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

Karten in diesem Lernkartenset

1. Karte 1

   Frage

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

   Antwort

   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. Karte 2

   Frage

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

   Antwort

   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. Karte 3

   Frage

   When does direct substitution evaluate a limit?

   Antwort

   When the function is continuous at the target input. Then

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

4. Karte 4

   Frage

   Three conditions for continuity at $x=a$?

   Antwort

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

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

5. Karte 5

   Frage

   Intermediate Value Theorem: hypotheses and conclusion?

   Antwort

   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. Karte 6

   Frage

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

   Antwort

   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. Karte 7

   Frage

   How do you read a finite limit from a graph?

   Antwort

   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. Karte 8

   Frage

   Limit law for a sum or difference?

   Antwort

   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. Karte 9

   Frage

   What makes a discontinuity removable?

   Antwort

   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. Karte 10

    Frage

    Squeeze Theorem: usable form?

    Antwort

    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. Karte 11

    Frage

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

    Antwort

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

12. Karte 12

    Frage

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

    Antwort

    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. Karte 13

    Frage

    Limit law for a product?

    Antwort

    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. Karte 14

    Frage

    Graph signature of a jump discontinuity?

    Antwort

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

15. Karte 15

    Frage

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

    Antwort

    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. Karte 16

    Frage

    Horizontal asymptote from a limit at infinity?

    Antwort

    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. Karte 17

    Frage

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

    Antwort

    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. Karte 18

    Frage

    Limit law for a quotient—and its condition?

    Antwort

    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. Karte 19

    Frage

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

    Antwort

    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. Karte 20

    Frage

    When is the Squeeze Theorem a natural choice?

    Antwort

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

21. Karte 21

    Frage

    Vertical asymptote from one-sided behavior?

    Antwort

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

22. Karte 22

    Frage

    Limit at infinity of equal-degree rational functions?

    Antwort

    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. Karte 23

    Frage

    When can a limit pass through a continuous outer function?

    Antwort

    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. Karte 24

    Frage

    What makes a discontinuity infinite?

    Antwort

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

25. Karte 25

    Frage

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

    Antwort

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

26. Karte 26

    Frage

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

    Antwort

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Karte 27

    Frage

    Continuity of a composition?

    Antwort

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

28. Karte 28

    Frage

    Limit at infinity when a rational numerator has lower degree?

    Antwort

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

29. Karte 29

    Frage

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

    Antwort

    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. Karte 30

    Frage

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

    Antwort

    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. Karte 31

    Frage

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

    Antwort

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

32. Karte 32

    Frage

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

    Antwort

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

33. Karte 33

    Frage

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

    Antwort

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

34. Karte 34

    Frage

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

    Antwort

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

35. Karte 35

    Frage

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

    Antwort

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

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

36. Karte 36

    Frage

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

    Antwort

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

    The derivative exists only if this finite limit exists.

37. Karte 37

    Frage

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

    Antwort

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

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

38. Karte 38

    Frage

    What does differentiability imply about continuity?

    Antwort

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

39. Karte 39

    Frage

    Power rule for derivatives?

    Antwort

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

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

40. Karte 40

    Frage

    Units of $f'(x)$?

    Antwort

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

41. Karte 41

    Frage

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

    Antwort

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

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

42. Karte 42

    Frage

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

    Antwort

    $f'(x)>0$ 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. Karte 43

    Frage

    Derivative of a constant?

    Antwort

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

    A constant function has zero rate of change.

44. Karte 44

    Frage

    Derivative of $\sin x$?

    Antwort

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

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

45. Karte 45

    Frage

    Product rule?

    Antwort

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

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

46. Karte 46

    Frage

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

    Antwort

    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. Karte 47

    Frage

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

    Antwort

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

48. Karte 48

    Frage

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

    Antwort

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

49. Karte 49

    Frage

    Derivative of a sum or difference?

    Antwort

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

50. Karte 50

    Frage

    Derivative of $\cos x$?

    Antwort

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

    The standard formula assumes radians.

51. Karte 51

    Frage

    Quotient rule?

    Antwort

    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. Karte 52

    Frage

    Common notations for the first derivative?

    Antwort

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

53. Karte 53

    Frage

    Derivative of $e^x$?

    Antwort

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

54. Karte 54

    Frage

    What graph features can make $f$ nondifferentiable?

    Antwort

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

55. Karte 55

    Frage

    Derivative of $\tan x$?

    Antwort

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Karte 56

    Frage

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

    Antwort

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

57. Karte 57

    Frage

    Derivative of $\ln x$?

    Antwort

    For $x>0$,

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

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

58. Karte 58

    Frage

    How does the power rule handle roots or negative powers?

    Antwort

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

59. Karte 59

    Frage

    Derivative of $\csc x$?

    Antwort

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Karte 60

    Frage

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

    Antwort

    $f$ is increasing on that interval.

61. Karte 61

    Frage

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

    Antwort

    For $a>0$,

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

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

62. Karte 62

    Frage

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

    Antwort

    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. Karte 63

    Frage

    Derivative of $\sec x$?

    Antwort

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Karte 64

    Frage

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

    Antwort

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

65. Karte 65

    Frage

    Derivative of $\log_a x$?

    Antwort

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

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

66. Karte 66

    Frage

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

    Antwort

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Karte 67

    Frage

    Derivative of $\cot x$?

    Antwort

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Karte 68

    Frage

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

    Antwort

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

69. Karte 69

    Frage

    Constant-multiple rule?

    Antwort

    For a constant $c$,

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

70. Karte 70

    Frage

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

    Antwort

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

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

71. Karte 71

    Frage

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

    Antwort

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

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

72. Karte 72

    Frage

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

    Antwort

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

73. Karte 73

    Frage

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

    Antwort

    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. Karte 74

    Frage

    Derivative of an inverse function at $x$?

    Antwort

    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. Karte 75

    Frage

    Derivative of $\arcsin x$?

    Antwort

    For $|x|<1$,

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

76. Karte 76

    Frage

    Notation for the third derivative of $f$?

    Antwort

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

77. Karte 77

    Frage

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

    Antwort

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

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

78. Karte 78

    Frage

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

    Antwort

    Where $y\ne0$,

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

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

79. Karte 79

    Frage

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

    Antwort

    Provided $f'(a)\ne0$,

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

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

80. Karte 80

    Frage

    Derivative of $\arctan x$?

    Antwort

    For every real $x$,

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

81. Karte 81

    Frage

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

    Antwort

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

    The extra factor is the chain rule.

82. Karte 82

    Frage

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

    Antwort

    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. Karte 83

    Frage

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

    Antwort

    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. Karte 84

    Frage

    Derivative of $\arccos x$?

    Antwort

    For $|x|<1$,

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

85. Karte 85

    Frage

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

    Antwort

    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. Karte 86

    Frage

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

    Antwort

    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. Karte 87

    Frage

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

    Antwort

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

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

88. Karte 88

    Frage

    How are tangent slopes of inverse graphs related?

    Antwort

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

89. Karte 89

    Frage

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

    Antwort

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

90. Karte 90

    Frage

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

    Antwort

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

91. Karte 91

    Frage

    Horizontal tangent on an implicit curve: derivative condition?

    Antwort

    $dy/dx=0$ 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. Karte 92

    Frage

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

    Antwort

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

93. Karte 93

    Frage

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

    Antwort

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

94. Karte 94

    Frage

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

    Antwort

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

    This combines the power rule with the chain rule.

95. Karte 95

    Frage

    Vertical tangent on an implicit curve: derivative clue?

    Antwort

    $dy/dx$ 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. Karte 96

    Frage

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

    Antwort

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

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

97. Karte 97

    Frage

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

    Antwort

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

98. Karte 98

    Frage

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

    Antwort

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

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

99. Karte 99

    Frage

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

    Antwort

    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. Karte 100

     Frage

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

     Antwort

     When the functions are differentiable, the chain rule gives

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

101. Karte 101

     Frage

     What local property lets a function have an inverse derivative?

     Antwort

     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. Karte 102

     Frage

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

     Antwort

     For $a>0$,

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

103. Karte 103

     Frage

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

     Antwort

     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. Karte 104

     Frage

     Position, velocity, and acceleration relationships?

     Antwort

     For position $s(t)$,

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

105. Karte 105

     Frage

     Central idea of a related-rates problem?

     Antwort

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

106. Karte 106

     Frage

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

     Antwort

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

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

107. Karte 107

     Frage

     L’Hospital’s Rule: basic conditions?

     Antwort

     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. Karte 108

     Frage

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

     Antwort

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

109. Karte 109

     Frage

     Speed in terms of velocity?

     Antwort

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Karte 110

     Frage

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

     Antwort

     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. Karte 111

     Frage

     Differential approximation connecting $dx$ and $dy$?

     Antwort

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

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

112. Karte 112

     Frage

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

     Antwort

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

113. Karte 113

     Frage

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

     Antwort

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

114. Karte 114

     Frage

     What does positive acceleration say about velocity?

     Antwort

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

115. Karte 115

     Frage

     Related rates: when should numerical values be substituted?

     Antwort

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

116. Karte 116

     Frage

     How does concavity predict linearization error?

     Antwort

     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. Karte 117

     Frage

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

     Antwort

     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. Karte 118

     Frage

     When is a particle moving in the positive direction?

     Antwort

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

119. Karte 119

     Frage

     How can velocity show a change of direction?

     Antwort

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

120. Karte 120

     Frage

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

     Antwort

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

121. Karte 121

     Frage

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

     Antwort

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

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

122. Karte 122

     Frage

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

     Antwort

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

123. Karte 123

     Frage

     What must a contextual derivative sentence include?

     Antwort

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

124. Karte 124

     Frage

     Velocity negative and acceleration positive: what happens?

     Antwort

     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. Karte 125

     Frage

     How should a negative related rate be interpreted?

     Antwort

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

126. Karte 126

     Frage

     When is local linearity a sound approximation tool?

     Antwort

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

127. Karte 127

     Frage

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

     Antwort

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

128. Karte 128

     Frage

     When is speed increasing?

     Antwort

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

129. Karte 129

     Frage

     Volume changes with time: notation for its rate?

     Antwort

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

130. Karte 130

     Frage

     Why are similar triangles useful in related rates?

     Antwort

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

131. Karte 131

     Frage

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

     Antwort

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

132. Karte 132

     Frage

     What conclusion does L’Hospital’s Rule permit?

     Antwort

     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. Karte 133

     Frage

     When is speed decreasing?

     Antwort

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

134. Karte 134

     Frage

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

     Antwort

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

135. Karte 135

     Frage

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

     Antwort

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

136. Karte 136

     Frage

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

     Antwort

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

137. Karte 137

     Frage

     Extreme Value Theorem: hypothesis and conclusion?

     Antwort

     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. Karte 138

     Frage

     What is a critical number of $f$?

     Antwort

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

139. Karte 139

     Frage

     First derivative test for a local maximum?

     Antwort

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

140. Karte 140

     Frage

     Second-derivative sign for concave up?

     Antwort

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

141. Karte 141

     Frage

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

     Antwort

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

142. Karte 142

     Frage

     First step in an optimization model?

     Antwort

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

143. Karte 143

     Frage

     Mean Value Theorem: hypotheses and conclusion?

     Antwort

     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. Karte 144

     Frage

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

     Antwort

     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. Karte 145

     Frage

     First derivative test for a local minimum?

     Antwort

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

146. Karte 146

     Frage

     What must happen at an inflection point?

     Antwort

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

147. Karte 147

     Frage

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

     Antwort

     $f''$ 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. Karte 148

     Frage

     How do you confirm an optimization answer is absolute?

     Antwort

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

149. Karte 149

     Frage

     Rolle’s Theorem: hypotheses and conclusion?

     Antwort

     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. Karte 150

     Frage

     Difference between absolute and relative extrema?

     Antwort

     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. Karte 151

     Frage

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

     Antwort

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

152. Karte 152

     Frage

     Second derivative test for a local minimum?

     Antwort

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

153. Karte 153

     Frage

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

     Antwort

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

154. Karte 154

     Frage

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

     Antwort

     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. Karte 155

     Frage

     Which theorem links an average slope to an instantaneous slope?

     Antwort

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

156. Karte 156

     Frage

     How can an implicit derivative locate a horizontal tangent?

     Antwort

     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. Karte 157

     Frage

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

     Antwort

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

158. Karte 158

     Frage

     Second derivative test for a local maximum?

     Antwort

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

159. Karte 159

     Frage

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

     Antwort

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

160. Karte 160

     Frage

     Why must an optimization domain be stated?

     Antwort

     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. Karte 161

     Frage

     Which theorem guarantees absolute extrema, not where they occur?

     Antwort

     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. Karte 162

     Frage

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

     Antwort

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

163. Karte 163

     Frage

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

     Antwort

     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. Karte 164

     Frage

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

     Antwort

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

165. Karte 165

     Frage

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

     Antwort

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

166. Karte 166

     Frage

     How can an implicit derivative locate a vertical tangent?

     Antwort

     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. Karte 167

     Frage

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

     Antwort

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

168. Karte 168

     Frage

     Why are endpoints included in the candidates test?

     Antwort

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

169. Karte 169

     Frage

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

     Antwort

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

170. Karte 170

     Frage

     Second-derivative sign for concave down?

     Antwort

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

171. Karte 171

     Frage

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

     Antwort

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

172. Karte 172

     Frage

     What should the final line of an optimization solution state?

     Antwort

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

173. Karte 173

     Frage

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

     Antwort

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

174. Karte 174

     Frage

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

     Antwort

     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. Karte 175

     Frage

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

     Antwort

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

176. Karte 176

     Frage

     Left Riemann sum on equal subintervals?

     Antwort

     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. Karte 177

     Frage

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

     Antwort

     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. Karte 178

     Frage

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Antwort

     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. Karte 179

     Frage

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

     Antwort

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Karte 180

     Frage

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

     Antwort

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

181. Karte 181

     Frage

     Right Riemann sum on equal subintervals?

     Antwort

     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. Karte 182

     Frage

     How does reversing integral bounds change the value?

     Antwort

     It changes the sign:

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

183. Karte 183

     Frage

     Net Change Theorem?

     Antwort

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

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

184. Karte 184

     Frage

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

     Antwort

     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. Karte 185

     Frage

     Power rule for antiderivatives?

     Antwort

     For $n\ne-1$,

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

186. Karte 186

     Frage

     Midpoint Riemann sum on equal subintervals?

     Antwort

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

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

187. Karte 187

     Frage

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

     Antwort

     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. Karte 188

     Frage

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

     Antwort

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Karte 189

     Frage

     Antiderivative of $1/x$?

     Antwort

     On any interval not crossing zero,

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

190. Karte 190

     Frage

     Trapezoidal approximation on equal subintervals?

     Antwort

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

191. Karte 191

     Frage

     How do geometric regions help evaluate a definite integral?

     Antwort

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

192. Karte 192

     Frage

     Basic antiderivatives of sine and cosine?

     Antwort

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

193. Karte 193

     Frage

     Definite integral as a limit of Riemann sums?

     Antwort

     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. Karte 194

     Frage

     Constant-multiple rule for integrals?

     Antwort

     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. Karte 195

     Frage

     What pattern suggests $u$-substitution?

     Antwort

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

196. Karte 196

     Frage

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

     Antwort

     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. Karte 197

     Frage

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

     Antwort

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

198. Karte 198

     Frage

     Sum-and-difference rule for definite integrals?

     Antwort

     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. Karte 199

     Frage

     Basic antiderivative of $e^x$?

     Antwort

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

200. Karte 200

     Frage

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

     Antwort

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

201. Karte 201

     Frage

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

     Antwort

     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. Karte 202

     Frage

     How does concavity predict trapezoidal and midpoint error?

     Antwort

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

203. Karte 203

     Frage

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

     Antwort

     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. Karte 204

     Frage

     What denominator pattern suggests an arctangent antiderivative?

     Antwort

     After completing the square and scaling, a form like

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

205. Karte 205

     Frage

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

     Antwort

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

206. Karte 206

     Frage

     How does an initial condition determine an antiderivative?

     Antwort

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

207. Karte 207

     Frage

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

     Antwort

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

208. Karte 208

     Frage

     Why does an indefinite integral include $+C$?

     Antwort

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

209. Karte 209

     Frage

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

     Antwort

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

210. Karte 210

     Frage

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

     Antwort

     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. Karte 211

     Frage

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

     Antwort

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

212. Karte 212

     Frage

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

     Antwort

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

213. Karte 213

     Frage

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

     Antwort

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

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

214. Karte 214

     Frage

     Riemann sum for unequal subinterval widths?

     Antwort

     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. Karte 215

     Frage

     Does continuity guarantee integrability on a closed interval?

     Antwort

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

216. Karte 216

     Frage

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

     Antwort

     Where $g(x)\ne0$,

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

217. Karte 217

     Frage

     What algebraic rewrites often reveal a basic antiderivative?

     Antwort

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

218. Karte 218

     Frage

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

     Antwort

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

219. Karte 219

     Frage

     What is a differential equation?

     Antwort

     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. Karte 220

     Frage

     How does a verbal rate statement become a differential equation?

     Antwort

     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. Karte 221

     Frage

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

     Antwort

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

222. Karte 222

     Frage

     General solution versus particular solution?

     Antwort

     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. Karte 223

     Frage

     What does one segment in a slope field show?

     Antwort

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

224. Karte 224

     Frage

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

     Antwort

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

225. Karte 225

     Frage

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

     Antwort

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

226. Karte 226

     Frage

     What makes a first-order differential equation separable?

     Antwort

     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. Karte 227

     Frage

     What is an initial value problem?

     Antwort

     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. Karte 228

     Frage

     What is an isocline in a slope field?

     Antwort

     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. Karte 229

     Frage

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

     Antwort

     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. Karte 230

     Frage

     General solution of $dy/dt=ky$?

     Antwort

     $$y=Ce^{kt}$$

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

231. Karte 231

     Frage

     Core method for solving a separable differential equation?

     Antwort

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

232. Karte 232

     Frage

     How should a solution curve follow a slope field?

     Antwort

     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. Karte 233

     Frage

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

     Antwort

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

234. Karte 234

     Frage

     Why is one integration constant enough after integrating both sides?

     Antwort

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

235. Karte 235

     Frage

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

     Antwort

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

236. Karte 236

     Frage

     Can one differential equation have infinitely many solutions?

     Antwort

     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. Karte 237

     Frage

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

     Antwort

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

238. Karte 238

     Frage

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

     Antwort

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

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

239. Karte 239

     Frage

     What can be lost when dividing to separate variables?

     Antwort

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

240. Karte 240

     Frage

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

     Antwort

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

241. Karte 241

     Frage

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

     Antwort

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

242. Karte 242

     Frage

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

     Antwort

     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. Karte 243

     Frage

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

     Antwort

     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. Karte 244

     Frage

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

     Antwort

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

245. Karte 245

     Frage

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

     Antwort

     For $k>0$,

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

     It is independent of the initial amount.

246. Karte 246

     Frage

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

     Antwort

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

247. Karte 247

     Frage

     How is an initial condition used after separation?

     Antwort

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

248. Karte 248

     Frage

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

     Antwort

     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. Karte 249

     Frage

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

     Antwort

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

250. Karte 250

     Frage

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

     Antwort

     For $k<0$,

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

251. Karte 251

     Frage

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

     Antwort

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

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

252. Karte 252

     Frage

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

     Antwort

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

     Velocity below zero contributes negative displacement.

253. Karte 253

     Frage

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

     Antwort

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

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

     Think top minus bottom.

254. Karte 254

     Frage

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

     Antwort

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

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

255. Karte 255

     Frage

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Antwort

     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. Karte 256

     Frage

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

     Antwort

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

257. Karte 257

     Frage

     Cross-sectional area when each slice is a square?

     Antwort

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

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

258. Karte 258

     Frage

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

     Antwort

     Integrate the net rate:

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

259. Karte 259

     Frage

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

     Antwort

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

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

     Think right minus left.

260. Karte 260

     Frage

     Disc-method volume formula?

     Antwort

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

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

261. Karte 261

     Frage

     What units does average value have?

     Antwort

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

262. Karte 262

     Frage

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

     Antwort

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

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

263. Karte 263

     Frage

     Cross-sectional area when each slice is a rectangle?

     Antwort

     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. Karte 264

     Frage

     How do you determine bounds for area between curves?

     Antwort

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

265. Karte 265

     Frage

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

     Antwort

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

266. Karte 266

     Frage

     When is a particle moving to the right or left?

     Antwort

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

267. Karte 267

     Frage

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

     Antwort

     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. Karte 268

     Frage

     Why must an area integral be split where curves intersect?

     Antwort

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

269. Karte 269

     Frage

     How can a velocity table approximate displacement?

     Antwort

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

270. Karte 270

     Frage

     Washer-method volume formula?

     Antwort

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

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

271. Karte 271

     Frage

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

     Antwort

     If $s(a)$ is known,

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

272. Karte 272

     Frage

     How do you choose between vertical and horizontal area slices?

     Antwort

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

273. Karte 273

     Frage

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

     Antwort

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

274. Karte 274

     Frage

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

     Antwort

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

275. Karte 275

     Frage

     Single expression for area between two curves?

     Antwort

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Karte 276

     Frage

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

     Antwort

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

277. Karte 277

     Frage

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

     Antwort

     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. Karte 278

     Frage

     When should a volume integral use $dy$?

     Antwort

     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. Karte 279

     Frage

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

     Antwort

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

280. Karte 280

     Frage

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

     Antwort

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

281. Karte 281

     Frage

     Why must total distance split at velocity sign changes?

     Antwort

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

282. Karte 282

     Frage

     When does an accumulated quantity reach a local maximum?

     Antwort

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

283. Karte 283

     Frage

     What distinguishes area from a definite integral?

     Antwort

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

284. Karte 284

     Frage

     How do position, velocity, and acceleration graphs correspond?

     Antwort

     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. Karte 285

     Frage

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

     Antwort

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

286. Karte 286

     Frage

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

     Antwort

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

287. Karte 287

     Frage

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

     Antwort

     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. Karte 288

     Frage

     Why should a contextual integral answer include a sentence?

     Antwort

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