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

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1. Κάρτα 1

   Ερώτηση

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

   Απάντηση

   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. Κάρτα 2

   Ερώτηση

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

   Απάντηση

   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. Κάρτα 3

   Ερώτηση

   When does direct substitution evaluate a limit?

   Απάντηση

   When the function is continuous at the target input. Then

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

4. Κάρτα 4

   Ερώτηση

   Three conditions for continuity at $x=a$?

   Απάντηση

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

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

5. Κάρτα 5

   Ερώτηση

   Intermediate Value Theorem: hypotheses and conclusion?

   Απάντηση

   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. Κάρτα 6

   Ερώτηση

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

   Απάντηση

   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. Κάρτα 7

   Ερώτηση

   How do you read a finite limit from a graph?

   Απάντηση

   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. Κάρτα 8

   Ερώτηση

   Limit law for a sum or difference?

   Απάντηση

   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. Κάρτα 9

   Ερώτηση

   What makes a discontinuity removable?

   Απάντηση

   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. Κάρτα 10

    Ερώτηση

    Squeeze Theorem: usable form?

    Απάντηση

    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. Κάρτα 11

    Ερώτηση

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

    Απάντηση

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

12. Κάρτα 12

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 13

    Ερώτηση

    Limit law for a product?

    Απάντηση

    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. Κάρτα 14

    Ερώτηση

    Graph signature of a jump discontinuity?

    Απάντηση

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

15. Κάρτα 15

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 16

    Ερώτηση

    Horizontal asymptote from a limit at infinity?

    Απάντηση

    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. Κάρτα 17

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 18

    Ερώτηση

    Limit law for a quotient—and its condition?

    Απάντηση

    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. Κάρτα 19

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 20

    Ερώτηση

    When is the Squeeze Theorem a natural choice?

    Απάντηση

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

21. Κάρτα 21

    Ερώτηση

    Vertical asymptote from one-sided behavior?

    Απάντηση

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

22. Κάρτα 22

    Ερώτηση

    Limit at infinity of equal-degree rational functions?

    Απάντηση

    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. Κάρτα 23

    Ερώτηση

    When can a limit pass through a continuous outer function?

    Απάντηση

    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. Κάρτα 24

    Ερώτηση

    What makes a discontinuity infinite?

    Απάντηση

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

25. Κάρτα 25

    Ερώτηση

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

    Απάντηση

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

26. Κάρτα 26

    Ερώτηση

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

    Απάντηση

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Κάρτα 27

    Ερώτηση

    Continuity of a composition?

    Απάντηση

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

28. Κάρτα 28

    Ερώτηση

    Limit at infinity when a rational numerator has lower degree?

    Απάντηση

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

29. Κάρτα 29

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 30

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 31

    Ερώτηση

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

    Απάντηση

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

32. Κάρτα 32

    Ερώτηση

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

    Απάντηση

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

33. Κάρτα 33

    Ερώτηση

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

    Απάντηση

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

34. Κάρτα 34

    Ερώτηση

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

    Απάντηση

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

35. Κάρτα 35

    Ερώτηση

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

    Απάντηση

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

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

36. Κάρτα 36

    Ερώτηση

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

    Απάντηση

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

    The derivative exists only if this finite limit exists.

37. Κάρτα 37

    Ερώτηση

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

    Απάντηση

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

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

38. Κάρτα 38

    Ερώτηση

    What does differentiability imply about continuity?

    Απάντηση

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

39. Κάρτα 39

    Ερώτηση

    Power rule for derivatives?

    Απάντηση

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

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

40. Κάρτα 40

    Ερώτηση

    Units of $f'(x)$?

    Απάντηση

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

41. Κάρτα 41

    Ερώτηση

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

    Απάντηση

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

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

42. Κάρτα 42

    Ερώτηση

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

    Απάντηση

    $f'(x)>0$ 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. Κάρτα 43

    Ερώτηση

    Derivative of a constant?

    Απάντηση

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

    A constant function has zero rate of change.

44. Κάρτα 44

    Ερώτηση

    Derivative of $\sin x$?

    Απάντηση

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

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

45. Κάρτα 45

    Ερώτηση

    Product rule?

    Απάντηση

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

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

46. Κάρτα 46

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 47

    Ερώτηση

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

    Απάντηση

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

48. Κάρτα 48

    Ερώτηση

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

    Απάντηση

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

49. Κάρτα 49

    Ερώτηση

    Derivative of a sum or difference?

    Απάντηση

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

50. Κάρτα 50

    Ερώτηση

    Derivative of $\cos x$?

    Απάντηση

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

    The standard formula assumes radians.

51. Κάρτα 51

    Ερώτηση

    Quotient rule?

    Απάντηση

    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. Κάρτα 52

    Ερώτηση

    Common notations for the first derivative?

    Απάντηση

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

53. Κάρτα 53

    Ερώτηση

    Derivative of $e^x$?

    Απάντηση

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

54. Κάρτα 54

    Ερώτηση

    What graph features can make $f$ nondifferentiable?

    Απάντηση

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

55. Κάρτα 55

    Ερώτηση

    Derivative of $\tan x$?

    Απάντηση

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Κάρτα 56

    Ερώτηση

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

    Απάντηση

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

57. Κάρτα 57

    Ερώτηση

    Derivative of $\ln x$?

    Απάντηση

    For $x>0$,

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

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

58. Κάρτα 58

    Ερώτηση

    How does the power rule handle roots or negative powers?

    Απάντηση

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

59. Κάρτα 59

    Ερώτηση

    Derivative of $\csc x$?

    Απάντηση

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Κάρτα 60

    Ερώτηση

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

    Απάντηση

    $f$ is increasing on that interval.

61. Κάρτα 61

    Ερώτηση

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

    Απάντηση

    For $a>0$,

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

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

62. Κάρτα 62

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 63

    Ερώτηση

    Derivative of $\sec x$?

    Απάντηση

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Κάρτα 64

    Ερώτηση

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

    Απάντηση

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

65. Κάρτα 65

    Ερώτηση

    Derivative of $\log_a x$?

    Απάντηση

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

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

66. Κάρτα 66

    Ερώτηση

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

    Απάντηση

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Κάρτα 67

    Ερώτηση

    Derivative of $\cot x$?

    Απάντηση

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Κάρτα 68

    Ερώτηση

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

    Απάντηση

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

69. Κάρτα 69

    Ερώτηση

    Constant-multiple rule?

    Απάντηση

    For a constant $c$,

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

70. Κάρτα 70

    Ερώτηση

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

    Απάντηση

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

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

71. Κάρτα 71

    Ερώτηση

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

    Απάντηση

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

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

72. Κάρτα 72

    Ερώτηση

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

    Απάντηση

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

73. Κάρτα 73

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 74

    Ερώτηση

    Derivative of an inverse function at $x$?

    Απάντηση

    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. Κάρτα 75

    Ερώτηση

    Derivative of $\arcsin x$?

    Απάντηση

    For $|x|<1$,

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

76. Κάρτα 76

    Ερώτηση

    Notation for the third derivative of $f$?

    Απάντηση

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

77. Κάρτα 77

    Ερώτηση

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

    Απάντηση

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

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

78. Κάρτα 78

    Ερώτηση

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

    Απάντηση

    Where $y\ne0$,

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

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

79. Κάρτα 79

    Ερώτηση

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

    Απάντηση

    Provided $f'(a)\ne0$,

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

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

80. Κάρτα 80

    Ερώτηση

    Derivative of $\arctan x$?

    Απάντηση

    For every real $x$,

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

81. Κάρτα 81

    Ερώτηση

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

    Απάντηση

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

    The extra factor is the chain rule.

82. Κάρτα 82

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 83

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 84

    Ερώτηση

    Derivative of $\arccos x$?

    Απάντηση

    For $|x|<1$,

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

85. Κάρτα 85

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 86

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 87

    Ερώτηση

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

    Απάντηση

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

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

88. Κάρτα 88

    Ερώτηση

    How are tangent slopes of inverse graphs related?

    Απάντηση

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

89. Κάρτα 89

    Ερώτηση

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

    Απάντηση

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

90. Κάρτα 90

    Ερώτηση

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

    Απάντηση

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

91. Κάρτα 91

    Ερώτηση

    Horizontal tangent on an implicit curve: derivative condition?

    Απάντηση

    $dy/dx=0$ 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. Κάρτα 92

    Ερώτηση

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

    Απάντηση

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

93. Κάρτα 93

    Ερώτηση

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

    Απάντηση

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

94. Κάρτα 94

    Ερώτηση

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

    Απάντηση

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

    This combines the power rule with the chain rule.

95. Κάρτα 95

    Ερώτηση

    Vertical tangent on an implicit curve: derivative clue?

    Απάντηση

    $dy/dx$ 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. Κάρτα 96

    Ερώτηση

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

    Απάντηση

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

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

97. Κάρτα 97

    Ερώτηση

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

    Απάντηση

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

98. Κάρτα 98

    Ερώτηση

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

    Απάντηση

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

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

99. Κάρτα 99

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 100

     Ερώτηση

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

     Απάντηση

     When the functions are differentiable, the chain rule gives

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

101. Κάρτα 101

     Ερώτηση

     What local property lets a function have an inverse derivative?

     Απάντηση

     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. Κάρτα 102

     Ερώτηση

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

     Απάντηση

     For $a>0$,

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

103. Κάρτα 103

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 104

     Ερώτηση

     Position, velocity, and acceleration relationships?

     Απάντηση

     For position $s(t)$,

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

105. Κάρτα 105

     Ερώτηση

     Central idea of a related-rates problem?

     Απάντηση

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

106. Κάρτα 106

     Ερώτηση

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

     Απάντηση

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

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

107. Κάρτα 107

     Ερώτηση

     L’Hospital’s Rule: basic conditions?

     Απάντηση

     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. Κάρτα 108

     Ερώτηση

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

     Απάντηση

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

109. Κάρτα 109

     Ερώτηση

     Speed in terms of velocity?

     Απάντηση

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Κάρτα 110

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 111

     Ερώτηση

     Differential approximation connecting $dx$ and $dy$?

     Απάντηση

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

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

112. Κάρτα 112

     Ερώτηση

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

     Απάντηση

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

113. Κάρτα 113

     Ερώτηση

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

     Απάντηση

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

114. Κάρτα 114

     Ερώτηση

     What does positive acceleration say about velocity?

     Απάντηση

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

115. Κάρτα 115

     Ερώτηση

     Related rates: when should numerical values be substituted?

     Απάντηση

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

116. Κάρτα 116

     Ερώτηση

     How does concavity predict linearization error?

     Απάντηση

     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. Κάρτα 117

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 118

     Ερώτηση

     When is a particle moving in the positive direction?

     Απάντηση

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

119. Κάρτα 119

     Ερώτηση

     How can velocity show a change of direction?

     Απάντηση

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

120. Κάρτα 120

     Ερώτηση

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

     Απάντηση

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

121. Κάρτα 121

     Ερώτηση

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

     Απάντηση

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

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

122. Κάρτα 122

     Ερώτηση

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

     Απάντηση

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

123. Κάρτα 123

     Ερώτηση

     What must a contextual derivative sentence include?

     Απάντηση

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

124. Κάρτα 124

     Ερώτηση

     Velocity negative and acceleration positive: what happens?

     Απάντηση

     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. Κάρτα 125

     Ερώτηση

     How should a negative related rate be interpreted?

     Απάντηση

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

126. Κάρτα 126

     Ερώτηση

     When is local linearity a sound approximation tool?

     Απάντηση

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

127. Κάρτα 127

     Ερώτηση

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

     Απάντηση

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

128. Κάρτα 128

     Ερώτηση

     When is speed increasing?

     Απάντηση

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

129. Κάρτα 129

     Ερώτηση

     Volume changes with time: notation for its rate?

     Απάντηση

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

130. Κάρτα 130

     Ερώτηση

     Why are similar triangles useful in related rates?

     Απάντηση

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

131. Κάρτα 131

     Ερώτηση

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

     Απάντηση

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

132. Κάρτα 132

     Ερώτηση

     What conclusion does L’Hospital’s Rule permit?

     Απάντηση

     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. Κάρτα 133

     Ερώτηση

     When is speed decreasing?

     Απάντηση

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

134. Κάρτα 134

     Ερώτηση

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

     Απάντηση

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

135. Κάρτα 135

     Ερώτηση

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

     Απάντηση

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

136. Κάρτα 136

     Ερώτηση

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

     Απάντηση

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

137. Κάρτα 137

     Ερώτηση

     Extreme Value Theorem: hypothesis and conclusion?

     Απάντηση

     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. Κάρτα 138

     Ερώτηση

     What is a critical number of $f$?

     Απάντηση

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

139. Κάρτα 139

     Ερώτηση

     First derivative test for a local maximum?

     Απάντηση

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

140. Κάρτα 140

     Ερώτηση

     Second-derivative sign for concave up?

     Απάντηση

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

141. Κάρτα 141

     Ερώτηση

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

     Απάντηση

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

142. Κάρτα 142

     Ερώτηση

     First step in an optimization model?

     Απάντηση

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

143. Κάρτα 143

     Ερώτηση

     Mean Value Theorem: hypotheses and conclusion?

     Απάντηση

     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. Κάρτα 144

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 145

     Ερώτηση

     First derivative test for a local minimum?

     Απάντηση

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

146. Κάρτα 146

     Ερώτηση

     What must happen at an inflection point?

     Απάντηση

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

147. Κάρτα 147

     Ερώτηση

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

     Απάντηση

     $f''$ 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. Κάρτα 148

     Ερώτηση

     How do you confirm an optimization answer is absolute?

     Απάντηση

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

149. Κάρτα 149

     Ερώτηση

     Rolle’s Theorem: hypotheses and conclusion?

     Απάντηση

     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. Κάρτα 150

     Ερώτηση

     Difference between absolute and relative extrema?

     Απάντηση

     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. Κάρτα 151

     Ερώτηση

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

     Απάντηση

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

152. Κάρτα 152

     Ερώτηση

     Second derivative test for a local minimum?

     Απάντηση

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

153. Κάρτα 153

     Ερώτηση

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

     Απάντηση

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

154. Κάρτα 154

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 155

     Ερώτηση

     Which theorem links an average slope to an instantaneous slope?

     Απάντηση

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

156. Κάρτα 156

     Ερώτηση

     How can an implicit derivative locate a horizontal tangent?

     Απάντηση

     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. Κάρτα 157

     Ερώτηση

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

     Απάντηση

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

158. Κάρτα 158

     Ερώτηση

     Second derivative test for a local maximum?

     Απάντηση

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

159. Κάρτα 159

     Ερώτηση

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

     Απάντηση

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

160. Κάρτα 160

     Ερώτηση

     Why must an optimization domain be stated?

     Απάντηση

     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. Κάρτα 161

     Ερώτηση

     Which theorem guarantees absolute extrema, not where they occur?

     Απάντηση

     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. Κάρτα 162

     Ερώτηση

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

     Απάντηση

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

163. Κάρτα 163

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 164

     Ερώτηση

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

     Απάντηση

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

165. Κάρτα 165

     Ερώτηση

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

     Απάντηση

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

166. Κάρτα 166

     Ερώτηση

     How can an implicit derivative locate a vertical tangent?

     Απάντηση

     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. Κάρτα 167

     Ερώτηση

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

     Απάντηση

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

168. Κάρτα 168

     Ερώτηση

     Why are endpoints included in the candidates test?

     Απάντηση

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

169. Κάρτα 169

     Ερώτηση

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

     Απάντηση

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

170. Κάρτα 170

     Ερώτηση

     Second-derivative sign for concave down?

     Απάντηση

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

171. Κάρτα 171

     Ερώτηση

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

     Απάντηση

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

172. Κάρτα 172

     Ερώτηση

     What should the final line of an optimization solution state?

     Απάντηση

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

173. Κάρτα 173

     Ερώτηση

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

     Απάντηση

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

174. Κάρτα 174

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 175

     Ερώτηση

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

     Απάντηση

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

176. Κάρτα 176

     Ερώτηση

     Left Riemann sum on equal subintervals?

     Απάντηση

     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. Κάρτα 177

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 178

     Ερώτηση

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Απάντηση

     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. Κάρτα 179

     Ερώτηση

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

     Απάντηση

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Κάρτα 180

     Ερώτηση

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

     Απάντηση

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

181. Κάρτα 181

     Ερώτηση

     Right Riemann sum on equal subintervals?

     Απάντηση

     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. Κάρτα 182

     Ερώτηση

     How does reversing integral bounds change the value?

     Απάντηση

     It changes the sign:

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

183. Κάρτα 183

     Ερώτηση

     Net Change Theorem?

     Απάντηση

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

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

184. Κάρτα 184

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 185

     Ερώτηση

     Power rule for antiderivatives?

     Απάντηση

     For $n\ne-1$,

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

186. Κάρτα 186

     Ερώτηση

     Midpoint Riemann sum on equal subintervals?

     Απάντηση

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

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

187. Κάρτα 187

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 188

     Ερώτηση

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

     Απάντηση

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Κάρτα 189

     Ερώτηση

     Antiderivative of $1/x$?

     Απάντηση

     On any interval not crossing zero,

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

190. Κάρτα 190

     Ερώτηση

     Trapezoidal approximation on equal subintervals?

     Απάντηση

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

191. Κάρτα 191

     Ερώτηση

     How do geometric regions help evaluate a definite integral?

     Απάντηση

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

192. Κάρτα 192

     Ερώτηση

     Basic antiderivatives of sine and cosine?

     Απάντηση

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

193. Κάρτα 193

     Ερώτηση

     Definite integral as a limit of Riemann sums?

     Απάντηση

     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. Κάρτα 194

     Ερώτηση

     Constant-multiple rule for integrals?

     Απάντηση

     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. Κάρτα 195

     Ερώτηση

     What pattern suggests $u$-substitution?

     Απάντηση

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

196. Κάρτα 196

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 197

     Ερώτηση

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

     Απάντηση

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

198. Κάρτα 198

     Ερώτηση

     Sum-and-difference rule for definite integrals?

     Απάντηση

     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. Κάρτα 199

     Ερώτηση

     Basic antiderivative of $e^x$?

     Απάντηση

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

200. Κάρτα 200

     Ερώτηση

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

     Απάντηση

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

201. Κάρτα 201

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 202

     Ερώτηση

     How does concavity predict trapezoidal and midpoint error?

     Απάντηση

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

203. Κάρτα 203

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 204

     Ερώτηση

     What denominator pattern suggests an arctangent antiderivative?

     Απάντηση

     After completing the square and scaling, a form like

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

205. Κάρτα 205

     Ερώτηση

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

     Απάντηση

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

206. Κάρτα 206

     Ερώτηση

     How does an initial condition determine an antiderivative?

     Απάντηση

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

207. Κάρτα 207

     Ερώτηση

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

     Απάντηση

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

208. Κάρτα 208

     Ερώτηση

     Why does an indefinite integral include $+C$?

     Απάντηση

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

209. Κάρτα 209

     Ερώτηση

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

     Απάντηση

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

210. Κάρτα 210

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 211

     Ερώτηση

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

     Απάντηση

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

212. Κάρτα 212

     Ερώτηση

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

     Απάντηση

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

213. Κάρτα 213

     Ερώτηση

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

     Απάντηση

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

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

214. Κάρτα 214

     Ερώτηση

     Riemann sum for unequal subinterval widths?

     Απάντηση

     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. Κάρτα 215

     Ερώτηση

     Does continuity guarantee integrability on a closed interval?

     Απάντηση

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

216. Κάρτα 216

     Ερώτηση

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

     Απάντηση

     Where $g(x)\ne0$,

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

217. Κάρτα 217

     Ερώτηση

     What algebraic rewrites often reveal a basic antiderivative?

     Απάντηση

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

218. Κάρτα 218

     Ερώτηση

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

     Απάντηση

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

219. Κάρτα 219

     Ερώτηση

     What is a differential equation?

     Απάντηση

     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. Κάρτα 220

     Ερώτηση

     How does a verbal rate statement become a differential equation?

     Απάντηση

     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. Κάρτα 221

     Ερώτηση

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

     Απάντηση

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

222. Κάρτα 222

     Ερώτηση

     General solution versus particular solution?

     Απάντηση

     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. Κάρτα 223

     Ερώτηση

     What does one segment in a slope field show?

     Απάντηση

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

224. Κάρτα 224

     Ερώτηση

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

     Απάντηση

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

225. Κάρτα 225

     Ερώτηση

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

     Απάντηση

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

226. Κάρτα 226

     Ερώτηση

     What makes a first-order differential equation separable?

     Απάντηση

     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. Κάρτα 227

     Ερώτηση

     What is an initial value problem?

     Απάντηση

     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. Κάρτα 228

     Ερώτηση

     What is an isocline in a slope field?

     Απάντηση

     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. Κάρτα 229

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 230

     Ερώτηση

     General solution of $dy/dt=ky$?

     Απάντηση

     $$y=Ce^{kt}$$

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

231. Κάρτα 231

     Ερώτηση

     Core method for solving a separable differential equation?

     Απάντηση

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

232. Κάρτα 232

     Ερώτηση

     How should a solution curve follow a slope field?

     Απάντηση

     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. Κάρτα 233

     Ερώτηση

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

     Απάντηση

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

234. Κάρτα 234

     Ερώτηση

     Why is one integration constant enough after integrating both sides?

     Απάντηση

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

235. Κάρτα 235

     Ερώτηση

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

     Απάντηση

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

236. Κάρτα 236

     Ερώτηση

     Can one differential equation have infinitely many solutions?

     Απάντηση

     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. Κάρτα 237

     Ερώτηση

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

     Απάντηση

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

238. Κάρτα 238

     Ερώτηση

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

     Απάντηση

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

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

239. Κάρτα 239

     Ερώτηση

     What can be lost when dividing to separate variables?

     Απάντηση

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

240. Κάρτα 240

     Ερώτηση

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

     Απάντηση

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

241. Κάρτα 241

     Ερώτηση

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

     Απάντηση

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

242. Κάρτα 242

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 243

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 244

     Ερώτηση

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

     Απάντηση

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

245. Κάρτα 245

     Ερώτηση

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

     Απάντηση

     For $k>0$,

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

     It is independent of the initial amount.

246. Κάρτα 246

     Ερώτηση

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

     Απάντηση

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

247. Κάρτα 247

     Ερώτηση

     How is an initial condition used after separation?

     Απάντηση

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

248. Κάρτα 248

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 249

     Ερώτηση

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

     Απάντηση

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

250. Κάρτα 250

     Ερώτηση

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

     Απάντηση

     For $k<0$,

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

251. Κάρτα 251

     Ερώτηση

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

     Απάντηση

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

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

252. Κάρτα 252

     Ερώτηση

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

     Απάντηση

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

     Velocity below zero contributes negative displacement.

253. Κάρτα 253

     Ερώτηση

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

     Απάντηση

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

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

     Think top minus bottom.

254. Κάρτα 254

     Ερώτηση

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

     Απάντηση

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

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

255. Κάρτα 255

     Ερώτηση

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Απάντηση

     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. Κάρτα 256

     Ερώτηση

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

     Απάντηση

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

257. Κάρτα 257

     Ερώτηση

     Cross-sectional area when each slice is a square?

     Απάντηση

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

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

258. Κάρτα 258

     Ερώτηση

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

     Απάντηση

     Integrate the net rate:

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

259. Κάρτα 259

     Ερώτηση

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

     Απάντηση

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

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

     Think right minus left.

260. Κάρτα 260

     Ερώτηση

     Disc-method volume formula?

     Απάντηση

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

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

261. Κάρτα 261

     Ερώτηση

     What units does average value have?

     Απάντηση

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

262. Κάρτα 262

     Ερώτηση

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

     Απάντηση

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

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

263. Κάρτα 263

     Ερώτηση

     Cross-sectional area when each slice is a rectangle?

     Απάντηση

     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. Κάρτα 264

     Ερώτηση

     How do you determine bounds for area between curves?

     Απάντηση

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

265. Κάρτα 265

     Ερώτηση

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

     Απάντηση

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

266. Κάρτα 266

     Ερώτηση

     When is a particle moving to the right or left?

     Απάντηση

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

267. Κάρτα 267

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 268

     Ερώτηση

     Why must an area integral be split where curves intersect?

     Απάντηση

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

269. Κάρτα 269

     Ερώτηση

     How can a velocity table approximate displacement?

     Απάντηση

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

270. Κάρτα 270

     Ερώτηση

     Washer-method volume formula?

     Απάντηση

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

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

271. Κάρτα 271

     Ερώτηση

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

     Απάντηση

     If $s(a)$ is known,

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

272. Κάρτα 272

     Ερώτηση

     How do you choose between vertical and horizontal area slices?

     Απάντηση

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

273. Κάρτα 273

     Ερώτηση

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

     Απάντηση

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

274. Κάρτα 274

     Ερώτηση

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

     Απάντηση

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

275. Κάρτα 275

     Ερώτηση

     Single expression for area between two curves?

     Απάντηση

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Κάρτα 276

     Ερώτηση

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

     Απάντηση

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

277. Κάρτα 277

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 278

     Ερώτηση

     When should a volume integral use $dy$?

     Απάντηση

     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. Κάρτα 279

     Ερώτηση

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

     Απάντηση

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

280. Κάρτα 280

     Ερώτηση

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

     Απάντηση

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

281. Κάρτα 281

     Ερώτηση

     Why must total distance split at velocity sign changes?

     Απάντηση

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

282. Κάρτα 282

     Ερώτηση

     When does an accumulated quantity reach a local maximum?

     Απάντηση

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

283. Κάρτα 283

     Ερώτηση

     What distinguishes area from a definite integral?

     Απάντηση

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

284. Κάρτα 284

     Ερώτηση

     How do position, velocity, and acceleration graphs correspond?

     Απάντηση

     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. Κάρτα 285

     Ερώτηση

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

     Απάντηση

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

286. Κάρτα 286

     Ερώτηση

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

     Απάντηση

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

287. Κάρτα 287

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 288

     Ερώτηση

     Why should a contextual integral answer include a sentence?

     Απάντηση

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