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

Kartice u ovom špilu

1. Kartica 1

   Pitanje

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

   Odgovor

   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. Kartica 2

   Pitanje

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

   Odgovor

   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. Kartica 3

   Pitanje

   When does direct substitution evaluate a limit?

   Odgovor

   When the function is continuous at the target input. Then

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

4. Kartica 4

   Pitanje

   Three conditions for continuity at $x=a$?

   Odgovor

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

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

5. Kartica 5

   Pitanje

   Intermediate Value Theorem: hypotheses and conclusion?

   Odgovor

   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. Kartica 6

   Pitanje

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

   Odgovor

   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. Kartica 7

   Pitanje

   How do you read a finite limit from a graph?

   Odgovor

   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. Kartica 8

   Pitanje

   Limit law for a sum or difference?

   Odgovor

   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. Kartica 9

   Pitanje

   What makes a discontinuity removable?

   Odgovor

   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. Kartica 10

    Pitanje

    Squeeze Theorem: usable form?

    Odgovor

    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. Kartica 11

    Pitanje

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

    Odgovor

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

12. Kartica 12

    Pitanje

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

    Odgovor

    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. Kartica 13

    Pitanje

    Limit law for a product?

    Odgovor

    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. Kartica 14

    Pitanje

    Graph signature of a jump discontinuity?

    Odgovor

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

15. Kartica 15

    Pitanje

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

    Odgovor

    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. Kartica 16

    Pitanje

    Horizontal asymptote from a limit at infinity?

    Odgovor

    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. Kartica 17

    Pitanje

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

    Odgovor

    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. Kartica 18

    Pitanje

    Limit law for a quotient—and its condition?

    Odgovor

    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. Kartica 19

    Pitanje

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

    Odgovor

    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. Kartica 20

    Pitanje

    When is the Squeeze Theorem a natural choice?

    Odgovor

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

21. Kartica 21

    Pitanje

    Vertical asymptote from one-sided behavior?

    Odgovor

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

22. Kartica 22

    Pitanje

    Limit at infinity of equal-degree rational functions?

    Odgovor

    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. Kartica 23

    Pitanje

    When can a limit pass through a continuous outer function?

    Odgovor

    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. Kartica 24

    Pitanje

    What makes a discontinuity infinite?

    Odgovor

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

25. Kartica 25

    Pitanje

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

    Odgovor

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

26. Kartica 26

    Pitanje

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

    Odgovor

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kartica 27

    Pitanje

    Continuity of a composition?

    Odgovor

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

28. Kartica 28

    Pitanje

    Limit at infinity when a rational numerator has lower degree?

    Odgovor

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

29. Kartica 29

    Pitanje

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

    Odgovor

    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. Kartica 30

    Pitanje

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

    Odgovor

    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. Kartica 31

    Pitanje

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

    Odgovor

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

32. Kartica 32

    Pitanje

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

    Odgovor

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

33. Kartica 33

    Pitanje

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

    Odgovor

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

34. Kartica 34

    Pitanje

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

    Odgovor

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

35. Kartica 35

    Pitanje

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

    Odgovor

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

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

36. Kartica 36

    Pitanje

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

    Odgovor

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

    The derivative exists only if this finite limit exists.

37. Kartica 37

    Pitanje

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

    Odgovor

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

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

38. Kartica 38

    Pitanje

    What does differentiability imply about continuity?

    Odgovor

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

39. Kartica 39

    Pitanje

    Power rule for derivatives?

    Odgovor

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

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

40. Kartica 40

    Pitanje

    Units of $f'(x)$?

    Odgovor

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

41. Kartica 41

    Pitanje

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

    Odgovor

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

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

42. Kartica 42

    Pitanje

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

    Odgovor

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

    Pitanje

    Derivative of a constant?

    Odgovor

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

    A constant function has zero rate of change.

44. Kartica 44

    Pitanje

    Derivative of $\sin x$?

    Odgovor

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

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

45. Kartica 45

    Pitanje

    Product rule?

    Odgovor

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

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

46. Kartica 46

    Pitanje

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

    Odgovor

    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. Kartica 47

    Pitanje

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

    Odgovor

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

48. Kartica 48

    Pitanje

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

    Odgovor

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

49. Kartica 49

    Pitanje

    Derivative of a sum or difference?

    Odgovor

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

50. Kartica 50

    Pitanje

    Derivative of $\cos x$?

    Odgovor

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

    The standard formula assumes radians.

51. Kartica 51

    Pitanje

    Quotient rule?

    Odgovor

    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. Kartica 52

    Pitanje

    Common notations for the first derivative?

    Odgovor

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

53. Kartica 53

    Pitanje

    Derivative of $e^x$?

    Odgovor

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

54. Kartica 54

    Pitanje

    What graph features can make $f$ nondifferentiable?

    Odgovor

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

55. Kartica 55

    Pitanje

    Derivative of $\tan x$?

    Odgovor

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kartica 56

    Pitanje

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

    Odgovor

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

57. Kartica 57

    Pitanje

    Derivative of $\ln x$?

    Odgovor

    For $x>0$,

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

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

58. Kartica 58

    Pitanje

    How does the power rule handle roots or negative powers?

    Odgovor

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

59. Kartica 59

    Pitanje

    Derivative of $\csc x$?

    Odgovor

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kartica 60

    Pitanje

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

    Odgovor

    $f$ is increasing on that interval.

61. Kartica 61

    Pitanje

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

    Odgovor

    For $a>0$,

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

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

62. Kartica 62

    Pitanje

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

    Odgovor

    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. Kartica 63

    Pitanje

    Derivative of $\sec x$?

    Odgovor

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kartica 64

    Pitanje

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

    Odgovor

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

65. Kartica 65

    Pitanje

    Derivative of $\log_a x$?

    Odgovor

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

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

66. Kartica 66

    Pitanje

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

    Odgovor

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kartica 67

    Pitanje

    Derivative of $\cot x$?

    Odgovor

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kartica 68

    Pitanje

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

    Odgovor

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

69. Kartica 69

    Pitanje

    Constant-multiple rule?

    Odgovor

    For a constant $c$,

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

70. Kartica 70

    Pitanje

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

    Odgovor

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

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

71. Kartica 71

    Pitanje

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

    Odgovor

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

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

72. Kartica 72

    Pitanje

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

    Odgovor

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

73. Kartica 73

    Pitanje

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

    Odgovor

    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. Kartica 74

    Pitanje

    Derivative of an inverse function at $x$?

    Odgovor

    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. Kartica 75

    Pitanje

    Derivative of $\arcsin x$?

    Odgovor

    For $|x|<1$,

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

76. Kartica 76

    Pitanje

    Notation for the third derivative of $f$?

    Odgovor

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

77. Kartica 77

    Pitanje

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

    Odgovor

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

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

78. Kartica 78

    Pitanje

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

    Odgovor

    Where $y\ne0$,

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

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

79. Kartica 79

    Pitanje

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

    Odgovor

    Provided $f'(a)\ne0$,

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

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

80. Kartica 80

    Pitanje

    Derivative of $\arctan x$?

    Odgovor

    For every real $x$,

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

81. Kartica 81

    Pitanje

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

    Odgovor

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

    The extra factor is the chain rule.

82. Kartica 82

    Pitanje

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

    Odgovor

    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. Kartica 83

    Pitanje

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

    Odgovor

    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. Kartica 84

    Pitanje

    Derivative of $\arccos x$?

    Odgovor

    For $|x|<1$,

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

85. Kartica 85

    Pitanje

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

    Odgovor

    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. Kartica 86

    Pitanje

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

    Odgovor

    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. Kartica 87

    Pitanje

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

    Odgovor

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

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

88. Kartica 88

    Pitanje

    How are tangent slopes of inverse graphs related?

    Odgovor

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

89. Kartica 89

    Pitanje

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

    Odgovor

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

90. Kartica 90

    Pitanje

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

    Odgovor

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

91. Kartica 91

    Pitanje

    Horizontal tangent on an implicit curve: derivative condition?

    Odgovor

    $dy/dx=0$ 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. Kartica 92

    Pitanje

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

    Odgovor

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

93. Kartica 93

    Pitanje

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

    Odgovor

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

94. Kartica 94

    Pitanje

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

    Odgovor

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

    This combines the power rule with the chain rule.

95. Kartica 95

    Pitanje

    Vertical tangent on an implicit curve: derivative clue?

    Odgovor

    $dy/dx$ 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. Kartica 96

    Pitanje

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

    Odgovor

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

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

97. Kartica 97

    Pitanje

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

    Odgovor

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

98. Kartica 98

    Pitanje

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

    Odgovor

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

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

99. Kartica 99

    Pitanje

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

    Odgovor

    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. Kartica 100

     Pitanje

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

     Odgovor

     When the functions are differentiable, the chain rule gives

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

101. Kartica 101

     Pitanje

     What local property lets a function have an inverse derivative?

     Odgovor

     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. Kartica 102

     Pitanje

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

     Odgovor

     For $a>0$,

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

103. Kartica 103

     Pitanje

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

     Odgovor

     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. Kartica 104

     Pitanje

     Position, velocity, and acceleration relationships?

     Odgovor

     For position $s(t)$,

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

105. Kartica 105

     Pitanje

     Central idea of a related-rates problem?

     Odgovor

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

106. Kartica 106

     Pitanje

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

     Odgovor

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

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

107. Kartica 107

     Pitanje

     L’Hospital’s Rule: basic conditions?

     Odgovor

     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. Kartica 108

     Pitanje

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

     Odgovor

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

109. Kartica 109

     Pitanje

     Speed in terms of velocity?

     Odgovor

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kartica 110

     Pitanje

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

     Odgovor

     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. Kartica 111

     Pitanje

     Differential approximation connecting $dx$ and $dy$?

     Odgovor

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

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

112. Kartica 112

     Pitanje

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

     Odgovor

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

113. Kartica 113

     Pitanje

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

     Odgovor

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

114. Kartica 114

     Pitanje

     What does positive acceleration say about velocity?

     Odgovor

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

115. Kartica 115

     Pitanje

     Related rates: when should numerical values be substituted?

     Odgovor

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

116. Kartica 116

     Pitanje

     How does concavity predict linearization error?

     Odgovor

     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. Kartica 117

     Pitanje

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

     Odgovor

     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. Kartica 118

     Pitanje

     When is a particle moving in the positive direction?

     Odgovor

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

119. Kartica 119

     Pitanje

     How can velocity show a change of direction?

     Odgovor

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

120. Kartica 120

     Pitanje

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

     Odgovor

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

121. Kartica 121

     Pitanje

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

     Odgovor

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

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

122. Kartica 122

     Pitanje

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

     Odgovor

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

123. Kartica 123

     Pitanje

     What must a contextual derivative sentence include?

     Odgovor

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

124. Kartica 124

     Pitanje

     Velocity negative and acceleration positive: what happens?

     Odgovor

     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. Kartica 125

     Pitanje

     How should a negative related rate be interpreted?

     Odgovor

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

126. Kartica 126

     Pitanje

     When is local linearity a sound approximation tool?

     Odgovor

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

127. Kartica 127

     Pitanje

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

     Odgovor

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

128. Kartica 128

     Pitanje

     When is speed increasing?

     Odgovor

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

129. Kartica 129

     Pitanje

     Volume changes with time: notation for its rate?

     Odgovor

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

130. Kartica 130

     Pitanje

     Why are similar triangles useful in related rates?

     Odgovor

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

131. Kartica 131

     Pitanje

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

     Odgovor

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

132. Kartica 132

     Pitanje

     What conclusion does L’Hospital’s Rule permit?

     Odgovor

     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. Kartica 133

     Pitanje

     When is speed decreasing?

     Odgovor

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

134. Kartica 134

     Pitanje

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

     Odgovor

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

135. Kartica 135

     Pitanje

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

     Odgovor

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

136. Kartica 136

     Pitanje

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

     Odgovor

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

137. Kartica 137

     Pitanje

     Extreme Value Theorem: hypothesis and conclusion?

     Odgovor

     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. Kartica 138

     Pitanje

     What is a critical number of $f$?

     Odgovor

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

139. Kartica 139

     Pitanje

     First derivative test for a local maximum?

     Odgovor

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

140. Kartica 140

     Pitanje

     Second-derivative sign for concave up?

     Odgovor

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

141. Kartica 141

     Pitanje

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

     Odgovor

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

142. Kartica 142

     Pitanje

     First step in an optimization model?

     Odgovor

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

143. Kartica 143

     Pitanje

     Mean Value Theorem: hypotheses and conclusion?

     Odgovor

     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. Kartica 144

     Pitanje

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

     Odgovor

     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. Kartica 145

     Pitanje

     First derivative test for a local minimum?

     Odgovor

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

146. Kartica 146

     Pitanje

     What must happen at an inflection point?

     Odgovor

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

147. Kartica 147

     Pitanje

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

     Odgovor

     $f''$ 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. Kartica 148

     Pitanje

     How do you confirm an optimization answer is absolute?

     Odgovor

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

149. Kartica 149

     Pitanje

     Rolle’s Theorem: hypotheses and conclusion?

     Odgovor

     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. Kartica 150

     Pitanje

     Difference between absolute and relative extrema?

     Odgovor

     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. Kartica 151

     Pitanje

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

     Odgovor

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

152. Kartica 152

     Pitanje

     Second derivative test for a local minimum?

     Odgovor

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

153. Kartica 153

     Pitanje

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

     Odgovor

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

154. Kartica 154

     Pitanje

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

     Odgovor

     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. Kartica 155

     Pitanje

     Which theorem links an average slope to an instantaneous slope?

     Odgovor

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

156. Kartica 156

     Pitanje

     How can an implicit derivative locate a horizontal tangent?

     Odgovor

     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. Kartica 157

     Pitanje

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

     Odgovor

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

158. Kartica 158

     Pitanje

     Second derivative test for a local maximum?

     Odgovor

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

159. Kartica 159

     Pitanje

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

     Odgovor

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

160. Kartica 160

     Pitanje

     Why must an optimization domain be stated?

     Odgovor

     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. Kartica 161

     Pitanje

     Which theorem guarantees absolute extrema, not where they occur?

     Odgovor

     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. Kartica 162

     Pitanje

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

     Odgovor

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

163. Kartica 163

     Pitanje

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

     Odgovor

     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. Kartica 164

     Pitanje

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

     Odgovor

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

165. Kartica 165

     Pitanje

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

     Odgovor

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

166. Kartica 166

     Pitanje

     How can an implicit derivative locate a vertical tangent?

     Odgovor

     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. Kartica 167

     Pitanje

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

     Odgovor

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

168. Kartica 168

     Pitanje

     Why are endpoints included in the candidates test?

     Odgovor

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

169. Kartica 169

     Pitanje

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

     Odgovor

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

170. Kartica 170

     Pitanje

     Second-derivative sign for concave down?

     Odgovor

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

171. Kartica 171

     Pitanje

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

     Odgovor

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

172. Kartica 172

     Pitanje

     What should the final line of an optimization solution state?

     Odgovor

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

173. Kartica 173

     Pitanje

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

     Odgovor

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

174. Kartica 174

     Pitanje

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

     Odgovor

     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. Kartica 175

     Pitanje

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

     Odgovor

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

176. Kartica 176

     Pitanje

     Left Riemann sum on equal subintervals?

     Odgovor

     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. Kartica 177

     Pitanje

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

     Odgovor

     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. Kartica 178

     Pitanje

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Odgovor

     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. Kartica 179

     Pitanje

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

     Odgovor

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kartica 180

     Pitanje

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

     Odgovor

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

181. Kartica 181

     Pitanje

     Right Riemann sum on equal subintervals?

     Odgovor

     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. Kartica 182

     Pitanje

     How does reversing integral bounds change the value?

     Odgovor

     It changes the sign:

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

183. Kartica 183

     Pitanje

     Net Change Theorem?

     Odgovor

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

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

184. Kartica 184

     Pitanje

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

     Odgovor

     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. Kartica 185

     Pitanje

     Power rule for antiderivatives?

     Odgovor

     For $n\ne-1$,

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

186. Kartica 186

     Pitanje

     Midpoint Riemann sum on equal subintervals?

     Odgovor

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

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

187. Kartica 187

     Pitanje

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

     Odgovor

     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. Kartica 188

     Pitanje

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

     Odgovor

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kartica 189

     Pitanje

     Antiderivative of $1/x$?

     Odgovor

     On any interval not crossing zero,

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

190. Kartica 190

     Pitanje

     Trapezoidal approximation on equal subintervals?

     Odgovor

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

191. Kartica 191

     Pitanje

     How do geometric regions help evaluate a definite integral?

     Odgovor

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

192. Kartica 192

     Pitanje

     Basic antiderivatives of sine and cosine?

     Odgovor

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

193. Kartica 193

     Pitanje

     Definite integral as a limit of Riemann sums?

     Odgovor

     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. Kartica 194

     Pitanje

     Constant-multiple rule for integrals?

     Odgovor

     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. Kartica 195

     Pitanje

     What pattern suggests $u$-substitution?

     Odgovor

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

196. Kartica 196

     Pitanje

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

     Odgovor

     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. Kartica 197

     Pitanje

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

     Odgovor

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

198. Kartica 198

     Pitanje

     Sum-and-difference rule for definite integrals?

     Odgovor

     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. Kartica 199

     Pitanje

     Basic antiderivative of $e^x$?

     Odgovor

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

200. Kartica 200

     Pitanje

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

     Odgovor

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

201. Kartica 201

     Pitanje

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

     Odgovor

     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. Kartica 202

     Pitanje

     How does concavity predict trapezoidal and midpoint error?

     Odgovor

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

203. Kartica 203

     Pitanje

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

     Odgovor

     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. Kartica 204

     Pitanje

     What denominator pattern suggests an arctangent antiderivative?

     Odgovor

     After completing the square and scaling, a form like

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

205. Kartica 205

     Pitanje

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

     Odgovor

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

206. Kartica 206

     Pitanje

     How does an initial condition determine an antiderivative?

     Odgovor

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

207. Kartica 207

     Pitanje

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

     Odgovor

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

208. Kartica 208

     Pitanje

     Why does an indefinite integral include $+C$?

     Odgovor

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

209. Kartica 209

     Pitanje

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

     Odgovor

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

210. Kartica 210

     Pitanje

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

     Odgovor

     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. Kartica 211

     Pitanje

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

     Odgovor

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

212. Kartica 212

     Pitanje

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

     Odgovor

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

213. Kartica 213

     Pitanje

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

     Odgovor

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

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

214. Kartica 214

     Pitanje

     Riemann sum for unequal subinterval widths?

     Odgovor

     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. Kartica 215

     Pitanje

     Does continuity guarantee integrability on a closed interval?

     Odgovor

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

216. Kartica 216

     Pitanje

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

     Odgovor

     Where $g(x)\ne0$,

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

217. Kartica 217

     Pitanje

     What algebraic rewrites often reveal a basic antiderivative?

     Odgovor

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

218. Kartica 218

     Pitanje

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

     Odgovor

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

219. Kartica 219

     Pitanje

     What is a differential equation?

     Odgovor

     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. Kartica 220

     Pitanje

     How does a verbal rate statement become a differential equation?

     Odgovor

     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. Kartica 221

     Pitanje

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

     Odgovor

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

222. Kartica 222

     Pitanje

     General solution versus particular solution?

     Odgovor

     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. Kartica 223

     Pitanje

     What does one segment in a slope field show?

     Odgovor

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

224. Kartica 224

     Pitanje

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

     Odgovor

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

225. Kartica 225

     Pitanje

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

     Odgovor

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

226. Kartica 226

     Pitanje

     What makes a first-order differential equation separable?

     Odgovor

     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. Kartica 227

     Pitanje

     What is an initial value problem?

     Odgovor

     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. Kartica 228

     Pitanje

     What is an isocline in a slope field?

     Odgovor

     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. Kartica 229

     Pitanje

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

     Odgovor

     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. Kartica 230

     Pitanje

     General solution of $dy/dt=ky$?

     Odgovor

     $$y=Ce^{kt}$$

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

231. Kartica 231

     Pitanje

     Core method for solving a separable differential equation?

     Odgovor

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

232. Kartica 232

     Pitanje

     How should a solution curve follow a slope field?

     Odgovor

     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. Kartica 233

     Pitanje

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

     Odgovor

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

234. Kartica 234

     Pitanje

     Why is one integration constant enough after integrating both sides?

     Odgovor

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

235. Kartica 235

     Pitanje

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

     Odgovor

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

236. Kartica 236

     Pitanje

     Can one differential equation have infinitely many solutions?

     Odgovor

     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. Kartica 237

     Pitanje

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

     Odgovor

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

238. Kartica 238

     Pitanje

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

     Odgovor

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

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

239. Kartica 239

     Pitanje

     What can be lost when dividing to separate variables?

     Odgovor

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

240. Kartica 240

     Pitanje

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

     Odgovor

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

241. Kartica 241

     Pitanje

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

     Odgovor

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

242. Kartica 242

     Pitanje

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

     Odgovor

     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. Kartica 243

     Pitanje

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

     Odgovor

     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. Kartica 244

     Pitanje

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

     Odgovor

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

245. Kartica 245

     Pitanje

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

     Odgovor

     For $k>0$,

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

     It is independent of the initial amount.

246. Kartica 246

     Pitanje

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

     Odgovor

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

247. Kartica 247

     Pitanje

     How is an initial condition used after separation?

     Odgovor

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

248. Kartica 248

     Pitanje

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

     Odgovor

     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. Kartica 249

     Pitanje

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

     Odgovor

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

250. Kartica 250

     Pitanje

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

     Odgovor

     For $k<0$,

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

251. Kartica 251

     Pitanje

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

     Odgovor

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

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

252. Kartica 252

     Pitanje

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

     Odgovor

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

     Velocity below zero contributes negative displacement.

253. Kartica 253

     Pitanje

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

     Odgovor

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

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

     Think top minus bottom.

254. Kartica 254

     Pitanje

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

     Odgovor

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

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

255. Kartica 255

     Pitanje

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Odgovor

     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. Kartica 256

     Pitanje

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

     Odgovor

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

257. Kartica 257

     Pitanje

     Cross-sectional area when each slice is a square?

     Odgovor

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

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

258. Kartica 258

     Pitanje

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

     Odgovor

     Integrate the net rate:

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

259. Kartica 259

     Pitanje

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

     Odgovor

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

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

     Think right minus left.

260. Kartica 260

     Pitanje

     Disc-method volume formula?

     Odgovor

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

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

261. Kartica 261

     Pitanje

     What units does average value have?

     Odgovor

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

262. Kartica 262

     Pitanje

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

     Odgovor

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

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

263. Kartica 263

     Pitanje

     Cross-sectional area when each slice is a rectangle?

     Odgovor

     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. Kartica 264

     Pitanje

     How do you determine bounds for area between curves?

     Odgovor

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

265. Kartica 265

     Pitanje

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

     Odgovor

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

266. Kartica 266

     Pitanje

     When is a particle moving to the right or left?

     Odgovor

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

267. Kartica 267

     Pitanje

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

     Odgovor

     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. Kartica 268

     Pitanje

     Why must an area integral be split where curves intersect?

     Odgovor

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

269. Kartica 269

     Pitanje

     How can a velocity table approximate displacement?

     Odgovor

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

270. Kartica 270

     Pitanje

     Washer-method volume formula?

     Odgovor

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

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

271. Kartica 271

     Pitanje

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

     Odgovor

     If $s(a)$ is known,

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

272. Kartica 272

     Pitanje

     How do you choose between vertical and horizontal area slices?

     Odgovor

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

273. Kartica 273

     Pitanje

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

     Odgovor

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

274. Kartica 274

     Pitanje

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

     Odgovor

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

275. Kartica 275

     Pitanje

     Single expression for area between two curves?

     Odgovor

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kartica 276

     Pitanje

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

     Odgovor

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

277. Kartica 277

     Pitanje

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

     Odgovor

     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. Kartica 278

     Pitanje

     When should a volume integral use $dy$?

     Odgovor

     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. Kartica 279

     Pitanje

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

     Odgovor

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

280. Kartica 280

     Pitanje

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

     Odgovor

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

281. Kartica 281

     Pitanje

     Why must total distance split at velocity sign changes?

     Odgovor

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

282. Kartica 282

     Pitanje

     When does an accumulated quantity reach a local maximum?

     Odgovor

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

283. Kartica 283

     Pitanje

     What distinguishes area from a definite integral?

     Odgovor

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

284. Kartica 284

     Pitanje

     How do position, velocity, and acceleration graphs correspond?

     Odgovor

     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. Kartica 285

     Pitanje

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

     Odgovor

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

286. Kartica 286

     Pitanje

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

     Odgovor

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

287. Kartica 287

     Pitanje

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

     Odgovor

     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. Kartica 288

     Pitanje

     Why should a contextual integral answer include a sentence?

     Odgovor

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