velocity, and acceleration graphs correspond?

     پاسخ

     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. کارت ۲۸۵

     پرسش

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

     پاسخ

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

286. کارت ۲۸۶

     پرسش

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

     پاسخ

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

287. کارت ۲۸۷

     پرسش

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

     پاسخ

     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. کارت ۲۸۸

     پرسش

     Why should a contextual integral answer include a sentence?

     پاسخ

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