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

بطاقات هذه الرزمة

1. البطاقة ١

   السؤال

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

   الإجابة

   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. البطاقة ٢

   السؤال

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

   الإجابة

   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. البطاقة ٣

   السؤال

   When does direct substitution evaluate a limit?

   الإجابة

   When the function is continuous at the target input. Then

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

4. البطاقة ٤

   السؤال

   Three conditions for continuity at $x=a$?

   الإجابة

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

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

5. البطاقة ٥

   السؤال

   Intermediate Value Theorem: hypotheses and conclusion?

   الإجابة

   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. البطاقة ٦

   السؤال

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

   الإجابة

   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. البطاقة ٧

   السؤال

   How do you read a finite limit from a graph?

   الإجابة

   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. البطاقة ٨

   السؤال

   Limit law for a sum or difference?

   الإجابة

   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. البطاقة ٩

   السؤال

   What makes a discontinuity removable?

   الإجابة

   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. البطاقة ١٠

    السؤال

    Squeeze Theorem: usable form?

    الإجابة

    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. البطاقة ١١

    السؤال

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

    الإجابة

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

12. البطاقة ١٢

    السؤال

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

    الإجابة

    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. البطاقة ١٣

    السؤال

    Limit law for a product?

    الإجابة

    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. البطاقة ١٤

    السؤال

    Graph signature of a jump discontinuity?

    الإجابة

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

15. البطاقة ١٥

    السؤال

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

    الإجابة

    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. البطاقة ١٦

    السؤال

    Horizontal asymptote from a limit at infinity?

    الإجابة

    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. البطاقة ١٧

    السؤال

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

    الإجابة

    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. البطاقة ١٨

    السؤال

    Limit law for a quotient—and its condition?

    الإجابة

    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. البطاقة ١٩

    السؤال

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

    الإجابة

    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. البطاقة ٢٠

    السؤال

    When is the Squeeze Theorem a natural choice?

    الإجابة

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

21. البطاقة ٢١

    السؤال

    Vertical asymptote from one-sided behavior?

    الإجابة

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

22. البطاقة ٢٢

    السؤال

    Limit at infinity of equal-degree rational functions?

    الإجابة

    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. البطاقة ٢٣

    السؤال

    When can a limit pass through a continuous outer function?

    الإجابة

    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. البطاقة ٢٤

    السؤال

    What makes a discontinuity infinite?

    الإجابة

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

25. البطاقة ٢٥

    السؤال

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

    الإجابة

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

26. البطاقة ٢٦

    السؤال

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

    الإجابة

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. البطاقة ٢٧

    السؤال

    Continuity of a composition?

    الإجابة

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

28. البطاقة ٢٨

    السؤال

    Limit at infinity when a rational numerator has lower degree?

    الإجابة

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

29. البطاقة ٢٩

    السؤال

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

    الإجابة

    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. البطاقة ٣٠

    السؤال

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

    الإجابة

    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. البطاقة ٣١

    السؤال

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

    الإجابة

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

32. البطاقة ٣٢

    السؤال

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

    الإجابة

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

33. البطاقة ٣٣

    السؤال

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

    الإجابة

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

34. البطاقة ٣٤

    السؤال

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

    الإجابة

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

35. البطاقة ٣٥

    السؤال

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

    الإجابة

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

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

36. البطاقة ٣٦

    السؤال

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

    الإجابة

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

    The derivative exists only if this finite limit exists.

37. البطاقة ٣٧

    السؤال

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

    الإجابة

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

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

38. البطاقة ٣٨

    السؤال

    What does differentiability imply about continuity?

    الإجابة

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

39. البطاقة ٣٩

    السؤال

    Power rule for derivatives?

    الإجابة

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

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

40. البطاقة ٤٠

    السؤال

    Units of $f'(x)$?

    الإجابة

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

41. البطاقة ٤١

    السؤال

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

    الإجابة

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

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

42. البطاقة ٤٢

    السؤال

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

    الإجابة

    $f'(x)>0$ 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. البطاقة ٤٣

    السؤال

    Derivative of a constant?

    الإجابة

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

    A constant function has zero rate of change.

44. البطاقة ٤٤

    السؤال

    Derivative of $\sin x$?

    الإجابة

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

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

45. البطاقة ٤٥

    السؤال

    Product rule?

    الإجابة

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

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

46. البطاقة ٤٦

    السؤال

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

    الإجابة

    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. البطاقة ٤٧

    السؤال

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

    الإجابة

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

48. البطاقة ٤٨

    السؤال

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

    الإجابة

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

49. البطاقة ٤٩

    السؤال

    Derivative of a sum or difference?

    الإجابة

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

50. البطاقة ٥٠

    السؤال

    Derivative of $\cos x$?

    الإجابة

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

    The standard formula assumes radians.

51. البطاقة ٥١

    السؤال

    Quotient rule?

    الإجابة

    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. البطاقة ٥٢

    السؤال

    Common notations for the first derivative?

    الإجابة

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

53. البطاقة ٥٣

    السؤال

    Derivative of $e^x$?

    الإجابة

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

54. البطاقة ٥٤

    السؤال

    What graph features can make $f$ nondifferentiable?

    الإجابة

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

55. البطاقة ٥٥

    السؤال

    Derivative of $\tan x$?

    الإجابة

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. البطاقة ٥٦

    السؤال

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

    الإجابة

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

57. البطاقة ٥٧

    السؤال

    Derivative of $\ln x$?

    الإجابة

    For $x>0$,

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

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

58. البطاقة ٥٨

    السؤال

    How does the power rule handle roots or negative powers?

    الإجابة

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

59. البطاقة ٥٩

    السؤال

    Derivative of $\csc x$?

    الإجابة

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. البطاقة ٦٠

    السؤال

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

    الإجابة

    $f$ is increasing on that interval.

61. البطاقة ٦١

    السؤال

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

    الإجابة

    For $a>0$,

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

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

62. البطاقة ٦٢

    السؤال

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

    الإجابة

    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. البطاقة ٦٣

    السؤال

    Derivative of $\sec x$?

    الإجابة

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. البطاقة ٦٤

    السؤال

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

    الإجابة

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

65. البطاقة ٦٥

    السؤال

    Derivative of $\log_a x$?

    الإجابة

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

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

66. البطاقة ٦٦

    السؤال

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

    الإجابة

    For $h=fg$,

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

    Use the four table entries at the same input.

67. البطاقة ٦٧

    السؤال

    Derivative of $\cot x$?

    الإجابة

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. البطاقة ٦٨

    السؤال

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

    الإجابة

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

69. البطاقة ٦٩

    السؤال

    Constant-multiple rule?

    الإجابة

    For a constant $c$,

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

70. البطاقة ٧٠

    السؤال

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

    الإجابة

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

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

71. البطاقة ٧١

    السؤال

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

    الإجابة

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

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

72. البطاقة ٧٢

    السؤال

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

    الإجابة

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

73. البطاقة ٧٣

    السؤال

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

    الإجابة

    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. البطاقة ٧٤

    السؤال

    Derivative of an inverse function at $x$?

    الإجابة

    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. البطاقة ٧٥

    السؤال

    Derivative of $\arcsin x$?

    الإجابة

    For $|x|<1$,

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

76. البطاقة ٧٦

    السؤال

    Notation for the third derivative of $f$?

    الإجابة

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

77. البطاقة ٧٧

    السؤال

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

    الإجابة

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

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

78. البطاقة ٧٨

    السؤال

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

    الإجابة

    Where $y\ne0$,

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

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

79. البطاقة ٧٩

    السؤال

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

    الإجابة

    Provided $f'(a)\ne0$,

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

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

80. البطاقة ٨٠

    السؤال

    Derivative of $\arctan x$?

    الإجابة

    For every real $x$,

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

81. البطاقة ٨١

    السؤال

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

    الإجابة

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

    The extra factor is the chain rule.

82. البطاقة ٨٢

    السؤال

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

    الإجابة

    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. البطاقة ٨٣

    السؤال

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

    الإجابة

    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. البطاقة ٨٤

    السؤال

    Derivative of $\arccos x$?

    الإجابة

    For $|x|<1$,

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

85. البطاقة ٨٥

    السؤال

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

    الإجابة

    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. البطاقة ٨٦

    السؤال

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

    الإجابة

    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. البطاقة ٨٧

    السؤال

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

    الإجابة

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

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

88. البطاقة ٨٨

    السؤال

    How are tangent slopes of inverse graphs related?

    الإجابة

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

89. البطاقة ٨٩

    السؤال

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

    الإجابة

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

90. البطاقة ٩٠

    السؤال

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

    الإجابة

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

91. البطاقة ٩١

    السؤال

    Horizontal tangent on an implicit curve: derivative condition?

    الإجابة

    $dy/dx=0$ 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. البطاقة ٩٢

    السؤال

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

    الإجابة

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

93. البطاقة ٩٣

    السؤال

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

    الإجابة

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

94. البطاقة ٩٤

    السؤال

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

    الإجابة

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

    This combines the power rule with the chain rule.

95. البطاقة ٩٥

    السؤال

    Vertical tangent on an implicit curve: derivative clue?

    الإجابة

    $dy/dx$ 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. البطاقة ٩٦

    السؤال

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

    الإجابة

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

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

97. البطاقة ٩٧

    السؤال

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

    الإجابة

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

98. البطاقة ٩٨

    السؤال

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

    الإجابة

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

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

99. البطاقة ٩٩

    السؤال

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

    الإجابة

    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. البطاقة ١٠٠

     السؤال

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

     الإجابة

     When the functions are differentiable, the chain rule gives

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

101. البطاقة ١٠١

     السؤال

     What local property lets a function have an inverse derivative?

     الإجابة

     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. البطاقة ١٠٢

     السؤال

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

     الإجابة

     For $a>0$,

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

103. البطاقة ١٠٣

     السؤال

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

     الإجابة

     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. البطاقة ١٠٤

     السؤال

     Position, velocity, and acceleration relationships?

     الإجابة

     For position $s(t)$,

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

105. البطاقة ١٠٥

     السؤال

     Central idea of a related-rates problem?

     الإجابة

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

106. البطاقة ١٠٦

     السؤال

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

     الإجابة

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

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

107. البطاقة ١٠٧

     السؤال

     L’Hospital’s Rule: basic conditions?

     الإجابة

     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. البطاقة ١٠٨

     السؤال

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

     الإجابة

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

109. البطاقة ١٠٩

     السؤال

     Speed in terms of velocity?

     الإجابة

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

     Velocity includes direction; speed is nonnegative magnitude.

110. البطاقة ١١٠

     السؤال

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

     الإجابة

     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. البطاقة ١١١

     السؤال

     Differential approximation connecting $dx$ and $dy$?

     الإجابة

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

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

112. البطاقة ١١٢

     السؤال

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

     الإجابة

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

113. البطاقة ١١٣

     السؤال

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

     الإجابة

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

114. البطاقة ١١٤

     السؤال

     What does positive acceleration say about velocity?

     الإجابة

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

115. البطاقة ١١٥

     السؤال

     Related rates: when should numerical values be substituted?

     الإجابة

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

116. البطاقة ١١٦

     السؤال

     How does concavity predict linearization error?

     الإجابة

     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. البطاقة ١١٧

     السؤال

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

     الإجابة

     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. البطاقة ١١٨

     السؤال

     When is a particle moving in the positive direction?

     الإجابة

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

119. البطاقة ١١٩

     السؤال

     How can velocity show a change of direction?

     الإجابة

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

120. البطاقة ١٢٠

     السؤال

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

     الإجابة

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

121. البطاقة ١٢١

     السؤال

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

     الإجابة

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

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

122. البطاقة ١٢٢

     السؤال

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

     الإجابة

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

123. البطاقة ١٢٣

     السؤال

     What must a contextual derivative sentence include?

     الإجابة

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

124. البطاقة ١٢٤

     السؤال

     Velocity negative and acceleration positive: what happens?

     الإجابة

     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. البطاقة ١٢٥

     السؤال

     How should a negative related rate be interpreted?

     الإجابة

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

126. البطاقة ١٢٦

     السؤال

     When is local linearity a sound approximation tool?

     الإجابة

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

127. البطاقة ١٢٧

     السؤال

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

     الإجابة

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

128. البطاقة ١٢٨

     السؤال

     When is speed increasing?

     الإجابة

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

129. البطاقة ١٢٩

     السؤال

     Volume changes with time: notation for its rate?

     الإجابة

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

130. البطاقة ١٣٠

     السؤال

     Why are similar triangles useful in related rates?

     الإجابة

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

131. البطاقة ١٣١

     السؤال

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

     الإجابة

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

132. البطاقة ١٣٢

     السؤال

     What conclusion does L’Hospital’s Rule permit?

     الإجابة

     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. البطاقة ١٣٣

     السؤال

     When is speed decreasing?

     الإجابة

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

134. البطاقة ١٣٤

     السؤال

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

     الإجابة

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

135. البطاقة ١٣٥

     السؤال

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

     الإجابة

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

136. البطاقة ١٣٦

     السؤال

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

     الإجابة

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

137. البطاقة ١٣٧

     السؤال

     Extreme Value Theorem: hypothesis and conclusion?

     الإجابة

     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. البطاقة ١٣٨

     السؤال

     What is a critical number of $f$?

     الإجابة

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

139. البطاقة ١٣٩

     السؤال

     First derivative test for a local maximum?

     الإجابة

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

140. البطاقة ١٤٠

     السؤال

     Second-derivative sign for concave up?

     الإجابة

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

141. البطاقة ١٤١

     السؤال

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

     الإجابة

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

142. البطاقة ١٤٢

     السؤال

     First step in an optimization model?

     الإجابة

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

143. البطاقة ١٤٣

     السؤال

     Mean Value Theorem: hypotheses and conclusion?

     الإجابة

     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. البطاقة ١٤٤

     السؤال

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

     الإجابة

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

     

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145. البطاقة ١٤٥

     السؤال

     First derivative test for a local minimum?

     الإجابة

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

146. البطاقة ١٤٦

     السؤال

     What must happen at an inflection point?

     الإجابة

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

147. البطاقة ١٤٧

     السؤال

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

     الإجابة

     $f''$ 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. البطاقة ١٤٨

     السؤال

     How do you confirm an optimization answer is absolute?

     الإجابة

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

149. البطاقة ١٤٩

     السؤال

     Rolle’s Theorem: hypotheses and conclusion?

     الإجابة

     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. البطاقة ١٥٠

     السؤال

     Difference between absolute and relative extrema?

     الإجابة

     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. البطاقة ١٥١

     السؤال

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

     الإجابة

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

152. البطاقة ١٥٢

     السؤال

     Second derivative test for a local minimum?

     الإجابة

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

153. البطاقة ١٥٣

     السؤال

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

     الإجابة

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

154. البطاقة ١٥٤

     السؤال

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

     الإجابة

     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. البطاقة ١٥٥

     السؤال

     Which theorem links an average slope to an instantaneous slope?

     الإجابة

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

156. البطاقة ١٥٦

     السؤال

     How can an implicit derivative locate a horizontal tangent?

     الإجابة

     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. البطاقة ١٥٧

     السؤال

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

     الإجابة

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

158. البطاقة ١٥٨

     السؤال

     Second derivative test for a local maximum?

     الإجابة

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

159. البطاقة ١٥٩

     السؤال

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

     الإجابة

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

160. البطاقة ١٦٠

     السؤال

     Why must an optimization domain be stated?

     الإجابة

     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. البطاقة ١٦١

     السؤال

     Which theorem guarantees absolute extrema, not where they occur?

     الإجابة

     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. البطاقة ١٦٢

     السؤال

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

     الإجابة

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

163. البطاقة ١٦٣

     السؤال

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

     الإجابة

     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. البطاقة ١٦٤

     السؤال

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

     الإجابة

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

165. البطاقة ١٦٥

     السؤال

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

     الإجابة

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

166. البطاقة ١٦٦

     السؤال

     How can an implicit derivative locate a vertical tangent?

     الإجابة

     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. البطاقة ١٦٧

     السؤال

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

     الإجابة

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

168. البطاقة ١٦٨

     السؤال

     Why are endpoints included in the candidates test?

     الإجابة

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

169. البطاقة ١٦٩

     السؤال

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

     الإجابة

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

170. البطاقة ١٧٠

     السؤال

     Second-derivative sign for concave down?

     الإجابة

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

171. البطاقة ١٧١

     السؤال

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

     الإجابة

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

172. البطاقة ١٧٢

     السؤال

     What should the final line of an optimization solution state?

     الإجابة

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

173. البطاقة ١٧٣

     السؤال

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

     الإجابة

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

174. البطاقة ١٧٤

     السؤال

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

     الإجابة

     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. البطاقة ١٧٥

     السؤال

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

     الإجابة

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

176. البطاقة ١٧٦

     السؤال

     Left Riemann sum on equal subintervals?

     الإجابة

     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. البطاقة ١٧٧

     السؤال

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

     الإجابة

     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. البطاقة ١٧٨

     السؤال

     Fundamental Theorem of Calculus: evaluate a definite integral?

     الإجابة

     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. البطاقة ١٧٩

     السؤال

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

     الإجابة

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. البطاقة ١٨٠

     السؤال

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

     الإجابة

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

181. البطاقة ١٨١

     السؤال

     Right Riemann sum on equal subintervals?

     الإجابة

     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. البطاقة ١٨٢

     السؤال

     How does reversing integral bounds change the value?

     الإجابة

     It changes the sign:

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

183. البطاقة ١٨٣

     السؤال

     Net Change Theorem?

     الإجابة

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

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

184. البطاقة ١٨٤

     السؤال

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

     الإجابة

     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. البطاقة ١٨٥

     السؤال

     Power rule for antiderivatives?

     الإجابة

     For $n\ne-1$,

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

186. البطاقة ١٨٦

     السؤال

     Midpoint Riemann sum on equal subintervals?

     الإجابة

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

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

187. البطاقة ١٨٧

     السؤال

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

     الإجابة

     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. البطاقة ١٨٨

     السؤال

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

     الإجابة

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. البطاقة ١٨٩

     السؤال

     Antiderivative of $1/x$?

     الإجابة

     On any interval not crossing zero,

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

190. البطاقة ١٩٠

     السؤال

     Trapezoidal approximation on equal subintervals?

     الإجابة

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

191. البطاقة ١٩١

     السؤال

     How do geometric regions help evaluate a definite integral?

     الإجابة

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

192. البطاقة ١٩٢

     السؤال

     Basic antiderivatives of sine and cosine?

     الإجابة

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

193. البطاقة ١٩٣

     السؤال

     Definite integral as a limit of Riemann sums?

     الإجابة

     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. البطاقة ١٩٤

     السؤال

     Constant-multiple rule for integrals?

     الإجابة

     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. البطاقة ١٩٥

     السؤال

     What pattern suggests $u$-substitution?

     الإجابة

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

196. البطاقة ١٩٦

     السؤال

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

     الإجابة

     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. البطاقة ١٩٧

     السؤال

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

     الإجابة

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

198. البطاقة ١٩٨

     السؤال

     Sum-and-difference rule for definite integrals?

     الإجابة

     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. البطاقة ١٩٩

     السؤال

     Basic antiderivative of $e^x$?

     الإجابة

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

200. البطاقة ٢٠٠

     السؤال

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

     الإجابة

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

201. البطاقة ٢٠١

     السؤال

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

     الإجابة

     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. البطاقة ٢٠٢

     السؤال

     How does concavity predict trapezoidal and midpoint error?

     الإجابة

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

203. البطاقة ٢٠٣

     السؤال

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

     الإجابة

     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. البطاقة ٢٠٤

     السؤال

     What denominator pattern suggests an arctangent antiderivative?

     الإجابة

     After completing the square and scaling, a form like

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

205. البطاقة ٢٠٥

     السؤال

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

     الإجابة

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

206. البطاقة ٢٠٦

     السؤال

     How does an initial condition determine an antiderivative?

     الإجابة

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

207. البطاقة ٢٠٧

     السؤال

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

     الإجابة

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

208. البطاقة ٢٠٨

     السؤال

     Why does an indefinite integral include $+C$?

     الإجابة

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

209. البطاقة ٢٠٩

     السؤال

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

     الإجابة

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

210. البطاقة ٢١٠

     السؤال

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

     الإجابة

     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. البطاقة ٢١١

     السؤال

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

     الإجابة

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

212. البطاقة ٢١٢

     السؤال

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

     الإجابة

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

213. البطاقة ٢١٣

     السؤال

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

     الإجابة

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

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

214. البطاقة ٢١٤

     السؤال

     Riemann sum for unequal subinterval widths?

     الإجابة

     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. البطاقة ٢١٥

     السؤال

     Does continuity guarantee integrability on a closed interval?

     الإجابة

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

216. البطاقة ٢١٦

     السؤال

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

     الإجابة

     Where $g(x)\ne0$,

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

217. البطاقة ٢١٧

     السؤال

     What algebraic rewrites often reveal a basic antiderivative?

     الإجابة

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

218. البطاقة ٢١٨

     السؤال

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

     الإجابة

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

219. البطاقة ٢١٩

     السؤال

     What is a differential equation?

     الإجابة

     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. البطاقة ٢٢٠

     السؤال

     How does a verbal rate statement become a differential equation?

     الإجابة

     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. البطاقة ٢٢١

     السؤال

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

     الإجابة

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

222. البطاقة ٢٢٢

     السؤال

     General solution versus particular solution?

     الإجابة

     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. البطاقة ٢٢٣

     السؤال

     What does one segment in a slope field show?

     الإجابة

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

224. البطاقة ٢٢٤

     السؤال

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

     الإجابة

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

225. البطاقة ٢٢٥

     السؤال

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

     الإجابة

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

226. البطاقة ٢٢٦

     السؤال

     What makes a first-order differential equation separable?

     الإجابة

     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. البطاقة ٢٢٧

     السؤال

     What is an initial value problem?

     الإجابة

     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. البطاقة ٢٢٨

     السؤال

     What is an isocline in a slope field?

     الإجابة

     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. البطاقة ٢٢٩

     السؤال

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

     الإجابة

     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. البطاقة ٢٣٠

     السؤال

     General solution of $dy/dt=ky$?

     الإجابة

     $$y=Ce^{kt}$$

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

231. البطاقة ٢٣١

     السؤال

     Core method for solving a separable differential equation?

     الإجابة

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

232. البطاقة ٢٣٢

     السؤال

     How should a solution curve follow a slope field?

     الإجابة

     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. البطاقة ٢٣٣

     السؤال

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

     الإجابة

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

234. البطاقة ٢٣٤

     السؤال

     Why is one integration constant enough after integrating both sides?

     الإجابة

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

235. البطاقة ٢٣٥

     السؤال

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

     الإجابة

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

236. البطاقة ٢٣٦

     السؤال

     Can one differential equation have infinitely many solutions?

     الإجابة

     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. البطاقة ٢٣٧

     السؤال

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

     الإجابة

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

238. البطاقة ٢٣٨

     السؤال

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

     الإجابة

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

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

239. البطاقة ٢٣٩

     السؤال

     What can be lost when dividing to separate variables?

     الإجابة

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

240. البطاقة ٢٤٠

     السؤال

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

     الإجابة

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

241. البطاقة ٢٤١

     السؤال

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

     الإجابة

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

242. البطاقة ٢٤٢

     السؤال

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

     الإجابة

     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. البطاقة ٢٤٣

     السؤال

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

     الإجابة

     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. البطاقة ٢٤٤

     السؤال

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

     الإجابة

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

245. البطاقة ٢٤٥

     السؤال

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

     الإجابة

     For $k>0$,

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

     It is independent of the initial amount.

246. البطاقة ٢٤٦

     السؤال

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

     الإجابة

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

247. البطاقة ٢٤٧

     السؤال

     How is an initial condition used after separation?

     الإجابة

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

248. البطاقة ٢٤٨

     السؤال

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

     الإجابة

     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. البطاقة ٢٤٩

     السؤال

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

     الإجابة

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

250. البطاقة ٢٥٠

     السؤال

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

     الإجابة

     For $k<0$,

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

251. البطاقة ٢٥١

     السؤال

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

     الإجابة

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

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

252. البطاقة ٢٥٢

     السؤال

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

     الإجابة

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

     Velocity below zero contributes negative displacement.

253. البطاقة ٢٥٣

     السؤال

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

     الإجابة

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

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

     Think top minus bottom.

254. البطاقة ٢٥٤

     السؤال

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

     الإجابة

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

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

255. البطاقة ٢٥٥

     السؤال

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     الإجابة

     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. البطاقة ٢٥٦

     السؤال

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

     الإجابة

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

257. البطاقة ٢٥٧

     السؤال

     Cross-sectional area when each slice is a square?

     الإجابة

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

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

258. البطاقة ٢٥٨

     السؤال

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

     الإجابة

     Integrate the net rate:

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

259. البطاقة ٢٥٩

     السؤال

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

     الإجابة

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

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

     Think right minus left.

260. البطاقة ٢٦٠

     السؤال

     Disc-method volume formula?

     الإجابة

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

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

261. البطاقة ٢٦١

     السؤال

     What units does average value have?

     الإجابة

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

262. البطاقة ٢٦٢

     السؤال

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

     الإجابة

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

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

263. البطاقة ٢٦٣

     السؤال

     Cross-sectional area when each slice is a rectangle?

     الإجابة

     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. البطاقة ٢٦٤

     السؤال

     How do you determine bounds for area between curves?

     الإجابة

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

265. البطاقة ٢٦٥

     السؤال

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

     الإجابة

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

266. البطاقة ٢٦٦

     السؤال

     When is a particle moving to the right or left?

     الإجابة

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

267. البطاقة ٢٦٧

     السؤال

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

     الإجابة

     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. البطاقة ٢٦٨

     السؤال

     Why must an area integral be split where curves intersect?

     الإجابة

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

269. البطاقة ٢٦٩

     السؤال

     How can a velocity table approximate displacement?

     الإجابة

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

270. البطاقة ٢٧٠

     السؤال

     Washer-method volume formula?

     الإجابة

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

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

271. البطاقة ٢٧١

     السؤال

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

     الإجابة

     If $s(a)$ is known,

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

272. البطاقة ٢٧٢

     السؤال

     How do you choose between vertical and horizontal area slices?

     الإجابة

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

273. البطاقة ٢٧٣

     السؤال

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

     الإجابة

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

274. البطاقة ٢٧٤

     السؤال

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

     الإجابة

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

275. البطاقة ٢٧٥

     السؤال

     Single expression for area between two curves?

     الإجابة

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. البطاقة ٢٧٦

     السؤال

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

     الإجابة

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

277. البطاقة ٢٧٧

     السؤال

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

     الإجابة

     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. البطاقة ٢٧٨

     السؤال

     When should a volume integral use $dy$?

     الإجابة

     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. البطاقة ٢٧٩

     السؤال

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

     الإجابة

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

280. البطاقة ٢٨٠

     السؤال

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

     الإجابة

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

281. البطاقة ٢٨١

     السؤال

     Why must total distance split at velocity sign changes?

     الإجابة

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

282. البطاقة ٢٨٢

     السؤال

     When does an accumulated quantity reach a local maximum?

     الإجابة

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

283. البطاقة ٢٨٣

     السؤال

     What distinguishes area from a definite integral?

     الإجابة

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

284. البطاقة ٢٨٤

     السؤال

     How do position, velocity, and acceleration graphs correspond?

     الإجابة

     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. البطاقة ٢٨٥

     السؤال

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

     الإجابة

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

286. البطاقة ٢٨٦

     السؤال

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

     الإجابة

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

287. البطاقة ٢٨٧

     السؤال

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

     الإجابة

     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. البطاقة ٢٨٨

     السؤال

     Why should a contextual integral answer include a sentence?

     الإجابة

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