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

Selle kaardipaki kaardid

1. Kaart 1

   Küsimus

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

   Vastus

   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. Kaart 2

   Küsimus

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

   Vastus

   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. Kaart 3

   Küsimus

   When does direct substitution evaluate a limit?

   Vastus

   When the function is continuous at the target input. Then

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

4. Kaart 4

   Küsimus

   Three conditions for continuity at $x=a$?

   Vastus

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

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

5. Kaart 5

   Küsimus

   Intermediate Value Theorem: hypotheses and conclusion?

   Vastus

   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. Kaart 6

   Küsimus

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

   Vastus

   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. Kaart 7

   Küsimus

   How do you read a finite limit from a graph?

   Vastus

   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. Kaart 8

   Küsimus

   Limit law for a sum or difference?

   Vastus

   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. Kaart 9

   Küsimus

   What makes a discontinuity removable?

   Vastus

   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. Kaart 10

    Küsimus

    Squeeze Theorem: usable form?

    Vastus

    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. Kaart 11

    Küsimus

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

    Vastus

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

12. Kaart 12

    Küsimus

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

    Vastus

    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. Kaart 13

    Küsimus

    Limit law for a product?

    Vastus

    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. Kaart 14

    Küsimus

    Graph signature of a jump discontinuity?

    Vastus

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

15. Kaart 15

    Küsimus

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

    Vastus

    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. Kaart 16

    Küsimus

    Horizontal asymptote from a limit at infinity?

    Vastus

    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. Kaart 17

    Küsimus

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

    Vastus

    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. Kaart 18

    Küsimus

    Limit law for a quotient—and its condition?

    Vastus

    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. Kaart 19

    Küsimus

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

    Vastus

    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. Kaart 20

    Küsimus

    When is the Squeeze Theorem a natural choice?

    Vastus

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

21. Kaart 21

    Küsimus

    Vertical asymptote from one-sided behavior?

    Vastus

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

22. Kaart 22

    Küsimus

    Limit at infinity of equal-degree rational functions?

    Vastus

    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. Kaart 23

    Küsimus

    When can a limit pass through a continuous outer function?

    Vastus

    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. Kaart 24

    Küsimus

    What makes a discontinuity infinite?

    Vastus

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

25. Kaart 25

    Küsimus

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

    Vastus

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

26. Kaart 26

    Küsimus

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

    Vastus

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kaart 27

    Küsimus

    Continuity of a composition?

    Vastus

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

28. Kaart 28

    Küsimus

    Limit at infinity when a rational numerator has lower degree?

    Vastus

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

29. Kaart 29

    Küsimus

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

    Vastus

    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. Kaart 30

    Küsimus

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

    Vastus

    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. Kaart 31

    Küsimus

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

    Vastus

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

32. Kaart 32

    Küsimus

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

    Vastus

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

33. Kaart 33

    Küsimus

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

    Vastus

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

34. Kaart 34

    Küsimus

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

    Vastus

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

35. Kaart 35

    Küsimus

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

    Vastus

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

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

36. Kaart 36

    Küsimus

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

    Vastus

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

    The derivative exists only if this finite limit exists.

37. Kaart 37

    Küsimus

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

    Vastus

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

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

38. Kaart 38

    Küsimus

    What does differentiability imply about continuity?

    Vastus

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

39. Kaart 39

    Küsimus

    Power rule for derivatives?

    Vastus

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

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

40. Kaart 40

    Küsimus

    Units of $f'(x)$?

    Vastus

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

41. Kaart 41

    Küsimus

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

    Vastus

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

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

42. Kaart 42

    Küsimus

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

    Vastus

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

    Küsimus

    Derivative of a constant?

    Vastus

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

    A constant function has zero rate of change.

44. Kaart 44

    Küsimus

    Derivative of $\sin x$?

    Vastus

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

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

45. Kaart 45

    Küsimus

    Product rule?

    Vastus

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

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

46. Kaart 46

    Küsimus

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

    Vastus

    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. Kaart 47

    Küsimus

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

    Vastus

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

48. Kaart 48

    Küsimus

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

    Vastus

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

49. Kaart 49

    Küsimus

    Derivative of a sum or difference?

    Vastus

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

50. Kaart 50

    Küsimus

    Derivative of $\cos x$?

    Vastus

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

    The standard formula assumes radians.

51. Kaart 51

    Küsimus

    Quotient rule?

    Vastus

    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. Kaart 52

    Küsimus

    Common notations for the first derivative?

    Vastus

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

53. Kaart 53

    Küsimus

    Derivative of $e^x$?

    Vastus

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

54. Kaart 54

    Küsimus

    What graph features can make $f$ nondifferentiable?

    Vastus

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

55. Kaart 55

    Küsimus

    Derivative of $\tan x$?

    Vastus

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kaart 56

    Küsimus

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

    Vastus

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

57. Kaart 57

    Küsimus

    Derivative of $\ln x$?

    Vastus

    For $x>0$,

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

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

58. Kaart 58

    Küsimus

    How does the power rule handle roots or negative powers?

    Vastus

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

59. Kaart 59

    Küsimus

    Derivative of $\csc x$?

    Vastus

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kaart 60

    Küsimus

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

    Vastus

    $f$ is increasing on that interval.

61. Kaart 61

    Küsimus

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

    Vastus

    For $a>0$,

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

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

62. Kaart 62

    Küsimus

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

    Vastus

    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. Kaart 63

    Küsimus

    Derivative of $\sec x$?

    Vastus

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kaart 64

    Küsimus

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

    Vastus

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

65. Kaart 65

    Küsimus

    Derivative of $\log_a x$?

    Vastus

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

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

66. Kaart 66

    Küsimus

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

    Vastus

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kaart 67

    Küsimus

    Derivative of $\cot x$?

    Vastus

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kaart 68

    Küsimus

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

    Vastus

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

69. Kaart 69

    Küsimus

    Constant-multiple rule?

    Vastus

    For a constant $c$,

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

70. Kaart 70

    Küsimus

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

    Vastus

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

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

71. Kaart 71

    Küsimus

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

    Vastus

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

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

72. Kaart 72

    Küsimus

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

    Vastus

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

73. Kaart 73

    Küsimus

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

    Vastus

    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. Kaart 74

    Küsimus

    Derivative of an inverse function at $x$?

    Vastus

    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. Kaart 75

    Küsimus

    Derivative of $\arcsin x$?

    Vastus

    For $|x|<1$,

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

76. Kaart 76

    Küsimus

    Notation for the third derivative of $f$?

    Vastus

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

77. Kaart 77

    Küsimus

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

    Vastus

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

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

78. Kaart 78

    Küsimus

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

    Vastus

    Where $y\ne0$,

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

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

79. Kaart 79

    Küsimus

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

    Vastus

    Provided $f'(a)\ne0$,

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

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

80. Kaart 80

    Küsimus

    Derivative of $\arctan x$?

    Vastus

    For every real $x$,

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

81. Kaart 81

    Küsimus

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

    Vastus

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

    The extra factor is the chain rule.

82. Kaart 82

    Küsimus

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

    Vastus

    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. Kaart 83

    Küsimus

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

    Vastus

    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. Kaart 84

    Küsimus

    Derivative of $\arccos x$?

    Vastus

    For $|x|<1$,

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

85. Kaart 85

    Küsimus

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

    Vastus

    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. Kaart 86

    Küsimus

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

    Vastus

    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. Kaart 87

    Küsimus

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

    Vastus

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

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

88. Kaart 88

    Küsimus

    How are tangent slopes of inverse graphs related?

    Vastus

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

89. Kaart 89

    Küsimus

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

    Vastus

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

90. Kaart 90

    Küsimus

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

    Vastus

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

91. Kaart 91

    Küsimus

    Horizontal tangent on an implicit curve: derivative condition?

    Vastus

    $dy/dx=0$ 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. Kaart 92

    Küsimus

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

    Vastus

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

93. Kaart 93

    Küsimus

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

    Vastus

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

94. Kaart 94

    Küsimus

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

    Vastus

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

    This combines the power rule with the chain rule.

95. Kaart 95

    Küsimus

    Vertical tangent on an implicit curve: derivative clue?

    Vastus

    $dy/dx$ 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. Kaart 96

    Küsimus

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

    Vastus

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

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

97. Kaart 97

    Küsimus

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

    Vastus

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

98. Kaart 98

    Küsimus

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

    Vastus

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

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

99. Kaart 99

    Küsimus

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

    Vastus

    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. Kaart 100

     Küsimus

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

     Vastus

     When the functions are differentiable, the chain rule gives

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

101. Kaart 101

     Küsimus

     What local property lets a function have an inverse derivative?

     Vastus

     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. Kaart 102

     Küsimus

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

     Vastus

     For $a>0$,

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

103. Kaart 103

     Küsimus

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

     Vastus

     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. Kaart 104

     Küsimus

     Position, velocity, and acceleration relationships?

     Vastus

     For position $s(t)$,

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

105. Kaart 105

     Küsimus

     Central idea of a related-rates problem?

     Vastus

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

106. Kaart 106

     Küsimus

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

     Vastus

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

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

107. Kaart 107

     Küsimus

     L’Hospital’s Rule: basic conditions?

     Vastus

     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. Kaart 108

     Küsimus

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

     Vastus

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

109. Kaart 109

     Küsimus

     Speed in terms of velocity?

     Vastus

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kaart 110

     Küsimus

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

     Vastus

     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. Kaart 111

     Küsimus

     Differential approximation connecting $dx$ and $dy$?

     Vastus

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

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

112. Kaart 112

     Küsimus

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

     Vastus

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

113. Kaart 113

     Küsimus

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

     Vastus

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

114. Kaart 114

     Küsimus

     What does positive acceleration say about velocity?

     Vastus

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

115. Kaart 115

     Küsimus

     Related rates: when should numerical values be substituted?

     Vastus

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

116. Kaart 116

     Küsimus

     How does concavity predict linearization error?

     Vastus

     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. Kaart 117

     Küsimus

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

     Vastus

     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. Kaart 118

     Küsimus

     When is a particle moving in the positive direction?

     Vastus

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

119. Kaart 119

     Küsimus

     How can velocity show a change of direction?

     Vastus

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

120. Kaart 120

     Küsimus

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

     Vastus

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

121. Kaart 121

     Küsimus

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

     Vastus

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

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

122. Kaart 122

     Küsimus

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

     Vastus

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

123. Kaart 123

     Küsimus

     What must a contextual derivative sentence include?

     Vastus

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

124. Kaart 124

     Küsimus

     Velocity negative and acceleration positive: what happens?

     Vastus

     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. Kaart 125

     Küsimus

     How should a negative related rate be interpreted?

     Vastus

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

126. Kaart 126

     Küsimus

     When is local linearity a sound approximation tool?

     Vastus

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

127. Kaart 127

     Küsimus

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

     Vastus

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

128. Kaart 128

     Küsimus

     When is speed increasing?

     Vastus

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

129. Kaart 129

     Küsimus

     Volume changes with time: notation for its rate?

     Vastus

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

130. Kaart 130

     Küsimus

     Why are similar triangles useful in related rates?

     Vastus

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

131. Kaart 131

     Küsimus

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

     Vastus

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

132. Kaart 132

     Küsimus

     What conclusion does L’Hospital’s Rule permit?

     Vastus

     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. Kaart 133

     Küsimus

     When is speed decreasing?

     Vastus

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

134. Kaart 134

     Küsimus

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

     Vastus

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

135. Kaart 135

     Küsimus

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

     Vastus

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

136. Kaart 136

     Küsimus

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

     Vastus

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

137. Kaart 137

     Küsimus

     Extreme Value Theorem: hypothesis and conclusion?

     Vastus

     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. Kaart 138

     Küsimus

     What is a critical number of $f$?

     Vastus

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

139. Kaart 139

     Küsimus

     First derivative test for a local maximum?

     Vastus

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

140. Kaart 140

     Küsimus

     Second-derivative sign for concave up?

     Vastus

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

141. Kaart 141

     Küsimus

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

     Vastus

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

142. Kaart 142

     Küsimus

     First step in an optimization model?

     Vastus

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

143. Kaart 143

     Küsimus

     Mean Value Theorem: hypotheses and conclusion?

     Vastus

     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. Kaart 144

     Küsimus

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

     Vastus

     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.

     

     288 kaarti

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

     Õpi seda kaardipakki tasuta

     Nibomo avaneb, et saaksid õppimist alustada.

145. Kaart 145

     Küsimus

     First derivative test for a local minimum?

     Vastus

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

146. Kaart 146

     Küsimus

     What must happen at an inflection point?

     Vastus

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

147. Kaart 147

     Küsimus

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

     Vastus

     $f''$ 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. Kaart 148

     Küsimus

     How do you confirm an optimization answer is absolute?

     Vastus

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

149. Kaart 149

     Küsimus

     Rolle’s Theorem: hypotheses and conclusion?

     Vastus

     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. Kaart 150

     Küsimus

     Difference between absolute and relative extrema?

     Vastus

     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. Kaart 151

     Küsimus

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

     Vastus

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

152. Kaart 152

     Küsimus

     Second derivative test for a local minimum?

     Vastus

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

153. Kaart 153

     Küsimus

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

     Vastus

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

154. Kaart 154

     Küsimus

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

     Vastus

     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. Kaart 155

     Küsimus

     Which theorem links an average slope to an instantaneous slope?

     Vastus

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

156. Kaart 156

     Küsimus

     How can an implicit derivative locate a horizontal tangent?

     Vastus

     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. Kaart 157

     Küsimus

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

     Vastus

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

158. Kaart 158

     Küsimus

     Second derivative test for a local maximum?

     Vastus

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

159. Kaart 159

     Küsimus

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

     Vastus

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

160. Kaart 160

     Küsimus

     Why must an optimization domain be stated?

     Vastus

     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. Kaart 161

     Küsimus

     Which theorem guarantees absolute extrema, not where they occur?

     Vastus

     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. Kaart 162

     Küsimus

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

     Vastus

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

163. Kaart 163

     Küsimus

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

     Vastus

     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. Kaart 164

     Küsimus

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

     Vastus

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

165. Kaart 165

     Küsimus

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

     Vastus

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

166. Kaart 166

     Küsimus

     How can an implicit derivative locate a vertical tangent?

     Vastus

     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. Kaart 167

     Küsimus

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

     Vastus

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

168. Kaart 168

     Küsimus

     Why are endpoints included in the candidates test?

     Vastus

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

169. Kaart 169

     Küsimus

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

     Vastus

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

170. Kaart 170

     Küsimus

     Second-derivative sign for concave down?

     Vastus

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

171. Kaart 171

     Küsimus

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

     Vastus

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

172. Kaart 172

     Küsimus

     What should the final line of an optimization solution state?

     Vastus

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

173. Kaart 173

     Küsimus

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

     Vastus

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

174. Kaart 174

     Küsimus

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

     Vastus

     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. Kaart 175

     Küsimus

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

     Vastus

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

176. Kaart 176

     Küsimus

     Left Riemann sum on equal subintervals?

     Vastus

     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. Kaart 177

     Küsimus

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

     Vastus

     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. Kaart 178

     Küsimus

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Vastus

     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. Kaart 179

     Küsimus

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

     Vastus

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kaart 180

     Küsimus

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

     Vastus

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

181. Kaart 181

     Küsimus

     Right Riemann sum on equal subintervals?

     Vastus

     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. Kaart 182

     Küsimus

     How does reversing integral bounds change the value?

     Vastus

     It changes the sign:

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

183. Kaart 183

     Küsimus

     Net Change Theorem?

     Vastus

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

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

184. Kaart 184

     Küsimus

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

     Vastus

     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. Kaart 185

     Küsimus

     Power rule for antiderivatives?

     Vastus

     For $n\ne-1$,

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

186. Kaart 186

     Küsimus

     Midpoint Riemann sum on equal subintervals?

     Vastus

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

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

187. Kaart 187

     Küsimus

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

     Vastus

     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. Kaart 188

     Küsimus

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

     Vastus

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kaart 189

     Küsimus

     Antiderivative of $1/x$?

     Vastus

     On any interval not crossing zero,

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

190. Kaart 190

     Küsimus

     Trapezoidal approximation on equal subintervals?

     Vastus

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

191. Kaart 191

     Küsimus

     How do geometric regions help evaluate a definite integral?

     Vastus

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

192. Kaart 192

     Küsimus

     Basic antiderivatives of sine and cosine?

     Vastus

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

193. Kaart 193

     Küsimus

     Definite integral as a limit of Riemann sums?

     Vastus

     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. Kaart 194

     Küsimus

     Constant-multiple rule for integrals?

     Vastus

     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. Kaart 195

     Küsimus

     What pattern suggests $u$-substitution?

     Vastus

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

196. Kaart 196

     Küsimus

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

     Vastus

     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. Kaart 197

     Küsimus

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

     Vastus

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

198. Kaart 198

     Küsimus

     Sum-and-difference rule for definite integrals?

     Vastus

     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. Kaart 199

     Küsimus

     Basic antiderivative of $e^x$?

     Vastus

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

200. Kaart 200

     Küsimus

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

     Vastus

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

201. Kaart 201

     Küsimus

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

     Vastus

     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. Kaart 202

     Küsimus

     How does concavity predict trapezoidal and midpoint error?

     Vastus

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

203. Kaart 203

     Küsimus

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

     Vastus

     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. Kaart 204

     Küsimus

     What denominator pattern suggests an arctangent antiderivative?

     Vastus

     After completing the square and scaling, a form like

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

205. Kaart 205

     Küsimus

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

     Vastus

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

206. Kaart 206

     Küsimus

     How does an initial condition determine an antiderivative?

     Vastus

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

207. Kaart 207

     Küsimus

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

     Vastus

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

208. Kaart 208

     Küsimus

     Why does an indefinite integral include $+C$?

     Vastus

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

209. Kaart 209

     Küsimus

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

     Vastus

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

210. Kaart 210

     Küsimus

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

     Vastus

     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. Kaart 211

     Küsimus

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

     Vastus

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

212. Kaart 212

     Küsimus

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

     Vastus

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

213. Kaart 213

     Küsimus

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

     Vastus

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

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

214. Kaart 214

     Küsimus

     Riemann sum for unequal subinterval widths?

     Vastus

     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. Kaart 215

     Küsimus

     Does continuity guarantee integrability on a closed interval?

     Vastus

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

216. Kaart 216

     Küsimus

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

     Vastus

     Where $g(x)\ne0$,

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

217. Kaart 217

     Küsimus

     What algebraic rewrites often reveal a basic antiderivative?

     Vastus

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

218. Kaart 218

     Küsimus

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

     Vastus

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

219. Kaart 219

     Küsimus

     What is a differential equation?

     Vastus

     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. Kaart 220

     Küsimus

     How does a verbal rate statement become a differential equation?

     Vastus

     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. Kaart 221

     Küsimus

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

     Vastus

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

222. Kaart 222

     Küsimus

     General solution versus particular solution?

     Vastus

     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. Kaart 223

     Küsimus

     What does one segment in a slope field show?

     Vastus

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

224. Kaart 224

     Küsimus

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

     Vastus

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

225. Kaart 225

     Küsimus

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

     Vastus

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

226. Kaart 226

     Küsimus

     What makes a first-order differential equation separable?

     Vastus

     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. Kaart 227

     Küsimus

     What is an initial value problem?

     Vastus

     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. Kaart 228

     Küsimus

     What is an isocline in a slope field?

     Vastus

     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. Kaart 229

     Küsimus

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

     Vastus

     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. Kaart 230

     Küsimus

     General solution of $dy/dt=ky$?

     Vastus

     $$y=Ce^{kt}$$

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

231. Kaart 231

     Küsimus

     Core method for solving a separable differential equation?

     Vastus

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

232. Kaart 232

     Küsimus

     How should a solution curve follow a slope field?

     Vastus

     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. Kaart 233

     Küsimus

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

     Vastus

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

234. Kaart 234

     Küsimus

     Why is one integration constant enough after integrating both sides?

     Vastus

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

235. Kaart 235

     Küsimus

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

     Vastus

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

236. Kaart 236

     Küsimus

     Can one differential equation have infinitely many solutions?

     Vastus

     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. Kaart 237

     Küsimus

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

     Vastus

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

238. Kaart 238

     Küsimus

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

     Vastus

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

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

239. Kaart 239

     Küsimus

     What can be lost when dividing to separate variables?

     Vastus

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

240. Kaart 240

     Küsimus

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

     Vastus

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

241. Kaart 241

     Küsimus

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

     Vastus

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

242. Kaart 242

     Küsimus

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

     Vastus

     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. Kaart 243

     Küsimus

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

     Vastus

     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. Kaart 244

     Küsimus

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

     Vastus

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

245. Kaart 245

     Küsimus

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

     Vastus

     For $k>0$,

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

     It is independent of the initial amount.

246. Kaart 246

     Küsimus

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

     Vastus

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

247. Kaart 247

     Küsimus

     How is an initial condition used after separation?

     Vastus

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

248. Kaart 248

     Küsimus

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

     Vastus

     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. Kaart 249

     Küsimus

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

     Vastus

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

250. Kaart 250

     Küsimus

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

     Vastus

     For $k<0$,

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

251. Kaart 251

     Küsimus

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

     Vastus

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

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

252. Kaart 252

     Küsimus

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

     Vastus

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

     Velocity below zero contributes negative displacement.

253. Kaart 253

     Küsimus

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

     Vastus

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

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

     Think top minus bottom.

254. Kaart 254

     Küsimus

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

     Vastus

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

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

255. Kaart 255

     Küsimus

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Vastus

     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. Kaart 256

     Küsimus

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

     Vastus

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

257. Kaart 257

     Küsimus

     Cross-sectional area when each slice is a square?

     Vastus

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

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

258. Kaart 258

     Küsimus

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

     Vastus

     Integrate the net rate:

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

259. Kaart 259

     Küsimus

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

     Vastus

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

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

     Think right minus left.

260. Kaart 260

     Küsimus

     Disc-method volume formula?

     Vastus

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

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

261. Kaart 261

     Küsimus

     What units does average value have?

     Vastus

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

262. Kaart 262

     Küsimus

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

     Vastus

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

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

263. Kaart 263

     Küsimus

     Cross-sectional area when each slice is a rectangle?

     Vastus

     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. Kaart 264

     Küsimus

     How do you determine bounds for area between curves?

     Vastus

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

265. Kaart 265

     Küsimus

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

     Vastus

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

266. Kaart 266

     Küsimus

     When is a particle moving to the right or left?

     Vastus

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

267. Kaart 267

     Küsimus

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

     Vastus

     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. Kaart 268

     Küsimus

     Why must an area integral be split where curves intersect?

     Vastus

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

269. Kaart 269

     Küsimus

     How can a velocity table approximate displacement?

     Vastus

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

270. Kaart 270

     Küsimus

     Washer-method volume formula?

     Vastus

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

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

271. Kaart 271

     Küsimus

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

     Vastus

     If $s(a)$ is known,

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

272. Kaart 272

     Küsimus

     How do you choose between vertical and horizontal area slices?

     Vastus

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

273. Kaart 273

     Küsimus

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

     Vastus

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

274. Kaart 274

     Küsimus

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

     Vastus

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

275. Kaart 275

     Küsimus

     Single expression for area between two curves?

     Vastus

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kaart 276

     Küsimus

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

     Vastus

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

277. Kaart 277

     Küsimus

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

     Vastus

     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. Kaart 278

     Küsimus

     When should a volume integral use $dy$?

     Vastus

     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. Kaart 279

     Küsimus

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

     Vastus

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

280. Kaart 280

     Küsimus

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

     Vastus

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

281. Kaart 281

     Küsimus

     Why must total distance split at velocity sign changes?

     Vastus

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

282. Kaart 282

     Küsimus

     When does an accumulated quantity reach a local maximum?

     Vastus

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

283. Kaart 283

     Küsimus

     What distinguishes area from a definite integral?

     Vastus

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

284. Kaart 284

     Küsimus

     How do position, velocity, and acceleration graphs correspond?

     Vastus

     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. Kaart 285

     Küsimus

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

     Vastus

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

286. Kaart 286

     Küsimus

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

     Vastus

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

287. Kaart 287

     Küsimus

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

     Vastus

     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. Kaart 288

     Küsimus

     Why should a contextual integral answer include a sentence?

     Vastus

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