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

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1. Fișa 1

   Întrebare

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

   Răspuns

   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. Fișa 2

   Întrebare

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

   Răspuns

   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. Fișa 3

   Întrebare

   When does direct substitution evaluate a limit?

   Răspuns

   When the function is continuous at the target input. Then

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

4. Fișa 4

   Întrebare

   Three conditions for continuity at $x=a$?

   Răspuns

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

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

5. Fișa 5

   Întrebare

   Intermediate Value Theorem: hypotheses and conclusion?

   Răspuns

   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. Fișa 6

   Întrebare

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

   Răspuns

   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. Fișa 7

   Întrebare

   How do you read a finite limit from a graph?

   Răspuns

   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. Fișa 8

   Întrebare

   Limit law for a sum or difference?

   Răspuns

   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. Fișa 9

   Întrebare

   What makes a discontinuity removable?

   Răspuns

   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. Fișa 10

    Întrebare

    Squeeze Theorem: usable form?

    Răspuns

    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. Fișa 11

    Întrebare

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

    Răspuns

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

12. Fișa 12

    Întrebare

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

    Răspuns

    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. Fișa 13

    Întrebare

    Limit law for a product?

    Răspuns

    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. Fișa 14

    Întrebare

    Graph signature of a jump discontinuity?

    Răspuns

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

15. Fișa 15

    Întrebare

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

    Răspuns

    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. Fișa 16

    Întrebare

    Horizontal asymptote from a limit at infinity?

    Răspuns

    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. Fișa 17

    Întrebare

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

    Răspuns

    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. Fișa 18

    Întrebare

    Limit law for a quotient—and its condition?

    Răspuns

    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. Fișa 19

    Întrebare

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

    Răspuns

    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. Fișa 20

    Întrebare

    When is the Squeeze Theorem a natural choice?

    Răspuns

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

21. Fișa 21

    Întrebare

    Vertical asymptote from one-sided behavior?

    Răspuns

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

22. Fișa 22

    Întrebare

    Limit at infinity of equal-degree rational functions?

    Răspuns

    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. Fișa 23

    Întrebare

    When can a limit pass through a continuous outer function?

    Răspuns

    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. Fișa 24

    Întrebare

    What makes a discontinuity infinite?

    Răspuns

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

25. Fișa 25

    Întrebare

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

    Răspuns

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

26. Fișa 26

    Întrebare

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

    Răspuns

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Fișa 27

    Întrebare

    Continuity of a composition?

    Răspuns

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

28. Fișa 28

    Întrebare

    Limit at infinity when a rational numerator has lower degree?

    Răspuns

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

29. Fișa 29

    Întrebare

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

    Răspuns

    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. Fișa 30

    Întrebare

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

    Răspuns

    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. Fișa 31

    Întrebare

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

    Răspuns

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

32. Fișa 32

    Întrebare

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

    Răspuns

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

33. Fișa 33

    Întrebare

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

    Răspuns

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

34. Fișa 34

    Întrebare

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

    Răspuns

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

35. Fișa 35

    Întrebare

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

    Răspuns

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

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

36. Fișa 36

    Întrebare

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

    Răspuns

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

    The derivative exists only if this finite limit exists.

37. Fișa 37

    Întrebare

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

    Răspuns

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

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

38. Fișa 38

    Întrebare

    What does differentiability imply about continuity?

    Răspuns

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

39. Fișa 39

    Întrebare

    Power rule for derivatives?

    Răspuns

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

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

40. Fișa 40

    Întrebare

    Units of $f'(x)$?

    Răspuns

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

41. Fișa 41

    Întrebare

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

    Răspuns

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

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

42. Fișa 42

    Întrebare

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

    Răspuns

    $f'(x)>0$ 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. Fișa 43

    Întrebare

    Derivative of a constant?

    Răspuns

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

    A constant function has zero rate of change.

44. Fișa 44

    Întrebare

    Derivative of $\sin x$?

    Răspuns

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

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

45. Fișa 45

    Întrebare

    Product rule?

    Răspuns

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

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

46. Fișa 46

    Întrebare

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

    Răspuns

    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. Fișa 47

    Întrebare

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

    Răspuns

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

48. Fișa 48

    Întrebare

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

    Răspuns

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

49. Fișa 49

    Întrebare

    Derivative of a sum or difference?

    Răspuns

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

50. Fișa 50

    Întrebare

    Derivative of $\cos x$?

    Răspuns

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

    The standard formula assumes radians.

51. Fișa 51

    Întrebare

    Quotient rule?

    Răspuns

    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. Fișa 52

    Întrebare

    Common notations for the first derivative?

    Răspuns

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

53. Fișa 53

    Întrebare

    Derivative of $e^x$?

    Răspuns

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

54. Fișa 54

    Întrebare

    What graph features can make $f$ nondifferentiable?

    Răspuns

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

55. Fișa 55

    Întrebare

    Derivative of $\tan x$?

    Răspuns

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Fișa 56

    Întrebare

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

    Răspuns

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

57. Fișa 57

    Întrebare

    Derivative of $\ln x$?

    Răspuns

    For $x>0$,

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

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

58. Fișa 58

    Întrebare

    How does the power rule handle roots or negative powers?

    Răspuns

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

59. Fișa 59

    Întrebare

    Derivative of $\csc x$?

    Răspuns

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Fișa 60

    Întrebare

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

    Răspuns

    $f$ is increasing on that interval.

61. Fișa 61

    Întrebare

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

    Răspuns

    For $a>0$,

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

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

62. Fișa 62

    Întrebare

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

    Răspuns

    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. Fișa 63

    Întrebare

    Derivative of $\sec x$?

    Răspuns

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Fișa 64

    Întrebare

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

    Răspuns

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

65. Fișa 65

    Întrebare

    Derivative of $\log_a x$?

    Răspuns

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

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

66. Fișa 66

    Întrebare

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

    Răspuns

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Fișa 67

    Întrebare

    Derivative of $\cot x$?

    Răspuns

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Fișa 68

    Întrebare

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

    Răspuns

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

69. Fișa 69

    Întrebare

    Constant-multiple rule?

    Răspuns

    For a constant $c$,

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

70. Fișa 70

    Întrebare

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

    Răspuns

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

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

71. Fișa 71

    Întrebare

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

    Răspuns

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

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

72. Fișa 72

    Întrebare

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

    Răspuns

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

73. Fișa 73

    Întrebare

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

    Răspuns

    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. Fișa 74

    Întrebare

    Derivative of an inverse function at $x$?

    Răspuns

    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. Fișa 75

    Întrebare

    Derivative of $\arcsin x$?

    Răspuns

    For $|x|<1$,

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

76. Fișa 76

    Întrebare

    Notation for the third derivative of $f$?

    Răspuns

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

77. Fișa 77

    Întrebare

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

    Răspuns

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

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

78. Fișa 78

    Întrebare

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

    Răspuns

    Where $y\ne0$,

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

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

79. Fișa 79

    Întrebare

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

    Răspuns

    Provided $f'(a)\ne0$,

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

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

80. Fișa 80

    Întrebare

    Derivative of $\arctan x$?

    Răspuns

    For every real $x$,

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

81. Fișa 81

    Întrebare

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

    Răspuns

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

    The extra factor is the chain rule.

82. Fișa 82

    Întrebare

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

    Răspuns

    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. Fișa 83

    Întrebare

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

    Răspuns

    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. Fișa 84

    Întrebare

    Derivative of $\arccos x$?

    Răspuns

    For $|x|<1$,

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

85. Fișa 85

    Întrebare

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

    Răspuns

    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. Fișa 86

    Întrebare

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

    Răspuns

    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. Fișa 87

    Întrebare

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

    Răspuns

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

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

88. Fișa 88

    Întrebare

    How are tangent slopes of inverse graphs related?

    Răspuns

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

89. Fișa 89

    Întrebare

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

    Răspuns

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

90. Fișa 90

    Întrebare

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

    Răspuns

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

91. Fișa 91

    Întrebare

    Horizontal tangent on an implicit curve: derivative condition?

    Răspuns

    $dy/dx=0$ 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. Fișa 92

    Întrebare

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

    Răspuns

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

93. Fișa 93

    Întrebare

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

    Răspuns

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

94. Fișa 94

    Întrebare

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

    Răspuns

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

    This combines the power rule with the chain rule.

95. Fișa 95

    Întrebare

    Vertical tangent on an implicit curve: derivative clue?

    Răspuns

    $dy/dx$ 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. Fișa 96

    Întrebare

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

    Răspuns

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

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

97. Fișa 97

    Întrebare

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

    Răspuns

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

98. Fișa 98

    Întrebare

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

    Răspuns

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

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

99. Fișa 99

    Întrebare

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

    Răspuns

    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. Fișa 100

     Întrebare

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

     Răspuns

     When the functions are differentiable, the chain rule gives

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

101. Fișa 101

     Întrebare

     What local property lets a function have an inverse derivative?

     Răspuns

     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. Fișa 102

     Întrebare

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

     Răspuns

     For $a>0$,

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

103. Fișa 103

     Întrebare

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

     Răspuns

     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. Fișa 104

     Întrebare

     Position, velocity, and acceleration relationships?

     Răspuns

     For position $s(t)$,

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

105. Fișa 105

     Întrebare

     Central idea of a related-rates problem?

     Răspuns

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

106. Fișa 106

     Întrebare

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

     Răspuns

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

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

107. Fișa 107

     Întrebare

     L’Hospital’s Rule: basic conditions?

     Răspuns

     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. Fișa 108

     Întrebare

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

     Răspuns

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

109. Fișa 109

     Întrebare

     Speed in terms of velocity?

     Răspuns

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Fișa 110

     Întrebare

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

     Răspuns

     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. Fișa 111

     Întrebare

     Differential approximation connecting $dx$ and $dy$?

     Răspuns

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

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

112. Fișa 112

     Întrebare

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

     Răspuns

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

113. Fișa 113

     Întrebare

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

     Răspuns

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

114. Fișa 114

     Întrebare

     What does positive acceleration say about velocity?

     Răspuns

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

115. Fișa 115

     Întrebare

     Related rates: when should numerical values be substituted?

     Răspuns

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

116. Fișa 116

     Întrebare

     How does concavity predict linearization error?

     Răspuns

     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. Fișa 117

     Întrebare

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

     Răspuns

     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. Fișa 118

     Întrebare

     When is a particle moving in the positive direction?

     Răspuns

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

119. Fișa 119

     Întrebare

     How can velocity show a change of direction?

     Răspuns

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

120. Fișa 120

     Întrebare

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

     Răspuns

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

121. Fișa 121

     Întrebare

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

     Răspuns

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

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

122. Fișa 122

     Întrebare

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

     Răspuns

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

123. Fișa 123

     Întrebare

     What must a contextual derivative sentence include?

     Răspuns

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

124. Fișa 124

     Întrebare

     Velocity negative and acceleration positive: what happens?

     Răspuns

     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. Fișa 125

     Întrebare

     How should a negative related rate be interpreted?

     Răspuns

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

126. Fișa 126

     Întrebare

     When is local linearity a sound approximation tool?

     Răspuns

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

127. Fișa 127

     Întrebare

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

     Răspuns

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

128. Fișa 128

     Întrebare

     When is speed increasing?

     Răspuns

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

129. Fișa 129

     Întrebare

     Volume changes with time: notation for its rate?

     Răspuns

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

130. Fișa 130

     Întrebare

     Why are similar triangles useful in related rates?

     Răspuns

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

131. Fișa 131

     Întrebare

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

     Răspuns

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

132. Fișa 132

     Întrebare

     What conclusion does L’Hospital’s Rule permit?

     Răspuns

     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. Fișa 133

     Întrebare

     When is speed decreasing?

     Răspuns

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

134. Fișa 134

     Întrebare

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

     Răspuns

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

135. Fișa 135

     Întrebare

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

     Răspuns

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

136. Fișa 136

     Întrebare

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

     Răspuns

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

137. Fișa 137

     Întrebare

     Extreme Value Theorem: hypothesis and conclusion?

     Răspuns

     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. Fișa 138

     Întrebare

     What is a critical number of $f$?

     Răspuns

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

139. Fișa 139

     Întrebare

     First derivative test for a local maximum?

     Răspuns

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

140. Fișa 140

     Întrebare

     Second-derivative sign for concave up?

     Răspuns

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

141. Fișa 141

     Întrebare

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

     Răspuns

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

142. Fișa 142

     Întrebare

     First step in an optimization model?

     Răspuns

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

143. Fișa 143

     Întrebare

     Mean Value Theorem: hypotheses and conclusion?

     Răspuns

     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. Fișa 144

     Întrebare

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

     Răspuns

     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. Fișa 145

     Întrebare

     First derivative test for a local minimum?

     Răspuns

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

146. Fișa 146

     Întrebare

     What must happen at an inflection point?

     Răspuns

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

147. Fișa 147

     Întrebare

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

     Răspuns

     $f''$ 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. Fișa 148

     Întrebare

     How do you confirm an optimization answer is absolute?

     Răspuns

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

149. Fișa 149

     Întrebare

     Rolle’s Theorem: hypotheses and conclusion?

     Răspuns

     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. Fișa 150

     Întrebare

     Difference between absolute and relative extrema?

     Răspuns

     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. Fișa 151

     Întrebare

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

     Răspuns

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

152. Fișa 152

     Întrebare

     Second derivative test for a local minimum?

     Răspuns

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

153. Fișa 153

     Întrebare

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

     Răspuns

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

154. Fișa 154

     Întrebare

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

     Răspuns

     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. Fișa 155

     Întrebare

     Which theorem links an average slope to an instantaneous slope?

     Răspuns

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

156. Fișa 156

     Întrebare

     How can an implicit derivative locate a horizontal tangent?

     Răspuns

     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. Fișa 157

     Întrebare

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

     Răspuns

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

158. Fișa 158

     Întrebare

     Second derivative test for a local maximum?

     Răspuns

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

159. Fișa 159

     Întrebare

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

     Răspuns

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

160. Fișa 160

     Întrebare

     Why must an optimization domain be stated?

     Răspuns

     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. Fișa 161

     Întrebare

     Which theorem guarantees absolute extrema, not where they occur?

     Răspuns

     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. Fișa 162

     Întrebare

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

     Răspuns

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

163. Fișa 163

     Întrebare

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

     Răspuns

     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. Fișa 164

     Întrebare

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

     Răspuns

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

165. Fișa 165

     Întrebare

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

     Răspuns

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

166. Fișa 166

     Întrebare

     How can an implicit derivative locate a vertical tangent?

     Răspuns

     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. Fișa 167

     Întrebare

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

     Răspuns

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

168. Fișa 168

     Întrebare

     Why are endpoints included in the candidates test?

     Răspuns

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

169. Fișa 169

     Întrebare

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

     Răspuns

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

170. Fișa 170

     Întrebare

     Second-derivative sign for concave down?

     Răspuns

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

171. Fișa 171

     Întrebare

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

     Răspuns

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

172. Fișa 172

     Întrebare

     What should the final line of an optimization solution state?

     Răspuns

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

173. Fișa 173

     Întrebare

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

     Răspuns

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

174. Fișa 174

     Întrebare

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

     Răspuns

     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. Fișa 175

     Întrebare

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

     Răspuns

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

176. Fișa 176

     Întrebare

     Left Riemann sum on equal subintervals?

     Răspuns

     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. Fișa 177

     Întrebare

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

     Răspuns

     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. Fișa 178

     Întrebare

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Răspuns

     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. Fișa 179

     Întrebare

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

     Răspuns

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Fișa 180

     Întrebare

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

     Răspuns

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

181. Fișa 181

     Întrebare

     Right Riemann sum on equal subintervals?

     Răspuns

     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. Fișa 182

     Întrebare

     How does reversing integral bounds change the value?

     Răspuns

     It changes the sign:

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

183. Fișa 183

     Întrebare

     Net Change Theorem?

     Răspuns

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

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

184. Fișa 184

     Întrebare

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

     Răspuns

     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. Fișa 185

     Întrebare

     Power rule for antiderivatives?

     Răspuns

     For $n\ne-1$,

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

186. Fișa 186

     Întrebare

     Midpoint Riemann sum on equal subintervals?

     Răspuns

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

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

187. Fișa 187

     Întrebare

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

     Răspuns

     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. Fișa 188

     Întrebare

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

     Răspuns

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Fișa 189

     Întrebare

     Antiderivative of $1/x$?

     Răspuns

     On any interval not crossing zero,

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

190. Fișa 190

     Întrebare

     Trapezoidal approximation on equal subintervals?

     Răspuns

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

191. Fișa 191

     Întrebare

     How do geometric regions help evaluate a definite integral?

     Răspuns

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

192. Fișa 192

     Întrebare

     Basic antiderivatives of sine and cosine?

     Răspuns

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

193. Fișa 193

     Întrebare

     Definite integral as a limit of Riemann sums?

     Răspuns

     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. Fișa 194

     Întrebare

     Constant-multiple rule for integrals?

     Răspuns

     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. Fișa 195

     Întrebare

     What pattern suggests $u$-substitution?

     Răspuns

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

196. Fișa 196

     Întrebare

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

     Răspuns

     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. Fișa 197

     Întrebare

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

     Răspuns

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

198. Fișa 198

     Întrebare

     Sum-and-difference rule for definite integrals?

     Răspuns

     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. Fișa 199

     Întrebare

     Basic antiderivative of $e^x$?

     Răspuns

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

200. Fișa 200

     Întrebare

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

     Răspuns

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

201. Fișa 201

     Întrebare

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

     Răspuns

     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. Fișa 202

     Întrebare

     How does concavity predict trapezoidal and midpoint error?

     Răspuns

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

203. Fișa 203

     Întrebare

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

     Răspuns

     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. Fișa 204

     Întrebare

     What denominator pattern suggests an arctangent antiderivative?

     Răspuns

     After completing the square and scaling, a form like

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

205. Fișa 205

     Întrebare

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

     Răspuns

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

206. Fișa 206

     Întrebare

     How does an initial condition determine an antiderivative?

     Răspuns

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

207. Fișa 207

     Întrebare

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

     Răspuns

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

208. Fișa 208

     Întrebare

     Why does an indefinite integral include $+C$?

     Răspuns

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

209. Fișa 209

     Întrebare

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

     Răspuns

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

210. Fișa 210

     Întrebare

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

     Răspuns

     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. Fișa 211

     Întrebare

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

     Răspuns

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

212. Fișa 212

     Întrebare

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

     Răspuns

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

213. Fișa 213

     Întrebare

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

     Răspuns

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

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

214. Fișa 214

     Întrebare

     Riemann sum for unequal subinterval widths?

     Răspuns

     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. Fișa 215

     Întrebare

     Does continuity guarantee integrability on a closed interval?

     Răspuns

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

216. Fișa 216

     Întrebare

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

     Răspuns

     Where $g(x)\ne0$,

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

217. Fișa 217

     Întrebare

     What algebraic rewrites often reveal a basic antiderivative?

     Răspuns

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

218. Fișa 218

     Întrebare

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

     Răspuns

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

219. Fișa 219

     Întrebare

     What is a differential equation?

     Răspuns

     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. Fișa 220

     Întrebare

     How does a verbal rate statement become a differential equation?

     Răspuns

     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. Fișa 221

     Întrebare

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

     Răspuns

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

222. Fișa 222

     Întrebare

     General solution versus particular solution?

     Răspuns

     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. Fișa 223

     Întrebare

     What does one segment in a slope field show?

     Răspuns

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

224. Fișa 224

     Întrebare

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

     Răspuns

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

225. Fișa 225

     Întrebare

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

     Răspuns

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

226. Fișa 226

     Întrebare

     What makes a first-order differential equation separable?

     Răspuns

     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. Fișa 227

     Întrebare

     What is an initial value problem?

     Răspuns

     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. Fișa 228

     Întrebare

     What is an isocline in a slope field?

     Răspuns

     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. Fișa 229

     Întrebare

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

     Răspuns

     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. Fișa 230

     Întrebare

     General solution of $dy/dt=ky$?

     Răspuns

     $$y=Ce^{kt}$$

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

231. Fișa 231

     Întrebare

     Core method for solving a separable differential equation?

     Răspuns

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

232. Fișa 232

     Întrebare

     How should a solution curve follow a slope field?

     Răspuns

     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. Fișa 233

     Întrebare

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

     Răspuns

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

234. Fișa 234

     Întrebare

     Why is one integration constant enough after integrating both sides?

     Răspuns

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

235. Fișa 235

     Întrebare

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

     Răspuns

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

236. Fișa 236

     Întrebare

     Can one differential equation have infinitely many solutions?

     Răspuns

     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. Fișa 237

     Întrebare

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

     Răspuns

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

238. Fișa 238

     Întrebare

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

     Răspuns

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

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

239. Fișa 239

     Întrebare

     What can be lost when dividing to separate variables?

     Răspuns

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

240. Fișa 240

     Întrebare

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

     Răspuns

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

241. Fișa 241

     Întrebare

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

     Răspuns

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

242. Fișa 242

     Întrebare

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

     Răspuns

     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. Fișa 243

     Întrebare

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

     Răspuns

     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. Fișa 244

     Întrebare

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

     Răspuns

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

245. Fișa 245

     Întrebare

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

     Răspuns

     For $k>0$,

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

     It is independent of the initial amount.

246. Fișa 246

     Întrebare

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

     Răspuns

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

247. Fișa 247

     Întrebare

     How is an initial condition used after separation?

     Răspuns

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

248. Fișa 248

     Întrebare

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

     Răspuns

     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. Fișa 249

     Întrebare

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

     Răspuns

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

250. Fișa 250

     Întrebare

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

     Răspuns

     For $k<0$,

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

251. Fișa 251

     Întrebare

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

     Răspuns

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

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

252. Fișa 252

     Întrebare

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

     Răspuns

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

     Velocity below zero contributes negative displacement.

253. Fișa 253

     Întrebare

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

     Răspuns

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

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

     Think top minus bottom.

254. Fișa 254

     Întrebare

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

     Răspuns

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

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

255. Fișa 255

     Întrebare

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Răspuns

     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. Fișa 256

     Întrebare

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

     Răspuns

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

257. Fișa 257

     Întrebare

     Cross-sectional area when each slice is a square?

     Răspuns

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

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

258. Fișa 258

     Întrebare

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

     Răspuns

     Integrate the net rate:

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

259. Fișa 259

     Întrebare

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

     Răspuns

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

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

     Think right minus left.

260. Fișa 260

     Întrebare

     Disc-method volume formula?

     Răspuns

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

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

261. Fișa 261

     Întrebare

     What units does average value have?

     Răspuns

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

262. Fișa 262

     Întrebare

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

     Răspuns

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

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

263. Fișa 263

     Întrebare

     Cross-sectional area when each slice is a rectangle?

     Răspuns

     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. Fișa 264

     Întrebare

     How do you determine bounds for area between curves?

     Răspuns

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

265. Fișa 265

     Întrebare

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

     Răspuns

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

266. Fișa 266

     Întrebare

     When is a particle moving to the right or left?

     Răspuns

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

267. Fișa 267

     Întrebare

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

     Răspuns

     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. Fișa 268

     Întrebare

     Why must an area integral be split where curves intersect?

     Răspuns

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

269. Fișa 269

     Întrebare

     How can a velocity table approximate displacement?

     Răspuns

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

270. Fișa 270

     Întrebare

     Washer-method volume formula?

     Răspuns

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

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

271. Fișa 271

     Întrebare

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

     Răspuns

     If $s(a)$ is known,

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

272. Fișa 272

     Întrebare

     How do you choose between vertical and horizontal area slices?

     Răspuns

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

273. Fișa 273

     Întrebare

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

     Răspuns

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

274. Fișa 274

     Întrebare

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

     Răspuns

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

275. Fișa 275

     Întrebare

     Single expression for area between two curves?

     Răspuns

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Fișa 276

     Întrebare

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

     Răspuns

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

277. Fișa 277

     Întrebare

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

     Răspuns

     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. Fișa 278

     Întrebare

     When should a volume integral use $dy$?

     Răspuns

     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. Fișa 279

     Întrebare

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

     Răspuns

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

280. Fișa 280

     Întrebare

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

     Răspuns

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

281. Fișa 281

     Întrebare

     Why must total distance split at velocity sign changes?

     Răspuns

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

282. Fișa 282

     Întrebare

     When does an accumulated quantity reach a local maximum?

     Răspuns

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

283. Fișa 283

     Întrebare

     What distinguishes area from a definite integral?

     Răspuns

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

284. Fișa 284

     Întrebare

     How do position, velocity, and acceleration graphs correspond?

     Răspuns

     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. Fișa 285

     Întrebare

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

     Răspuns

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

286. Fișa 286

     Întrebare

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

     Răspuns

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

287. Fișa 287

     Întrebare

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

     Răspuns

     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. Fișa 288

     Întrebare

     Why should a contextual integral answer include a sentence?

     Răspuns

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