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

Kartičky v tomto balíčku

1. Kartička 1

   Otázka

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

   Odpoveď

   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. Kartička 2

   Otázka

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

   Odpoveď

   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. Kartička 3

   Otázka

   When does direct substitution evaluate a limit?

   Odpoveď

   When the function is continuous at the target input. Then

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

4. Kartička 4

   Otázka

   Three conditions for continuity at $x=a$?

   Odpoveď

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

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

5. Kartička 5

   Otázka

   Intermediate Value Theorem: hypotheses and conclusion?

   Odpoveď

   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. Kartička 6

   Otázka

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

   Odpoveď

   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. Kartička 7

   Otázka

   How do you read a finite limit from a graph?

   Odpoveď

   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. Kartička 8

   Otázka

   Limit law for a sum or difference?

   Odpoveď

   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. Kartička 9

   Otázka

   What makes a discontinuity removable?

   Odpoveď

   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. Kartička 10

    Otázka

    Squeeze Theorem: usable form?

    Odpoveď

    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. Kartička 11

    Otázka

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

    Odpoveď

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

12. Kartička 12

    Otázka

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

    Odpoveď

    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. Kartička 13

    Otázka

    Limit law for a product?

    Odpoveď

    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. Kartička 14

    Otázka

    Graph signature of a jump discontinuity?

    Odpoveď

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

15. Kartička 15

    Otázka

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

    Odpoveď

    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. Kartička 16

    Otázka

    Horizontal asymptote from a limit at infinity?

    Odpoveď

    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. Kartička 17

    Otázka

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

    Odpoveď

    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. Kartička 18

    Otázka

    Limit law for a quotient—and its condition?

    Odpoveď

    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. Kartička 19

    Otázka

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

    Odpoveď

    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. Kartička 20

    Otázka

    When is the Squeeze Theorem a natural choice?

    Odpoveď

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

21. Kartička 21

    Otázka

    Vertical asymptote from one-sided behavior?

    Odpoveď

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

22. Kartička 22

    Otázka

    Limit at infinity of equal-degree rational functions?

    Odpoveď

    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. Kartička 23

    Otázka

    When can a limit pass through a continuous outer function?

    Odpoveď

    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. Kartička 24

    Otázka

    What makes a discontinuity infinite?

    Odpoveď

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

25. Kartička 25

    Otázka

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

    Odpoveď

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

26. Kartička 26

    Otázka

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

    Odpoveď

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kartička 27

    Otázka

    Continuity of a composition?

    Odpoveď

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

28. Kartička 28

    Otázka

    Limit at infinity when a rational numerator has lower degree?

    Odpoveď

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

29. Kartička 29

    Otázka

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

    Odpoveď

    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. Kartička 30

    Otázka

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

    Odpoveď

    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. Kartička 31

    Otázka

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

    Odpoveď

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

32. Kartička 32

    Otázka

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

    Odpoveď

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

33. Kartička 33

    Otázka

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

    Odpoveď

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

34. Kartička 34

    Otázka

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

    Odpoveď

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

35. Kartička 35

    Otázka

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

    Odpoveď

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

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

36. Kartička 36

    Otázka

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

    Odpoveď

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

    The derivative exists only if this finite limit exists.

37. Kartička 37

    Otázka

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

    Odpoveď

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

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

38. Kartička 38

    Otázka

    What does differentiability imply about continuity?

    Odpoveď

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

39. Kartička 39

    Otázka

    Power rule for derivatives?

    Odpoveď

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

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

40. Kartička 40

    Otázka

    Units of $f'(x)$?

    Odpoveď

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

41. Kartička 41

    Otázka

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

    Odpoveď

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

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

42. Kartička 42

    Otázka

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

    Odpoveď

    $f'(x)>0$ 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. Kartička 43

    Otázka

    Derivative of a constant?

    Odpoveď

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

    A constant function has zero rate of change.

44. Kartička 44

    Otázka

    Derivative of $\sin x$?

    Odpoveď

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

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

45. Kartička 45

    Otázka

    Product rule?

    Odpoveď

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

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

46. Kartička 46

    Otázka

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

    Odpoveď

    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. Kartička 47

    Otázka

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

    Odpoveď

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

48. Kartička 48

    Otázka

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

    Odpoveď

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

49. Kartička 49

    Otázka

    Derivative of a sum or difference?

    Odpoveď

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

50. Kartička 50

    Otázka

    Derivative of $\cos x$?

    Odpoveď

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

    The standard formula assumes radians.

51. Kartička 51

    Otázka

    Quotient rule?

    Odpoveď

    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. Kartička 52

    Otázka

    Common notations for the first derivative?

    Odpoveď

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

53. Kartička 53

    Otázka

    Derivative of $e^x$?

    Odpoveď

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

54. Kartička 54

    Otázka

    What graph features can make $f$ nondifferentiable?

    Odpoveď

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

55. Kartička 55

    Otázka

    Derivative of $\tan x$?

    Odpoveď

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kartička 56

    Otázka

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

    Odpoveď

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

57. Kartička 57

    Otázka

    Derivative of $\ln x$?

    Odpoveď

    For $x>0$,

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

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

58. Kartička 58

    Otázka

    How does the power rule handle roots or negative powers?

    Odpoveď

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

59. Kartička 59

    Otázka

    Derivative of $\csc x$?

    Odpoveď

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kartička 60

    Otázka

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

    Odpoveď

    $f$ is increasing on that interval.

61. Kartička 61

    Otázka

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

    Odpoveď

    For $a>0$,

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

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

62. Kartička 62

    Otázka

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

    Odpoveď

    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. Kartička 63

    Otázka

    Derivative of $\sec x$?

    Odpoveď

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kartička 64

    Otázka

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

    Odpoveď

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

65. Kartička 65

    Otázka

    Derivative of $\log_a x$?

    Odpoveď

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

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

66. Kartička 66

    Otázka

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

    Odpoveď

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kartička 67

    Otázka

    Derivative of $\cot x$?

    Odpoveď

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kartička 68

    Otázka

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

    Odpoveď

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

69. Kartička 69

    Otázka

    Constant-multiple rule?

    Odpoveď

    For a constant $c$,

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

70. Kartička 70

    Otázka

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

    Odpoveď

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

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

71. Kartička 71

    Otázka

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

    Odpoveď

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

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

72. Kartička 72

    Otázka

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

    Odpoveď

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

73. Kartička 73

    Otázka

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

    Odpoveď

    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. Kartička 74

    Otázka

    Derivative of an inverse function at $x$?

    Odpoveď

    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. Kartička 75

    Otázka

    Derivative of $\arcsin x$?

    Odpoveď

    For $|x|<1$,

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

76. Kartička 76

    Otázka

    Notation for the third derivative of $f$?

    Odpoveď

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

77. Kartička 77

    Otázka

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

    Odpoveď

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

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

78. Kartička 78

    Otázka

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

    Odpoveď

    Where $y\ne0$,

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

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

79. Kartička 79

    Otázka

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

    Odpoveď

    Provided $f'(a)\ne0$,

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

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

80. Kartička 80

    Otázka

    Derivative of $\arctan x$?

    Odpoveď

    For every real $x$,

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

81. Kartička 81

    Otázka

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

    Odpoveď

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

    The extra factor is the chain rule.

82. Kartička 82

    Otázka

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

    Odpoveď

    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. Kartička 83

    Otázka

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

    Odpoveď

    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. Kartička 84

    Otázka

    Derivative of $\arccos x$?

    Odpoveď

    For $|x|<1$,

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

85. Kartička 85

    Otázka

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

    Odpoveď

    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. Kartička 86

    Otázka

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

    Odpoveď

    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. Kartička 87

    Otázka

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

    Odpoveď

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

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

88. Kartička 88

    Otázka

    How are tangent slopes of inverse graphs related?

    Odpoveď

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

89. Kartička 89

    Otázka

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

    Odpoveď

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

90. Kartička 90

    Otázka

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

    Odpoveď

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

91. Kartička 91

    Otázka

    Horizontal tangent on an implicit curve: derivative condition?

    Odpoveď

    $dy/dx=0$ 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. Kartička 92

    Otázka

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

    Odpoveď

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

93. Kartička 93

    Otázka

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

    Odpoveď

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

94. Kartička 94

    Otázka

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

    Odpoveď

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

    This combines the power rule with the chain rule.

95. Kartička 95

    Otázka

    Vertical tangent on an implicit curve: derivative clue?

    Odpoveď

    $dy/dx$ 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. Kartička 96

    Otázka

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

    Odpoveď

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

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

97. Kartička 97

    Otázka

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

    Odpoveď

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

98. Kartička 98

    Otázka

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

    Odpoveď

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

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

99. Kartička 99

    Otázka

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

    Odpoveď

    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. Kartička 100

     Otázka

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

     Odpoveď

     When the functions are differentiable, the chain rule gives

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

101. Kartička 101

     Otázka

     What local property lets a function have an inverse derivative?

     Odpoveď

     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. Kartička 102

     Otázka

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

     Odpoveď

     For $a>0$,

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

103. Kartička 103

     Otázka

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

     Odpoveď

     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. Kartička 104

     Otázka

     Position, velocity, and acceleration relationships?

     Odpoveď

     For position $s(t)$,

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

105. Kartička 105

     Otázka

     Central idea of a related-rates problem?

     Odpoveď

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

106. Kartička 106

     Otázka

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

     Odpoveď

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

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

107. Kartička 107

     Otázka

     L’Hospital’s Rule: basic conditions?

     Odpoveď

     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. Kartička 108

     Otázka

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

     Odpoveď

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

109. Kartička 109

     Otázka

     Speed in terms of velocity?

     Odpoveď

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kartička 110

     Otázka

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

     Odpoveď

     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. Kartička 111

     Otázka

     Differential approximation connecting $dx$ and $dy$?

     Odpoveď

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

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

112. Kartička 112

     Otázka

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

     Odpoveď

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

113. Kartička 113

     Otázka

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

     Odpoveď

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

114. Kartička 114

     Otázka

     What does positive acceleration say about velocity?

     Odpoveď

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

115. Kartička 115

     Otázka

     Related rates: when should numerical values be substituted?

     Odpoveď

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

116. Kartička 116

     Otázka

     How does concavity predict linearization error?

     Odpoveď

     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. Kartička 117

     Otázka

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

     Odpoveď

     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. Kartička 118

     Otázka

     When is a particle moving in the positive direction?

     Odpoveď

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

119. Kartička 119

     Otázka

     How can velocity show a change of direction?

     Odpoveď

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

120. Kartička 120

     Otázka

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

     Odpoveď

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

121. Kartička 121

     Otázka

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

     Odpoveď

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

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

122. Kartička 122

     Otázka

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

     Odpoveď

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

123. Kartička 123

     Otázka

     What must a contextual derivative sentence include?

     Odpoveď

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

124. Kartička 124

     Otázka

     Velocity negative and acceleration positive: what happens?

     Odpoveď

     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. Kartička 125

     Otázka

     How should a negative related rate be interpreted?

     Odpoveď

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

126. Kartička 126

     Otázka

     When is local linearity a sound approximation tool?

     Odpoveď

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

127. Kartička 127

     Otázka

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

     Odpoveď

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

128. Kartička 128

     Otázka

     When is speed increasing?

     Odpoveď

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

129. Kartička 129

     Otázka

     Volume changes with time: notation for its rate?

     Odpoveď

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

130. Kartička 130

     Otázka

     Why are similar triangles useful in related rates?

     Odpoveď

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

131. Kartička 131

     Otázka

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

     Odpoveď

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

132. Kartička 132

     Otázka

     What conclusion does L’Hospital’s Rule permit?

     Odpoveď

     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. Kartička 133

     Otázka

     When is speed decreasing?

     Odpoveď

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

134. Kartička 134

     Otázka

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

     Odpoveď

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

135. Kartička 135

     Otázka

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

     Odpoveď

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

136. Kartička 136

     Otázka

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

     Odpoveď

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

137. Kartička 137

     Otázka

     Extreme Value Theorem: hypothesis and conclusion?

     Odpoveď

     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. Kartička 138

     Otázka

     What is a critical number of $f$?

     Odpoveď

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

139. Kartička 139

     Otázka

     First derivative test for a local maximum?

     Odpoveď

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

140. Kartička 140

     Otázka

     Second-derivative sign for concave up?

     Odpoveď

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

141. Kartička 141

     Otázka

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

     Odpoveď

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

142. Kartička 142

     Otázka

     First step in an optimization model?

     Odpoveď

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

143. Kartička 143

     Otázka

     Mean Value Theorem: hypotheses and conclusion?

     Odpoveď

     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. Kartička 144

     Otázka

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

     Odpoveď

     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. Kartička 145

     Otázka

     First derivative test for a local minimum?

     Odpoveď

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

146. Kartička 146

     Otázka

     What must happen at an inflection point?

     Odpoveď

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

147. Kartička 147

     Otázka

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

     Odpoveď

     $f''$ 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. Kartička 148

     Otázka

     How do you confirm an optimization answer is absolute?

     Odpoveď

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

149. Kartička 149

     Otázka

     Rolle’s Theorem: hypotheses and conclusion?

     Odpoveď

     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. Kartička 150

     Otázka

     Difference between absolute and relative extrema?

     Odpoveď

     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. Kartička 151

     Otázka

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

     Odpoveď

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

152. Kartička 152

     Otázka

     Second derivative test for a local minimum?

     Odpoveď

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

153. Kartička 153

     Otázka

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

     Odpoveď

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

154. Kartička 154

     Otázka

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

     Odpoveď

     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. Kartička 155

     Otázka

     Which theorem links an average slope to an instantaneous slope?

     Odpoveď

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

156. Kartička 156

     Otázka

     How can an implicit derivative locate a horizontal tangent?

     Odpoveď

     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. Kartička 157

     Otázka

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

     Odpoveď

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

158. Kartička 158

     Otázka

     Second derivative test for a local maximum?

     Odpoveď

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

159. Kartička 159

     Otázka

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

     Odpoveď

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

160. Kartička 160

     Otázka

     Why must an optimization domain be stated?

     Odpoveď

     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. Kartička 161

     Otázka

     Which theorem guarantees absolute extrema, not where they occur?

     Odpoveď

     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. Kartička 162

     Otázka

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

     Odpoveď

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

163. Kartička 163

     Otázka

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

     Odpoveď

     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. Kartička 164

     Otázka

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

     Odpoveď

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

165. Kartička 165

     Otázka

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

     Odpoveď

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

166. Kartička 166

     Otázka

     How can an implicit derivative locate a vertical tangent?

     Odpoveď

     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. Kartička 167

     Otázka

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

     Odpoveď

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

168. Kartička 168

     Otázka

     Why are endpoints included in the candidates test?

     Odpoveď

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

169. Kartička 169

     Otázka

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

     Odpoveď

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

170. Kartička 170

     Otázka

     Second-derivative sign for concave down?

     Odpoveď

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

171. Kartička 171

     Otázka

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

     Odpoveď

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

172. Kartička 172

     Otázka

     What should the final line of an optimization solution state?

     Odpoveď

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

173. Kartička 173

     Otázka

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

     Odpoveď

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

174. Kartička 174

     Otázka

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

     Odpoveď

     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. Kartička 175

     Otázka

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

     Odpoveď

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

176. Kartička 176

     Otázka

     Left Riemann sum on equal subintervals?

     Odpoveď

     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. Kartička 177

     Otázka

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

     Odpoveď

     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. Kartička 178

     Otázka

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Odpoveď

     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. Kartička 179

     Otázka

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

     Odpoveď

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kartička 180

     Otázka

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

     Odpoveď

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

181. Kartička 181

     Otázka

     Right Riemann sum on equal subintervals?

     Odpoveď

     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. Kartička 182

     Otázka

     How does reversing integral bounds change the value?

     Odpoveď

     It changes the sign:

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

183. Kartička 183

     Otázka

     Net Change Theorem?

     Odpoveď

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

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

184. Kartička 184

     Otázka

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

     Odpoveď

     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. Kartička 185

     Otázka

     Power rule for antiderivatives?

     Odpoveď

     For $n\ne-1$,

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

186. Kartička 186

     Otázka

     Midpoint Riemann sum on equal subintervals?

     Odpoveď

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

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

187. Kartička 187

     Otázka

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

     Odpoveď

     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. Kartička 188

     Otázka

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

     Odpoveď

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kartička 189

     Otázka

     Antiderivative of $1/x$?

     Odpoveď

     On any interval not crossing zero,

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

190. Kartička 190

     Otázka

     Trapezoidal approximation on equal subintervals?

     Odpoveď

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

191. Kartička 191

     Otázka

     How do geometric regions help evaluate a definite integral?

     Odpoveď

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

192. Kartička 192

     Otázka

     Basic antiderivatives of sine and cosine?

     Odpoveď

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

193. Kartička 193

     Otázka

     Definite integral as a limit of Riemann sums?

     Odpoveď

     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. Kartička 194

     Otázka

     Constant-multiple rule for integrals?

     Odpoveď

     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. Kartička 195

     Otázka

     What pattern suggests $u$-substitution?

     Odpoveď

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

196. Kartička 196

     Otázka

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

     Odpoveď

     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. Kartička 197

     Otázka

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

     Odpoveď

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

198. Kartička 198

     Otázka

     Sum-and-difference rule for definite integrals?

     Odpoveď

     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. Kartička 199

     Otázka

     Basic antiderivative of $e^x$?

     Odpoveď

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

200. Kartička 200

     Otázka

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

     Odpoveď

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

201. Kartička 201

     Otázka

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

     Odpoveď

     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. Kartička 202

     Otázka

     How does concavity predict trapezoidal and midpoint error?

     Odpoveď

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

203. Kartička 203

     Otázka

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

     Odpoveď

     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. Kartička 204

     Otázka

     What denominator pattern suggests an arctangent antiderivative?

     Odpoveď

     After completing the square and scaling, a form like

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

205. Kartička 205

     Otázka

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

     Odpoveď

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

206. Kartička 206

     Otázka

     How does an initial condition determine an antiderivative?

     Odpoveď

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

207. Kartička 207

     Otázka

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

     Odpoveď

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

208. Kartička 208

     Otázka

     Why does an indefinite integral include $+C$?

     Odpoveď

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

209. Kartička 209

     Otázka

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

     Odpoveď

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

210. Kartička 210

     Otázka

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

     Odpoveď

     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. Kartička 211

     Otázka

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

     Odpoveď

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

212. Kartička 212

     Otázka

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

     Odpoveď

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

213. Kartička 213

     Otázka

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

     Odpoveď

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

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

214. Kartička 214

     Otázka

     Riemann sum for unequal subinterval widths?

     Odpoveď

     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. Kartička 215

     Otázka

     Does continuity guarantee integrability on a closed interval?

     Odpoveď

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

216. Kartička 216

     Otázka

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

     Odpoveď

     Where $g(x)\ne0$,

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

217. Kartička 217

     Otázka

     What algebraic rewrites often reveal a basic antiderivative?

     Odpoveď

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

218. Kartička 218

     Otázka

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

     Odpoveď

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

219. Kartička 219

     Otázka

     What is a differential equation?

     Odpoveď

     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. Kartička 220

     Otázka

     How does a verbal rate statement become a differential equation?

     Odpoveď

     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. Kartička 221

     Otázka

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

     Odpoveď

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

222. Kartička 222

     Otázka

     General solution versus particular solution?

     Odpoveď

     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. Kartička 223

     Otázka

     What does one segment in a slope field show?

     Odpoveď

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

224. Kartička 224

     Otázka

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

     Odpoveď

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

225. Kartička 225

     Otázka

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

     Odpoveď

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

226. Kartička 226

     Otázka

     What makes a first-order differential equation separable?

     Odpoveď

     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. Kartička 227

     Otázka

     What is an initial value problem?

     Odpoveď

     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. Kartička 228

     Otázka

     What is an isocline in a slope field?

     Odpoveď

     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. Kartička 229

     Otázka

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

     Odpoveď

     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. Kartička 230

     Otázka

     General solution of $dy/dt=ky$?

     Odpoveď

     $$y=Ce^{kt}$$

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

231. Kartička 231

     Otázka

     Core method for solving a separable differential equation?

     Odpoveď

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

232. Kartička 232

     Otázka

     How should a solution curve follow a slope field?

     Odpoveď

     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. Kartička 233

     Otázka

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

     Odpoveď

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

234. Kartička 234

     Otázka

     Why is one integration constant enough after integrating both sides?

     Odpoveď

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

235. Kartička 235

     Otázka

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

     Odpoveď

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

236. Kartička 236

     Otázka

     Can one differential equation have infinitely many solutions?

     Odpoveď

     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. Kartička 237

     Otázka

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

     Odpoveď

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

238. Kartička 238

     Otázka

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

     Odpoveď

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

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

239. Kartička 239

     Otázka

     What can be lost when dividing to separate variables?

     Odpoveď

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

240. Kartička 240

     Otázka

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

     Odpoveď

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

241. Kartička 241

     Otázka

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

     Odpoveď

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

242. Kartička 242

     Otázka

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

     Odpoveď

     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. Kartička 243

     Otázka

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

     Odpoveď

     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. Kartička 244

     Otázka

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

     Odpoveď

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

245. Kartička 245

     Otázka

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

     Odpoveď

     For $k>0$,

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

     It is independent of the initial amount.

246. Kartička 246

     Otázka

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

     Odpoveď

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

247. Kartička 247

     Otázka

     How is an initial condition used after separation?

     Odpoveď

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

248. Kartička 248

     Otázka

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

     Odpoveď

     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. Kartička 249

     Otázka

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

     Odpoveď

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

250. Kartička 250

     Otázka

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

     Odpoveď

     For $k<0$,

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

251. Kartička 251

     Otázka

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

     Odpoveď

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

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

252. Kartička 252

     Otázka

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

     Odpoveď

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

     Velocity below zero contributes negative displacement.

253. Kartička 253

     Otázka

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

     Odpoveď

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

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

     Think top minus bottom.

254. Kartička 254

     Otázka

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

     Odpoveď

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

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

255. Kartička 255

     Otázka

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Odpoveď

     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. Kartička 256

     Otázka

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

     Odpoveď

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

257. Kartička 257

     Otázka

     Cross-sectional area when each slice is a square?

     Odpoveď

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

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

258. Kartička 258

     Otázka

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

     Odpoveď

     Integrate the net rate:

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

259. Kartička 259

     Otázka

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

     Odpoveď

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

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

     Think right minus left.

260. Kartička 260

     Otázka

     Disc-method volume formula?

     Odpoveď

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

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

261. Kartička 261

     Otázka

     What units does average value have?

     Odpoveď

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

262. Kartička 262

     Otázka

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

     Odpoveď

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

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

263. Kartička 263

     Otázka

     Cross-sectional area when each slice is a rectangle?

     Odpoveď

     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. Kartička 264

     Otázka

     How do you determine bounds for area between curves?

     Odpoveď

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

265. Kartička 265

     Otázka

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

     Odpoveď

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

266. Kartička 266

     Otázka

     When is a particle moving to the right or left?

     Odpoveď

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

267. Kartička 267

     Otázka

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

     Odpoveď

     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. Kartička 268

     Otázka

     Why must an area integral be split where curves intersect?

     Odpoveď

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

269. Kartička 269

     Otázka

     How can a velocity table approximate displacement?

     Odpoveď

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

270. Kartička 270

     Otázka

     Washer-method volume formula?

     Odpoveď

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

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

271. Kartička 271

     Otázka

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

     Odpoveď

     If $s(a)$ is known,

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

272. Kartička 272

     Otázka

     How do you choose between vertical and horizontal area slices?

     Odpoveď

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

273. Kartička 273

     Otázka

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

     Odpoveď

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

274. Kartička 274

     Otázka

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

     Odpoveď

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

275. Kartička 275

     Otázka

     Single expression for area between two curves?

     Odpoveď

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kartička 276

     Otázka

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

     Odpoveď

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

277. Kartička 277

     Otázka

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

     Odpoveď

     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. Kartička 278

     Otázka

     When should a volume integral use $dy$?

     Odpoveď

     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. Kartička 279

     Otázka

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

     Odpoveď

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

280. Kartička 280

     Otázka

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

     Odpoveď

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

281. Kartička 281

     Otázka

     Why must total distance split at velocity sign changes?

     Odpoveď

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

282. Kartička 282

     Otázka

     When does an accumulated quantity reach a local maximum?

     Odpoveď

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

283. Kartička 283

     Otázka

     What distinguishes area from a definite integral?

     Odpoveď

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

284. Kartička 284

     Otázka

     How do position, velocity, and acceleration graphs correspond?

     Odpoveď

     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. Kartička 285

     Otázka

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

     Odpoveď

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

286. Kartička 286

     Otázka

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

     Odpoveď

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

287. Kartička 287

     Otázka

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

     Odpoveď

     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. Kartička 288

     Otázka

     Why should a contextual integral answer include a sentence?

     Odpoveď

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