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

Karty w tej talii

1. Karta 1

   Pytanie

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

   Odpowiedź

   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. Karta 2

   Pytanie

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

   Odpowiedź

   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. Karta 3

   Pytanie

   When does direct substitution evaluate a limit?

   Odpowiedź

   When the function is continuous at the target input. Then

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

4. Karta 4

   Pytanie

   Three conditions for continuity at $x=a$?

   Odpowiedź

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

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

5. Karta 5

   Pytanie

   Intermediate Value Theorem: hypotheses and conclusion?

   Odpowiedź

   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. Karta 6

   Pytanie

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

   Odpowiedź

   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. Karta 7

   Pytanie

   How do you read a finite limit from a graph?

   Odpowiedź

   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. Karta 8

   Pytanie

   Limit law for a sum or difference?

   Odpowiedź

   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. Karta 9

   Pytanie

   What makes a discontinuity removable?

   Odpowiedź

   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. Karta 10

    Pytanie

    Squeeze Theorem: usable form?

    Odpowiedź

    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. Karta 11

    Pytanie

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

    Odpowiedź

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

12. Karta 12

    Pytanie

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

    Odpowiedź

    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. Karta 13

    Pytanie

    Limit law for a product?

    Odpowiedź

    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. Karta 14

    Pytanie

    Graph signature of a jump discontinuity?

    Odpowiedź

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

15. Karta 15

    Pytanie

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

    Odpowiedź

    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. Karta 16

    Pytanie

    Horizontal asymptote from a limit at infinity?

    Odpowiedź

    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. Karta 17

    Pytanie

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

    Odpowiedź

    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. Karta 18

    Pytanie

    Limit law for a quotient—and its condition?

    Odpowiedź

    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. Karta 19

    Pytanie

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

    Odpowiedź

    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. Karta 20

    Pytanie

    When is the Squeeze Theorem a natural choice?

    Odpowiedź

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

21. Karta 21

    Pytanie

    Vertical asymptote from one-sided behavior?

    Odpowiedź

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

22. Karta 22

    Pytanie

    Limit at infinity of equal-degree rational functions?

    Odpowiedź

    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. Karta 23

    Pytanie

    When can a limit pass through a continuous outer function?

    Odpowiedź

    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. Karta 24

    Pytanie

    What makes a discontinuity infinite?

    Odpowiedź

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

25. Karta 25

    Pytanie

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

    Odpowiedź

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

26. Karta 26

    Pytanie

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

    Odpowiedź

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Karta 27

    Pytanie

    Continuity of a composition?

    Odpowiedź

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

28. Karta 28

    Pytanie

    Limit at infinity when a rational numerator has lower degree?

    Odpowiedź

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

29. Karta 29

    Pytanie

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

    Odpowiedź

    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. Karta 30

    Pytanie

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

    Odpowiedź

    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. Karta 31

    Pytanie

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

    Odpowiedź

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

32. Karta 32

    Pytanie

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

    Odpowiedź

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

33. Karta 33

    Pytanie

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

    Odpowiedź

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

34. Karta 34

    Pytanie

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

    Odpowiedź

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

35. Karta 35

    Pytanie

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

    Odpowiedź

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

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

36. Karta 36

    Pytanie

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

    Odpowiedź

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

    The derivative exists only if this finite limit exists.

37. Karta 37

    Pytanie

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

    Odpowiedź

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

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

38. Karta 38

    Pytanie

    What does differentiability imply about continuity?

    Odpowiedź

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

39. Karta 39

    Pytanie

    Power rule for derivatives?

    Odpowiedź

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

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

40. Karta 40

    Pytanie

    Units of $f'(x)$?

    Odpowiedź

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

41. Karta 41

    Pytanie

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

    Odpowiedź

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

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

42. Karta 42

    Pytanie

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

    Odpowiedź

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

    Pytanie

    Derivative of a constant?

    Odpowiedź

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

    A constant function has zero rate of change.

44. Karta 44

    Pytanie

    Derivative of $\sin x$?

    Odpowiedź

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

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

45. Karta 45

    Pytanie

    Product rule?

    Odpowiedź

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

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

46. Karta 46

    Pytanie

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

    Odpowiedź

    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. Karta 47

    Pytanie

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

    Odpowiedź

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

48. Karta 48

    Pytanie

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

    Odpowiedź

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

49. Karta 49

    Pytanie

    Derivative of a sum or difference?

    Odpowiedź

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

50. Karta 50

    Pytanie

    Derivative of $\cos x$?

    Odpowiedź

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

    The standard formula assumes radians.

51. Karta 51

    Pytanie

    Quotient rule?

    Odpowiedź

    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. Karta 52

    Pytanie

    Common notations for the first derivative?

    Odpowiedź

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

53. Karta 53

    Pytanie

    Derivative of $e^x$?

    Odpowiedź

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

54. Karta 54

    Pytanie

    What graph features can make $f$ nondifferentiable?

    Odpowiedź

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

55. Karta 55

    Pytanie

    Derivative of $\tan x$?

    Odpowiedź

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Karta 56

    Pytanie

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

    Odpowiedź

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

57. Karta 57

    Pytanie

    Derivative of $\ln x$?

    Odpowiedź

    For $x>0$,

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

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

58. Karta 58

    Pytanie

    How does the power rule handle roots or negative powers?

    Odpowiedź

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

59. Karta 59

    Pytanie

    Derivative of $\csc x$?

    Odpowiedź

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Karta 60

    Pytanie

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

    Odpowiedź

    $f$ is increasing on that interval.

61. Karta 61

    Pytanie

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

    Odpowiedź

    For $a>0$,

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

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

62. Karta 62

    Pytanie

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

    Odpowiedź

    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. Karta 63

    Pytanie

    Derivative of $\sec x$?

    Odpowiedź

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Karta 64

    Pytanie

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

    Odpowiedź

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

65. Karta 65

    Pytanie

    Derivative of $\log_a x$?

    Odpowiedź

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

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

66. Karta 66

    Pytanie

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

    Odpowiedź

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Karta 67

    Pytanie

    Derivative of $\cot x$?

    Odpowiedź

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Karta 68

    Pytanie

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

    Odpowiedź

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

69. Karta 69

    Pytanie

    Constant-multiple rule?

    Odpowiedź

    For a constant $c$,

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

70. Karta 70

    Pytanie

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

    Odpowiedź

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

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

71. Karta 71

    Pytanie

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

    Odpowiedź

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

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

72. Karta 72

    Pytanie

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

    Odpowiedź

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

73. Karta 73

    Pytanie

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

    Odpowiedź

    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. Karta 74

    Pytanie

    Derivative of an inverse function at $x$?

    Odpowiedź

    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. Karta 75

    Pytanie

    Derivative of $\arcsin x$?

    Odpowiedź

    For $|x|<1$,

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

76. Karta 76

    Pytanie

    Notation for the third derivative of $f$?

    Odpowiedź

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

77. Karta 77

    Pytanie

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

    Odpowiedź

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

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

78. Karta 78

    Pytanie

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

    Odpowiedź

    Where $y\ne0$,

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

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

79. Karta 79

    Pytanie

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

    Odpowiedź

    Provided $f'(a)\ne0$,

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

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

80. Karta 80

    Pytanie

    Derivative of $\arctan x$?

    Odpowiedź

    For every real $x$,

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

81. Karta 81

    Pytanie

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

    Odpowiedź

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

    The extra factor is the chain rule.

82. Karta 82

    Pytanie

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

    Odpowiedź

    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. Karta 83

    Pytanie

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

    Odpowiedź

    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. Karta 84

    Pytanie

    Derivative of $\arccos x$?

    Odpowiedź

    For $|x|<1$,

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

85. Karta 85

    Pytanie

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

    Odpowiedź

    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. Karta 86

    Pytanie

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

    Odpowiedź

    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. Karta 87

    Pytanie

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

    Odpowiedź

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

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

88. Karta 88

    Pytanie

    How are tangent slopes of inverse graphs related?

    Odpowiedź

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

89. Karta 89

    Pytanie

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

    Odpowiedź

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

90. Karta 90

    Pytanie

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

    Odpowiedź

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

91. Karta 91

    Pytanie

    Horizontal tangent on an implicit curve: derivative condition?

    Odpowiedź

    $dy/dx=0$ 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. Karta 92

    Pytanie

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

    Odpowiedź

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

93. Karta 93

    Pytanie

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

    Odpowiedź

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

94. Karta 94

    Pytanie

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

    Odpowiedź

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

    This combines the power rule with the chain rule.

95. Karta 95

    Pytanie

    Vertical tangent on an implicit curve: derivative clue?

    Odpowiedź

    $dy/dx$ 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. Karta 96

    Pytanie

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

    Odpowiedź

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

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

97. Karta 97

    Pytanie

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

    Odpowiedź

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

98. Karta 98

    Pytanie

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

    Odpowiedź

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

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

99. Karta 99

    Pytanie

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

    Odpowiedź

    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. Karta 100

     Pytanie

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

     Odpowiedź

     When the functions are differentiable, the chain rule gives

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

101. Karta 101

     Pytanie

     What local property lets a function have an inverse derivative?

     Odpowiedź

     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. Karta 102

     Pytanie

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

     Odpowiedź

     For $a>0$,

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

103. Karta 103

     Pytanie

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

     Odpowiedź

     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. Karta 104

     Pytanie

     Position, velocity, and acceleration relationships?

     Odpowiedź

     For position $s(t)$,

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

105. Karta 105

     Pytanie

     Central idea of a related-rates problem?

     Odpowiedź

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

106. Karta 106

     Pytanie

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

     Odpowiedź

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

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

107. Karta 107

     Pytanie

     L’Hospital’s Rule: basic conditions?

     Odpowiedź

     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. Karta 108

     Pytanie

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

     Odpowiedź

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

109. Karta 109

     Pytanie

     Speed in terms of velocity?

     Odpowiedź

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Karta 110

     Pytanie

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

     Odpowiedź

     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. Karta 111

     Pytanie

     Differential approximation connecting $dx$ and $dy$?

     Odpowiedź

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

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

112. Karta 112

     Pytanie

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

     Odpowiedź

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

113. Karta 113

     Pytanie

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

     Odpowiedź

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

114. Karta 114

     Pytanie

     What does positive acceleration say about velocity?

     Odpowiedź

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

115. Karta 115

     Pytanie

     Related rates: when should numerical values be substituted?

     Odpowiedź

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

116. Karta 116

     Pytanie

     How does concavity predict linearization error?

     Odpowiedź

     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. Karta 117

     Pytanie

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

     Odpowiedź

     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. Karta 118

     Pytanie

     When is a particle moving in the positive direction?

     Odpowiedź

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

119. Karta 119

     Pytanie

     How can velocity show a change of direction?

     Odpowiedź

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

120. Karta 120

     Pytanie

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

     Odpowiedź

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

121. Karta 121

     Pytanie

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

     Odpowiedź

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

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

122. Karta 122

     Pytanie

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

     Odpowiedź

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

123. Karta 123

     Pytanie

     What must a contextual derivative sentence include?

     Odpowiedź

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

124. Karta 124

     Pytanie

     Velocity negative and acceleration positive: what happens?

     Odpowiedź

     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. Karta 125

     Pytanie

     How should a negative related rate be interpreted?

     Odpowiedź

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

126. Karta 126

     Pytanie

     When is local linearity a sound approximation tool?

     Odpowiedź

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

127. Karta 127

     Pytanie

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

     Odpowiedź

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

128. Karta 128

     Pytanie

     When is speed increasing?

     Odpowiedź

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

129. Karta 129

     Pytanie

     Volume changes with time: notation for its rate?

     Odpowiedź

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

130. Karta 130

     Pytanie

     Why are similar triangles useful in related rates?

     Odpowiedź

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

131. Karta 131

     Pytanie

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

     Odpowiedź

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

132. Karta 132

     Pytanie

     What conclusion does L’Hospital’s Rule permit?

     Odpowiedź

     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. Karta 133

     Pytanie

     When is speed decreasing?

     Odpowiedź

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

134. Karta 134

     Pytanie

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

     Odpowiedź

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

135. Karta 135

     Pytanie

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

     Odpowiedź

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

136. Karta 136

     Pytanie

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

     Odpowiedź

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

137. Karta 137

     Pytanie

     Extreme Value Theorem: hypothesis and conclusion?

     Odpowiedź

     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. Karta 138

     Pytanie

     What is a critical number of $f$?

     Odpowiedź

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

139. Karta 139

     Pytanie

     First derivative test for a local maximum?

     Odpowiedź

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

140. Karta 140

     Pytanie

     Second-derivative sign for concave up?

     Odpowiedź

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

141. Karta 141

     Pytanie

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

     Odpowiedź

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

142. Karta 142

     Pytanie

     First step in an optimization model?

     Odpowiedź

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

143. Karta 143

     Pytanie

     Mean Value Theorem: hypotheses and conclusion?

     Odpowiedź

     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. Karta 144

     Pytanie

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

     Odpowiedź

     Assuming $f$ is continuous on $[a,b]$, evaluate $f$ at every critical number in $(a,b)$ and at both endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.

     

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145. Karta 145

     Pytanie

     First derivative test for a local minimum?

     Odpowiedź

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

146. Karta 146

     Pytanie

     What must happen at an inflection point?

     Odpowiedź

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

147. Karta 147

     Pytanie

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

     Odpowiedź

     $f''$ 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. Karta 148

     Pytanie

     How do you confirm an optimization answer is absolute?

     Odpowiedź

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

149. Karta 149

     Pytanie

     Rolle’s Theorem: hypotheses and conclusion?

     Odpowiedź

     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. Karta 150

     Pytanie

     Difference between absolute and relative extrema?

     Odpowiedź

     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. Karta 151

     Pytanie

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

     Odpowiedź

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

152. Karta 152

     Pytanie

     Second derivative test for a local minimum?

     Odpowiedź

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

153. Karta 153

     Pytanie

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

     Odpowiedź

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

154. Karta 154

     Pytanie

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

     Odpowiedź

     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. Karta 155

     Pytanie

     Which theorem links an average slope to an instantaneous slope?

     Odpowiedź

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

156. Karta 156

     Pytanie

     How can an implicit derivative locate a horizontal tangent?

     Odpowiedź

     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. Karta 157

     Pytanie

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

     Odpowiedź

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

158. Karta 158

     Pytanie

     Second derivative test for a local maximum?

     Odpowiedź

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

159. Karta 159

     Pytanie

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

     Odpowiedź

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

160. Karta 160

     Pytanie

     Why must an optimization domain be stated?

     Odpowiedź

     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. Karta 161

     Pytanie

     Which theorem guarantees absolute extrema, not where they occur?

     Odpowiedź

     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. Karta 162

     Pytanie

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

     Odpowiedź

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

163. Karta 163

     Pytanie

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

     Odpowiedź

     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. Karta 164

     Pytanie

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

     Odpowiedź

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

165. Karta 165

     Pytanie

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

     Odpowiedź

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

166. Karta 166

     Pytanie

     How can an implicit derivative locate a vertical tangent?

     Odpowiedź

     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. Karta 167

     Pytanie

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

     Odpowiedź

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

168. Karta 168

     Pytanie

     Why are endpoints included in the candidates test?

     Odpowiedź

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

169. Karta 169

     Pytanie

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

     Odpowiedź

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

170. Karta 170

     Pytanie

     Second-derivative sign for concave down?

     Odpowiedź

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

171. Karta 171

     Pytanie

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

     Odpowiedź

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

172. Karta 172

     Pytanie

     What should the final line of an optimization solution state?

     Odpowiedź

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

173. Karta 173

     Pytanie

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

     Odpowiedź

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

174. Karta 174

     Pytanie

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

     Odpowiedź

     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. Karta 175

     Pytanie

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

     Odpowiedź

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

176. Karta 176

     Pytanie

     Left Riemann sum on equal subintervals?

     Odpowiedź

     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. Karta 177

     Pytanie

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

     Odpowiedź

     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. Karta 178

     Pytanie

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Odpowiedź

     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. Karta 179

     Pytanie

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

     Odpowiedź

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Karta 180

     Pytanie

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

     Odpowiedź

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

181. Karta 181

     Pytanie

     Right Riemann sum on equal subintervals?

     Odpowiedź

     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. Karta 182

     Pytanie

     How does reversing integral bounds change the value?

     Odpowiedź

     It changes the sign:

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

183. Karta 183

     Pytanie

     Net Change Theorem?

     Odpowiedź

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

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

184. Karta 184

     Pytanie

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

     Odpowiedź

     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. Karta 185

     Pytanie

     Power rule for antiderivatives?

     Odpowiedź

     For $n\ne-1$,

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

186. Karta 186

     Pytanie

     Midpoint Riemann sum on equal subintervals?

     Odpowiedź

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

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

187. Karta 187

     Pytanie

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

     Odpowiedź

     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. Karta 188

     Pytanie

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

     Odpowiedź

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Karta 189

     Pytanie

     Antiderivative of $1/x$?

     Odpowiedź

     On any interval not crossing zero,

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

190. Karta 190

     Pytanie

     Trapezoidal approximation on equal subintervals?

     Odpowiedź

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

191. Karta 191

     Pytanie

     How do geometric regions help evaluate a definite integral?

     Odpowiedź

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

192. Karta 192

     Pytanie

     Basic antiderivatives of sine and cosine?

     Odpowiedź

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

193. Karta 193

     Pytanie

     Definite integral as a limit of Riemann sums?

     Odpowiedź

     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. Karta 194

     Pytanie

     Constant-multiple rule for integrals?

     Odpowiedź

     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. Karta 195

     Pytanie

     What pattern suggests $u$-substitution?

     Odpowiedź

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

196. Karta 196

     Pytanie

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

     Odpowiedź

     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. Karta 197

     Pytanie

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

     Odpowiedź

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

198. Karta 198

     Pytanie

     Sum-and-difference rule for definite integrals?

     Odpowiedź

     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. Karta 199

     Pytanie

     Basic antiderivative of $e^x$?

     Odpowiedź

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

200. Karta 200

     Pytanie

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

     Odpowiedź

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

201. Karta 201

     Pytanie

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

     Odpowiedź

     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. Karta 202

     Pytanie

     How does concavity predict trapezoidal and midpoint error?

     Odpowiedź

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

203. Karta 203

     Pytanie

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

     Odpowiedź

     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. Karta 204

     Pytanie

     What denominator pattern suggests an arctangent antiderivative?

     Odpowiedź

     After completing the square and scaling, a form like

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

205. Karta 205

     Pytanie

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

     Odpowiedź

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

206. Karta 206

     Pytanie

     How does an initial condition determine an antiderivative?

     Odpowiedź

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

207. Karta 207

     Pytanie

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

     Odpowiedź

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

208. Karta 208

     Pytanie

     Why does an indefinite integral include $+C$?

     Odpowiedź

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

209. Karta 209

     Pytanie

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

     Odpowiedź

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

210. Karta 210

     Pytanie

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

     Odpowiedź

     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. Karta 211

     Pytanie

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

     Odpowiedź

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

212. Karta 212

     Pytanie

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

     Odpowiedź

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

213. Karta 213

     Pytanie

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

     Odpowiedź

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

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

214. Karta 214

     Pytanie

     Riemann sum for unequal subinterval widths?

     Odpowiedź

     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. Karta 215

     Pytanie

     Does continuity guarantee integrability on a closed interval?

     Odpowiedź

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

216. Karta 216

     Pytanie

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

     Odpowiedź

     Where $g(x)\ne0$,

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

217. Karta 217

     Pytanie

     What algebraic rewrites often reveal a basic antiderivative?

     Odpowiedź

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

218. Karta 218

     Pytanie

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

     Odpowiedź

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

219. Karta 219

     Pytanie

     What is a differential equation?

     Odpowiedź

     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. Karta 220

     Pytanie

     How does a verbal rate statement become a differential equation?

     Odpowiedź

     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. Karta 221

     Pytanie

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

     Odpowiedź

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

222. Karta 222

     Pytanie

     General solution versus particular solution?

     Odpowiedź

     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. Karta 223

     Pytanie

     What does one segment in a slope field show?

     Odpowiedź

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

224. Karta 224

     Pytanie

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

     Odpowiedź

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

225. Karta 225

     Pytanie

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

     Odpowiedź

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

226. Karta 226

     Pytanie

     What makes a first-order differential equation separable?

     Odpowiedź

     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. Karta 227

     Pytanie

     What is an initial value problem?

     Odpowiedź

     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. Karta 228

     Pytanie

     What is an isocline in a slope field?

     Odpowiedź

     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. Karta 229

     Pytanie

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

     Odpowiedź

     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. Karta 230

     Pytanie

     General solution of $dy/dt=ky$?

     Odpowiedź

     $$y=Ce^{kt}$$

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

231. Karta 231

     Pytanie

     Core method for solving a separable differential equation?

     Odpowiedź

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

232. Karta 232

     Pytanie

     How should a solution curve follow a slope field?

     Odpowiedź

     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. Karta 233

     Pytanie

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

     Odpowiedź

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

234. Karta 234

     Pytanie

     Why is one integration constant enough after integrating both sides?

     Odpowiedź

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

235. Karta 235

     Pytanie

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

     Odpowiedź

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

236. Karta 236

     Pytanie

     Can one differential equation have infinitely many solutions?

     Odpowiedź

     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. Karta 237

     Pytanie

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

     Odpowiedź

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

238. Karta 238

     Pytanie

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

     Odpowiedź

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

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

239. Karta 239

     Pytanie

     What can be lost when dividing to separate variables?

     Odpowiedź

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

240. Karta 240

     Pytanie

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

     Odpowiedź

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

241. Karta 241

     Pytanie

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

     Odpowiedź

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

242. Karta 242

     Pytanie

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

     Odpowiedź

     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. Karta 243

     Pytanie

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

     Odpowiedź

     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. Karta 244

     Pytanie

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

     Odpowiedź

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

245. Karta 245

     Pytanie

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

     Odpowiedź

     For $k>0$,

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

     It is independent of the initial amount.

246. Karta 246

     Pytanie

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

     Odpowiedź

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

247. Karta 247

     Pytanie

     How is an initial condition used after separation?

     Odpowiedź

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

248. Karta 248

     Pytanie

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

     Odpowiedź

     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. Karta 249

     Pytanie

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

     Odpowiedź

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

250. Karta 250

     Pytanie

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

     Odpowiedź

     For $k<0$,

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

251. Karta 251

     Pytanie

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

     Odpowiedź

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

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

252. Karta 252

     Pytanie

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

     Odpowiedź

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

     Velocity below zero contributes negative displacement.

253. Karta 253

     Pytanie

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

     Odpowiedź

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

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

     Think top minus bottom.

254. Karta 254

     Pytanie

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

     Odpowiedź

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

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

255. Karta 255

     Pytanie

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Odpowiedź

     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. Karta 256

     Pytanie

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

     Odpowiedź

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

257. Karta 257

     Pytanie

     Cross-sectional area when each slice is a square?

     Odpowiedź

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

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

258. Karta 258

     Pytanie

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

     Odpowiedź

     Integrate the net rate:

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

259. Karta 259

     Pytanie

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

     Odpowiedź

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

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

     Think right minus left.

260. Karta 260

     Pytanie

     Disc-method volume formula?

     Odpowiedź

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

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

261. Karta 261

     Pytanie

     What units does average value have?

     Odpowiedź

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

262. Karta 262

     Pytanie

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

     Odpowiedź

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

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

263. Karta 263

     Pytanie

     Cross-sectional area when each slice is a rectangle?

     Odpowiedź

     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. Karta 264

     Pytanie

     How do you determine bounds for area between curves?

     Odpowiedź

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

265. Karta 265

     Pytanie

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

     Odpowiedź

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

266. Karta 266

     Pytanie

     When is a particle moving to the right or left?

     Odpowiedź

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

267. Karta 267

     Pytanie

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

     Odpowiedź

     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. Karta 268

     Pytanie

     Why must an area integral be split where curves intersect?

     Odpowiedź

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

269. Karta 269

     Pytanie

     How can a velocity table approximate displacement?

     Odpowiedź

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

270. Karta 270

     Pytanie

     Washer-method volume formula?

     Odpowiedź

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

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

271. Karta 271

     Pytanie

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

     Odpowiedź

     If $s(a)$ is known,

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

272. Karta 272

     Pytanie

     How do you choose between vertical and horizontal area slices?

     Odpowiedź

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

273. Karta 273

     Pytanie

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

     Odpowiedź

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

274. Karta 274

     Pytanie

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

     Odpowiedź

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

275. Karta 275

     Pytanie

     Single expression for area between two curves?

     Odpowiedź

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Karta 276

     Pytanie

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

     Odpowiedź

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

277. Karta 277

     Pytanie

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

     Odpowiedź

     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. Karta 278

     Pytanie

     When should a volume integral use $dy$?

     Odpowiedź

     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. Karta 279

     Pytanie

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

     Odpowiedź

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

280. Karta 280

     Pytanie

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

     Odpowiedź

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

281. Karta 281

     Pytanie

     Why must total distance split at velocity sign changes?

     Odpowiedź

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

282. Karta 282

     Pytanie

     When does an accumulated quantity reach a local maximum?

     Odpowiedź

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

283. Karta 283

     Pytanie

     What distinguishes area from a definite integral?

     Odpowiedź

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

284. Karta 284

     Pytanie

     How do position, velocity, and acceleration graphs correspond?

     Odpowiedź

     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. Karta 285

     Pytanie

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

     Odpowiedź

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

286. Karta 286

     Pytanie

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

     Odpowiedź

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

287. Karta 287

     Pytanie

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

     Odpowiedź

     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. Karta 288

     Pytanie

     Why should a contextual integral answer include a sentence?

     Odpowiedź

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