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

הכרטיסים בחפיסה הזו

1. כרטיס 1

   שאלה

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

   תשובה

   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. כרטיס 2

   שאלה

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

   תשובה

   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. כרטיס 3

   שאלה

   When does direct substitution evaluate a limit?

   תשובה

   When the function is continuous at the target input. Then

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

4. כרטיס 4

   שאלה

   Three conditions for continuity at $x=a$?

   תשובה

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

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

5. כרטיס 5

   שאלה

   Intermediate Value Theorem: hypotheses and conclusion?

   תשובה

   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. כרטיס 6

   שאלה

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

   תשובה

   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. כרטיס 7

   שאלה

   How do you read a finite limit from a graph?

   תשובה

   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. כרטיס 8

   שאלה

   Limit law for a sum or difference?

   תשובה

   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. כרטיס 9

   שאלה

   What makes a discontinuity removable?

   תשובה

   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. כרטיס 10

    שאלה

    Squeeze Theorem: usable form?

    תשובה

    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. כרטיס 11

    שאלה

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

    תשובה

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

12. כרטיס 12

    שאלה

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

    תשובה

    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. כרטיס 13

    שאלה

    Limit law for a product?

    תשובה

    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. כרטיס 14

    שאלה

    Graph signature of a jump discontinuity?

    תשובה

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

15. כרטיס 15

    שאלה

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

    תשובה

    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. כרטיס 16

    שאלה

    Horizontal asymptote from a limit at infinity?

    תשובה

    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. כרטיס 17

    שאלה

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

    תשובה

    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. כרטיס 18

    שאלה

    Limit law for a quotient—and its condition?

    תשובה

    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. כרטיס 19

    שאלה

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

    תשובה

    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. כרטיס 20

    שאלה

    When is the Squeeze Theorem a natural choice?

    תשובה

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

21. כרטיס 21

    שאלה

    Vertical asymptote from one-sided behavior?

    תשובה

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

22. כרטיס 22

    שאלה

    Limit at infinity of equal-degree rational functions?

    תשובה

    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. כרטיס 23

    שאלה

    When can a limit pass through a continuous outer function?

    תשובה

    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. כרטיס 24

    שאלה

    What makes a discontinuity infinite?

    תשובה

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

25. כרטיס 25

    שאלה

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

    תשובה

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

26. כרטיס 26

    שאלה

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

    תשובה

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. כרטיס 27

    שאלה

    Continuity of a composition?

    תשובה

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

28. כרטיס 28

    שאלה

    Limit at infinity when a rational numerator has lower degree?

    תשובה

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

29. כרטיס 29

    שאלה

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

    תשובה

    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. כרטיס 30

    שאלה

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

    תשובה

    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. כרטיס 31

    שאלה

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

    תשובה

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

32. כרטיס 32

    שאלה

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

    תשובה

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

33. כרטיס 33

    שאלה

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

    תשובה

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

34. כרטיס 34

    שאלה

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

    תשובה

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

35. כרטיס 35

    שאלה

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

    תשובה

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

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

36. כרטיס 36

    שאלה

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

    תשובה

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

    The derivative exists only if this finite limit exists.

37. כרטיס 37

    שאלה

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

    תשובה

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

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

38. כרטיס 38

    שאלה

    What does differentiability imply about continuity?

    תשובה

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

39. כרטיס 39

    שאלה

    Power rule for derivatives?

    תשובה

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

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

40. כרטיס 40

    שאלה

    Units of $f'(x)$?

    תשובה

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

41. כרטיס 41

    שאלה

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

    תשובה

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

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

42. כרטיס 42

    שאלה

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

    תשובה

    $f'(x)>0$ 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. כרטיס 43

    שאלה

    Derivative of a constant?

    תשובה

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

    A constant function has zero rate of change.

44. כרטיס 44

    שאלה

    Derivative of $\sin x$?

    תשובה

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

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

45. כרטיס 45

    שאלה

    Product rule?

    תשובה

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

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

46. כרטיס 46

    שאלה

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

    תשובה

    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. כרטיס 47

    שאלה

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

    תשובה

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

48. כרטיס 48

    שאלה

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

    תשובה

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

49. כרטיס 49

    שאלה

    Derivative of a sum or difference?

    תשובה

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

50. כרטיס 50

    שאלה

    Derivative of $\cos x$?

    תשובה

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

    The standard formula assumes radians.

51. כרטיס 51

    שאלה

    Quotient rule?

    תשובה

    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. כרטיס 52

    שאלה

    Common notations for the first derivative?

    תשובה

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

53. כרטיס 53

    שאלה

    Derivative of $e^x$?

    תשובה

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

54. כרטיס 54

    שאלה

    What graph features can make $f$ nondifferentiable?

    תשובה

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

55. כרטיס 55

    שאלה

    Derivative of $\tan x$?

    תשובה

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. כרטיס 56

    שאלה

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

    תשובה

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

57. כרטיס 57

    שאלה

    Derivative of $\ln x$?

    תשובה

    For $x>0$,

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

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

58. כרטיס 58

    שאלה

    How does the power rule handle roots or negative powers?

    תשובה

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

59. כרטיס 59

    שאלה

    Derivative of $\csc x$?

    תשובה

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. כרטיס 60

    שאלה

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

    תשובה

    $f$ is increasing on that interval.

61. כרטיס 61

    שאלה

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

    תשובה

    For $a>0$,

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

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

62. כרטיס 62

    שאלה

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

    תשובה

    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. כרטיס 63

    שאלה

    Derivative of $\sec x$?

    תשובה

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. כרטיס 64

    שאלה

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

    תשובה

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

65. כרטיס 65

    שאלה

    Derivative of $\log_a x$?

    תשובה

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

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

66. כרטיס 66

    שאלה

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

    תשובה

    For $h=fg$,

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

    Use the four table entries at the same input.

67. כרטיס 67

    שאלה

    Derivative of $\cot x$?

    תשובה

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. כרטיס 68

    שאלה

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

    תשובה

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

69. כרטיס 69

    שאלה

    Constant-multiple rule?

    תשובה

    For a constant $c$,

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

70. כרטיס 70

    שאלה

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

    תשובה

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

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

71. כרטיס 71

    שאלה

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

    תשובה

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

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

72. כרטיס 72

    שאלה

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

    תשובה

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

73. כרטיס 73

    שאלה

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

    תשובה

    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. כרטיס 74

    שאלה

    Derivative of an inverse function at $x$?

    תשובה

    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. כרטיס 75

    שאלה

    Derivative of $\arcsin x$?

    תשובה

    For $|x|<1$,

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

76. כרטיס 76

    שאלה

    Notation for the third derivative of $f$?

    תשובה

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

77. כרטיס 77

    שאלה

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

    תשובה

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

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

78. כרטיס 78

    שאלה

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

    תשובה

    Where $y\ne0$,

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

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

79. כרטיס 79

    שאלה

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

    תשובה

    Provided $f'(a)\ne0$,

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

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

80. כרטיס 80

    שאלה

    Derivative of $\arctan x$?

    תשובה

    For every real $x$,

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

81. כרטיס 81

    שאלה

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

    תשובה

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

    The extra factor is the chain rule.

82. כרטיס 82

    שאלה

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

    תשובה

    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. כרטיס 83

    שאלה

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

    תשובה

    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. כרטיס 84

    שאלה

    Derivative of $\arccos x$?

    תשובה

    For $|x|<1$,

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

85. כרטיס 85

    שאלה

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

    תשובה

    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. כרטיס 86

    שאלה

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

    תשובה

    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. כרטיס 87

    שאלה

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

    תשובה

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

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

88. כרטיס 88

    שאלה

    How are tangent slopes of inverse graphs related?

    תשובה

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

89. כרטיס 89

    שאלה

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

    תשובה

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

90. כרטיס 90

    שאלה

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

    תשובה

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

91. כרטיס 91

    שאלה

    Horizontal tangent on an implicit curve: derivative condition?

    תשובה

    $dy/dx=0$ 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. כרטיס 92

    שאלה

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

    תשובה

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

93. כרטיס 93

    שאלה

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

    תשובה

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

94. כרטיס 94

    שאלה

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

    תשובה

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

    This combines the power rule with the chain rule.

95. כרטיס 95

    שאלה

    Vertical tangent on an implicit curve: derivative clue?

    תשובה

    $dy/dx$ 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. כרטיס 96

    שאלה

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

    תשובה

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

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

97. כרטיס 97

    שאלה

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

    תשובה

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

98. כרטיס 98

    שאלה

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

    תשובה

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

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

99. כרטיס 99

    שאלה

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

    תשובה

    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. כרטיס 100

     שאלה

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

     תשובה

     When the functions are differentiable, the chain rule gives

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

101. כרטיס 101

     שאלה

     What local property lets a function have an inverse derivative?

     תשובה

     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. כרטיס 102

     שאלה

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

     תשובה

     For $a>0$,

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

103. כרטיס 103

     שאלה

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

     תשובה

     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. כרטיס 104

     שאלה

     Position, velocity, and acceleration relationships?

     תשובה

     For position $s(t)$,

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

105. כרטיס 105

     שאלה

     Central idea of a related-rates problem?

     תשובה

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

106. כרטיס 106

     שאלה

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

     תשובה

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

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

107. כרטיס 107

     שאלה

     L’Hospital’s Rule: basic conditions?

     תשובה

     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. כרטיס 108

     שאלה

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

     תשובה

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

109. כרטיס 109

     שאלה

     Speed in terms of velocity?

     תשובה

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

     Velocity includes direction; speed is nonnegative magnitude.

110. כרטיס 110

     שאלה

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

     תשובה

     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. כרטיס 111

     שאלה

     Differential approximation connecting $dx$ and $dy$?

     תשובה

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

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

112. כרטיס 112

     שאלה

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

     תשובה

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

113. כרטיס 113

     שאלה

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

     תשובה

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

114. כרטיס 114

     שאלה

     What does positive acceleration say about velocity?

     תשובה

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

115. כרטיס 115

     שאלה

     Related rates: when should numerical values be substituted?

     תשובה

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

116. כרטיס 116

     שאלה

     How does concavity predict linearization error?

     תשובה

     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. כרטיס 117

     שאלה

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

     תשובה

     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. כרטיס 118

     שאלה

     When is a particle moving in the positive direction?

     תשובה

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

119. כרטיס 119

     שאלה

     How can velocity show a change of direction?

     תשובה

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

120. כרטיס 120

     שאלה

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

     תשובה

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

121. כרטיס 121

     שאלה

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

     תשובה

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

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

122. כרטיס 122

     שאלה

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

     תשובה

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

123. כרטיס 123

     שאלה

     What must a contextual derivative sentence include?

     תשובה

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

124. כרטיס 124

     שאלה

     Velocity negative and acceleration positive: what happens?

     תשובה

     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. כרטיס 125

     שאלה

     How should a negative related rate be interpreted?

     תשובה

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

126. כרטיס 126

     שאלה

     When is local linearity a sound approximation tool?

     תשובה

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

127. כרטיס 127

     שאלה

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

     תשובה

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

128. כרטיס 128

     שאלה

     When is speed increasing?

     תשובה

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

129. כרטיס 129

     שאלה

     Volume changes with time: notation for its rate?

     תשובה

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

130. כרטיס 130

     שאלה

     Why are similar triangles useful in related rates?

     תשובה

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

131. כרטיס 131

     שאלה

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

     תשובה

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

132. כרטיס 132

     שאלה

     What conclusion does L’Hospital’s Rule permit?

     תשובה

     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. כרטיס 133

     שאלה

     When is speed decreasing?

     תשובה

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

134. כרטיס 134

     שאלה

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

     תשובה

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

135. כרטיס 135

     שאלה

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

     תשובה

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

136. כרטיס 136

     שאלה

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

     תשובה

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

137. כרטיס 137

     שאלה

     Extreme Value Theorem: hypothesis and conclusion?

     תשובה

     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. כרטיס 138

     שאלה

     What is a critical number of $f$?

     תשובה

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

139. כרטיס 139

     שאלה

     First derivative test for a local maximum?

     תשובה

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

140. כרטיס 140

     שאלה

     Second-derivative sign for concave up?

     תשובה

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

141. כרטיס 141

     שאלה

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

     תשובה

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

142. כרטיס 142

     שאלה

     First step in an optimization model?

     תשובה

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

143. כרטיס 143

     שאלה

     Mean Value Theorem: hypotheses and conclusion?

     תשובה

     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. כרטיס 144

     שאלה

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

     תשובה

     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 כרטיסים

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

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145. כרטיס 145

     שאלה

     First derivative test for a local minimum?

     תשובה

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

146. כרטיס 146

     שאלה

     What must happen at an inflection point?

     תשובה

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

147. כרטיס 147

     שאלה

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

     תשובה

     $f''$ 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. כרטיס 148

     שאלה

     How do you confirm an optimization answer is absolute?

     תשובה

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

149. כרטיס 149

     שאלה

     Rolle’s Theorem: hypotheses and conclusion?

     תשובה

     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. כרטיס 150

     שאלה

     Difference between absolute and relative extrema?

     תשובה

     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. כרטיס 151

     שאלה

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

     תשובה

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

152. כרטיס 152

     שאלה

     Second derivative test for a local minimum?

     תשובה

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

153. כרטיס 153

     שאלה

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

     תשובה

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

154. כרטיס 154

     שאלה

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

     תשובה

     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. כרטיס 155

     שאלה

     Which theorem links an average slope to an instantaneous slope?

     תשובה

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

156. כרטיס 156

     שאלה

     How can an implicit derivative locate a horizontal tangent?

     תשובה

     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. כרטיס 157

     שאלה

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

     תשובה

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

158. כרטיס 158

     שאלה

     Second derivative test for a local maximum?

     תשובה

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

159. כרטיס 159

     שאלה

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

     תשובה

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

160. כרטיס 160

     שאלה

     Why must an optimization domain be stated?

     תשובה

     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. כרטיס 161

     שאלה

     Which theorem guarantees absolute extrema, not where they occur?

     תשובה

     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. כרטיס 162

     שאלה

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

     תשובה

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

163. כרטיס 163

     שאלה

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

     תשובה

     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. כרטיס 164

     שאלה

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

     תשובה

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

165. כרטיס 165

     שאלה

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

     תשובה

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

166. כרטיס 166

     שאלה

     How can an implicit derivative locate a vertical tangent?

     תשובה

     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. כרטיס 167

     שאלה

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

     תשובה

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

168. כרטיס 168

     שאלה

     Why are endpoints included in the candidates test?

     תשובה

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

169. כרטיס 169

     שאלה

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

     תשובה

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

170. כרטיס 170

     שאלה

     Second-derivative sign for concave down?

     תשובה

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

171. כרטיס 171

     שאלה

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

     תשובה

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

172. כרטיס 172

     שאלה

     What should the final line of an optimization solution state?

     תשובה

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

173. כרטיס 173

     שאלה

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

     תשובה

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

174. כרטיס 174

     שאלה

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

     תשובה

     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. כרטיס 175

     שאלה

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

     תשובה

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

176. כרטיס 176

     שאלה

     Left Riemann sum on equal subintervals?

     תשובה

     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. כרטיס 177

     שאלה

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

     תשובה

     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. כרטיס 178

     שאלה

     Fundamental Theorem of Calculus: evaluate a definite integral?

     תשובה

     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. כרטיס 179

     שאלה

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

     תשובה

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. כרטיס 180

     שאלה

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

     תשובה

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

181. כרטיס 181

     שאלה

     Right Riemann sum on equal subintervals?

     תשובה

     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. כרטיס 182

     שאלה

     How does reversing integral bounds change the value?

     תשובה

     It changes the sign:

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

183. כרטיס 183

     שאלה

     Net Change Theorem?

     תשובה

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

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

184. כרטיס 184

     שאלה

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

     תשובה

     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. כרטיס 185

     שאלה

     Power rule for antiderivatives?

     תשובה

     For $n\ne-1$,

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

186. כרטיס 186

     שאלה

     Midpoint Riemann sum on equal subintervals?

     תשובה

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

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

187. כרטיס 187

     שאלה

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

     תשובה

     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. כרטיס 188

     שאלה

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

     תשובה

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. כרטיס 189

     שאלה

     Antiderivative of $1/x$?

     תשובה

     On any interval not crossing zero,

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

190. כרטיס 190

     שאלה

     Trapezoidal approximation on equal subintervals?

     תשובה

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

191. כרטיס 191

     שאלה

     How do geometric regions help evaluate a definite integral?

     תשובה

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

192. כרטיס 192

     שאלה

     Basic antiderivatives of sine and cosine?

     תשובה

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

193. כרטיס 193

     שאלה

     Definite integral as a limit of Riemann sums?

     תשובה

     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. כרטיס 194

     שאלה

     Constant-multiple rule for integrals?

     תשובה

     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. כרטיס 195

     שאלה

     What pattern suggests $u$-substitution?

     תשובה

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

196. כרטיס 196

     שאלה

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

     תשובה

     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. כרטיס 197

     שאלה

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

     תשובה

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

198. כרטיס 198

     שאלה

     Sum-and-difference rule for definite integrals?

     תשובה

     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. כרטיס 199

     שאלה

     Basic antiderivative of $e^x$?

     תשובה

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

200. כרטיס 200

     שאלה

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

     תשובה

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

201. כרטיס 201

     שאלה

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

     תשובה

     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. כרטיס 202

     שאלה

     How does concavity predict trapezoidal and midpoint error?

     תשובה

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

203. כרטיס 203

     שאלה

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

     תשובה

     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. כרטיס 204

     שאלה

     What denominator pattern suggests an arctangent antiderivative?

     תשובה

     After completing the square and scaling, a form like

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

205. כרטיס 205

     שאלה

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

     תשובה

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

206. כרטיס 206

     שאלה

     How does an initial condition determine an antiderivative?

     תשובה

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

207. כרטיס 207

     שאלה

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

     תשובה

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

208. כרטיס 208

     שאלה

     Why does an indefinite integral include $+C$?

     תשובה

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

209. כרטיס 209

     שאלה

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

     תשובה

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

210. כרטיס 210

     שאלה

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

     תשובה

     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. כרטיס 211

     שאלה

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

     תשובה

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

212. כרטיס 212

     שאלה

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

     תשובה

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

213. כרטיס 213

     שאלה

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

     תשובה

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

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

214. כרטיס 214

     שאלה

     Riemann sum for unequal subinterval widths?

     תשובה

     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. כרטיס 215

     שאלה

     Does continuity guarantee integrability on a closed interval?

     תשובה

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

216. כרטיס 216

     שאלה

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

     תשובה

     Where $g(x)\ne0$,

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

217. כרטיס 217

     שאלה

     What algebraic rewrites often reveal a basic antiderivative?

     תשובה

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

218. כרטיס 218

     שאלה

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

     תשובה

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

219. כרטיס 219

     שאלה

     What is a differential equation?

     תשובה

     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. כרטיס 220

     שאלה

     How does a verbal rate statement become a differential equation?

     תשובה

     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. כרטיס 221

     שאלה

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

     תשובה

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

222. כרטיס 222

     שאלה

     General solution versus particular solution?

     תשובה

     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. כרטיס 223

     שאלה

     What does one segment in a slope field show?

     תשובה

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

224. כרטיס 224

     שאלה

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

     תשובה

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

225. כרטיס 225

     שאלה

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

     תשובה

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

226. כרטיס 226

     שאלה

     What makes a first-order differential equation separable?

     תשובה

     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. כרטיס 227

     שאלה

     What is an initial value problem?

     תשובה

     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. כרטיס 228

     שאלה

     What is an isocline in a slope field?

     תשובה

     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. כרטיס 229

     שאלה

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

     תשובה

     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. כרטיס 230

     שאלה

     General solution of $dy/dt=ky$?

     תשובה

     $$y=Ce^{kt}$$

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

231. כרטיס 231

     שאלה

     Core method for solving a separable differential equation?

     תשובה

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

232. כרטיס 232

     שאלה

     How should a solution curve follow a slope field?

     תשובה

     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. כרטיס 233

     שאלה

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

     תשובה

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

234. כרטיס 234

     שאלה

     Why is one integration constant enough after integrating both sides?

     תשובה

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

235. כרטיס 235

     שאלה

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

     תשובה

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

236. כרטיס 236

     שאלה

     Can one differential equation have infinitely many solutions?

     תשובה

     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. כרטיס 237

     שאלה

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

     תשובה

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

238. כרטיס 238

     שאלה

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

     תשובה

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

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

239. כרטיס 239

     שאלה

     What can be lost when dividing to separate variables?

     תשובה

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

240. כרטיס 240

     שאלה

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

     תשובה

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

241. כרטיס 241

     שאלה

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

     תשובה

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

242. כרטיס 242

     שאלה

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

     תשובה

     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. כרטיס 243

     שאלה

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

     תשובה

     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. כרטיס 244

     שאלה

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

     תשובה

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

245. כרטיס 245

     שאלה

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

     תשובה

     For $k>0$,

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

     It is independent of the initial amount.

246. כרטיס 246

     שאלה

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

     תשובה

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

247. כרטיס 247

     שאלה

     How is an initial condition used after separation?

     תשובה

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

248. כרטיס 248

     שאלה

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

     תשובה

     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. כרטיס 249

     שאלה

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

     תשובה

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

250. כרטיס 250

     שאלה

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

     תשובה

     For $k<0$,

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

251. כרטיס 251

     שאלה

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

     תשובה

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

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

252. כרטיס 252

     שאלה

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

     תשובה

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

     Velocity below zero contributes negative displacement.

253. כרטיס 253

     שאלה

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

     תשובה

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

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

     Think top minus bottom.

254. כרטיס 254

     שאלה

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

     תשובה

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

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

255. כרטיס 255

     שאלה

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     תשובה

     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. כרטיס 256

     שאלה

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

     תשובה

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

257. כרטיס 257

     שאלה

     Cross-sectional area when each slice is a square?

     תשובה

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

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

258. כרטיס 258

     שאלה

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

     תשובה

     Integrate the net rate:

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

259. כרטיס 259

     שאלה

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

     תשובה

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

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

     Think right minus left.

260. כרטיס 260

     שאלה

     Disc-method volume formula?

     תשובה

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

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

261. כרטיס 261

     שאלה

     What units does average value have?

     תשובה

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

262. כרטיס 262

     שאלה

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

     תשובה

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

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

263. כרטיס 263

     שאלה

     Cross-sectional area when each slice is a rectangle?

     תשובה

     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. כרטיס 264

     שאלה

     How do you determine bounds for area between curves?

     תשובה

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

265. כרטיס 265

     שאלה

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

     תשובה

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

266. כרטיס 266

     שאלה

     When is a particle moving to the right or left?

     תשובה

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

267. כרטיס 267

     שאלה

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

     תשובה

     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. כרטיס 268

     שאלה

     Why must an area integral be split where curves intersect?

     תשובה

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

269. כרטיס 269

     שאלה

     How can a velocity table approximate displacement?

     תשובה

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

270. כרטיס 270

     שאלה

     Washer-method volume formula?

     תשובה

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

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

271. כרטיס 271

     שאלה

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

     תשובה

     If $s(a)$ is known,

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

272. כרטיס 272

     שאלה

     How do you choose between vertical and horizontal area slices?

     תשובה

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

273. כרטיס 273

     שאלה

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

     תשובה

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

274. כרטיס 274

     שאלה

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

     תשובה

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

275. כרטיס 275

     שאלה

     Single expression for area between two curves?

     תשובה

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. כרטיס 276

     שאלה

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

     תשובה

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

277. כרטיס 277

     שאלה

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

     תשובה

     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. כרטיס 278

     שאלה

     When should a volume integral use $dy$?

     תשובה

     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. כרטיס 279

     שאלה

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

     תשובה

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

280. כרטיס 280

     שאלה

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

     תשובה

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

281. כרטיס 281

     שאלה

     Why must total distance split at velocity sign changes?

     תשובה

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

282. כרטיס 282

     שאלה

     When does an accumulated quantity reach a local maximum?

     תשובה

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

283. כרטיס 283

     שאלה

     What distinguishes area from a definite integral?

     תשובה

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

284. כרטיס 284

     שאלה

     How do position, velocity, and acceleration graphs correspond?

     תשובה

     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. כרטיס 285

     שאלה

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

     תשובה

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

286. כרטיס 286

     שאלה

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

     תשובה

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

287. כרטיס 287

     שאלה

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

     תשובה

     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. כרטיס 288

     שאלה

     Why should a contextual integral answer include a sentence?

     תשובה

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