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

Kaarten in dit deck

1. Kaart 1

   Vraag

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

   Antwoord

   The values of $f(x)$ approach $L$ as $x$ approaches $a$ from both sides. The statement doesn't require $f(a)=L$ or even require $f(a)$ to exist.

2. Kaart 2

   Vraag

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

   Antwoord

   Use inputs approaching $a$ from below and above, then look for a common output value. Values exactly at $x=a$ don't determine the limit.

3. Kaart 3

   Vraag

   When does direct substitution evaluate a limit?

   Antwoord

   When the function is continuous at the target input. Then

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

4. Kaart 4

   Vraag

   Three conditions for continuity at $x=a$?

   Antwoord

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

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

5. Kaart 5

   Vraag

   Intermediate Value Theorem: hypotheses and conclusion?

   Antwoord

   If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then some $c$ in $[a,b]$ satisfies $f(c)=N$. If $N$ is strictly between the endpoint values, $c$ lies in $(a,b)$.

6. Kaart 6

   Vraag

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

   Antwoord

   Exactly when both one-sided limits equal $L$:

   $$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.$$

   If the one-sided limits differ, the two-sided limit doesn't exist.

7. Kaart 7

   Vraag

   How do you read a finite limit from a graph?

   Antwoord

   Follow the graph toward the target $x$-value from both sides. The common approached $y$-value is the limit, regardless of a hole or a differently placed filled point.

8. Kaart 8

   Vraag

   Limit law for a sum or difference?

   Antwoord

   If both component limits exist,

   $$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$$

9. Kaart 9

   Vraag

   What makes a discontinuity removable?

   Antwoord

   The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.

10. Kaart 10

    Vraag

    Squeeze Theorem: usable form?

    Antwoord

    If $g(x)\le f(x)\le h(x)$ near $a$ and

    $$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$$

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

11. Kaart 11

    Vraag

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

    Antwoord

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

12. Kaart 12

    Vraag

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

    Antwoord

    Inputs less than the target and moving toward it. For $\lim_{x\to a^-}f(x)$, use $x<a$ with $x$ getting closer to $a$.

13. Kaart 13

    Vraag

    Limit law for a product?

    Antwoord

    If both limits exist,

    $$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$$

14. Kaart 14

    Vraag

    Graph signature of a jump discontinuity?

    Antwoord

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

15. Kaart 15

    Vraag

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

    Antwoord

    The Intermediate Value Theorem. If $f$ is continuous on $[a,b]$ and $0$ lies between $f(a)$ and $f(b)$, then $f(c)=0$ for some $c$ in the interval.

16. Kaart 16

    Vraag

    Horizontal asymptote from a limit at infinity?

    Antwoord

    If $\lim_{x\to\infty}f(x)=L$ or $\lim_{x\to-\infty}f(x)=L$, then $y=L$ is a horizontal asymptote in that direction.

17. Kaart 17

    Vraag

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

    Antwoord

    Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.

18. Kaart 18

    Vraag

    Limit law for a quotient—and its condition?

    Antwoord

    If both limits exist and the denominator limit is nonzero,

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

    The law doesn't apply when the denominator limit is $0$.

19. Kaart 19

    Vraag

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

    Antwoord

    Continuity on $(a,b)$, right-continuity at $a$, and left-continuity at $b$:

    $$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$$

20. Kaart 20

    Vraag

    When is the Squeeze Theorem a natural choice?

    Antwoord

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

21. Kaart 21

    Vraag

    Vertical asymptote from one-sided behavior?

    Antwoord

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

22. Kaart 22

    Vraag

    Limit at infinity of equal-degree rational functions?

    Antwoord

    The ratio of the leading coefficients:

    $$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}.$$

    This assumes $b_n\ne0$.

23. Kaart 23

    Vraag

    When can a limit pass through a continuous outer function?

    Antwoord

    If $\lim_{x\to a}g(x)=L$ and $f$ is continuous at $L$, then

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

24. Kaart 24

    Vraag

    What makes a discontinuity infinite?

    Antwoord

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

25. Kaart 25

    Vraag

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

    Antwoord

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

26. Kaart 26

    Vraag

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

    Antwoord

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kaart 27

    Vraag

    Continuity of a composition?

    Antwoord

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

28. Kaart 28

    Vraag

    Limit at infinity when a rational numerator has lower degree?

    Antwoord

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

29. Kaart 29

    Vraag

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

    Antwoord

    Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation; $0/0$ isn't the limit value.

30. Kaart 30

    Vraag

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

    Antwoord

    They don't. For example, $+\infty$ from the left and $-\infty$ from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.

31. Kaart 31

    Vraag

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

    Antwoord

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

32. Kaart 32

    Vraag

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

    Antwoord

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

33. Kaart 33

    Vraag

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

    Antwoord

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

34. Kaart 34

    Vraag

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

    Antwoord

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

35. Kaart 35

    Vraag

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

    Antwoord

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

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

36. Kaart 36

    Vraag

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

    Antwoord

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

    The derivative exists only if this finite limit exists.

37. Kaart 37

    Vraag

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

    Antwoord

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

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

38. Kaart 38

    Vraag

    What does differentiability imply about continuity?

    Antwoord

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

39. Kaart 39

    Vraag

    Power rule for derivatives?

    Antwoord

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

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

40. Kaart 40

    Vraag

    Units of $f'(x)$?

    Antwoord

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

41. Kaart 41

    Vraag

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

    Antwoord

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

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

42. Kaart 42

    Vraag

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

    Antwoord

    $f'(x)>0$ where $f$ rises as $x$ increases, and $f'(x)<0$ where $f$ falls. A horizontal tangent gives $f'(x)=0$ when the derivative exists.

43. Kaart 43

    Vraag

    Derivative of a constant?

    Antwoord

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

    A constant function has zero rate of change.

44. Kaart 44

    Vraag

    Derivative of $\sin x$?

    Antwoord

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

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

45. Kaart 45

    Vraag

    Product rule?

    Antwoord

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

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

46. Kaart 46

    Vraag

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

    Antwoord

    Use a difference quotient with inputs close to $a$. A symmetric estimate is

    $$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}.$$

    Smaller $h$ often helps, subject to the data's precision.

47. Kaart 47

    Vraag

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

    Antwoord

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

48. Kaart 48

    Vraag

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

    Antwoord

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

49. Kaart 49

    Vraag

    Derivative of a sum or difference?

    Antwoord

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

50. Kaart 50

    Vraag

    Derivative of $\cos x$?

    Antwoord

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

    The standard formula assumes radians.

51. Kaart 51

    Vraag

    Quotient rule?

    Antwoord

    For $g(x)\ne0$,

    $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.$$

    The order in the numerator matters.

52. Kaart 52

    Vraag

    Common notations for the first derivative?

    Antwoord

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

53. Kaart 53

    Vraag

    Derivative of $e^x$?

    Antwoord

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

54. Kaart 54

    Vraag

    What graph features can make $f$ nondifferentiable?

    Antwoord

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

55. Kaart 55

    Vraag

    Derivative of $\tan x$?

    Antwoord

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kaart 56

    Vraag

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

    Antwoord

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

57. Kaart 57

    Vraag

    Derivative of $\ln x$?

    Antwoord

    For $x>0$,

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

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

58. Kaart 58

    Vraag

    How does the power rule handle roots or negative powers?

    Antwoord

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

59. Kaart 59

    Vraag

    Derivative of $\csc x$?

    Antwoord

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kaart 60

    Vraag

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

    Antwoord

    $f$ is increasing on that interval.

61. Kaart 61

    Vraag

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

    Antwoord

    For $a>0$,

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

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

62. Kaart 62

    Vraag

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

    Antwoord

    Estimate the slope of the tangent line at $x=a$, using two convenient points on a drawn tangent when available. A steeper tangent has a derivative with larger magnitude.

63. Kaart 63

    Vraag

    Derivative of $\sec x$?

    Antwoord

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kaart 64

    Vraag

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

    Antwoord

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

65. Kaart 65

    Vraag

    Derivative of $\log_a x$?

    Antwoord

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

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

66. Kaart 66

    Vraag

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

    Antwoord

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kaart 67

    Vraag

    Derivative of $\cot x$?

    Antwoord

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kaart 68

    Vraag

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

    Antwoord

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

69. Kaart 69

    Vraag

    Constant-multiple rule?

    Antwoord

    For a constant $c$,

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

70. Kaart 70

    Vraag

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

    Antwoord

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

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

71. Kaart 71

    Vraag

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

    Antwoord

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

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

72. Kaart 72

    Vraag

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

    Antwoord

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

73. Kaart 73

    Vraag

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

    Antwoord

    Treat $y$ as a differentiable function of $x$. Every derivative of an expression involving $y$ gains a factor of $dy/dx$ by the chain rule.

74. Kaart 74

    Vraag

    Derivative of an inverse function at $x$?

    Antwoord

    If $f$ is differentiable and one-to-one near $f^{-1}(x)$, with $f'(f^{-1}(x))\ne0$,

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

75. Kaart 75

    Vraag

    Derivative of $\arcsin x$?

    Antwoord

    For $|x|<1$,

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

76. Kaart 76

    Vraag

    Notation for the third derivative of $f$?

    Antwoord

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

77. Kaart 77

    Vraag

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

    Antwoord

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

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

78. Kaart 78

    Vraag

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

    Antwoord

    Where $y\ne0$,

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

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

79. Kaart 79

    Vraag

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

    Antwoord

    Provided $f'(a)\ne0$,

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

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

80. Kaart 80

    Vraag

    Derivative of $\arctan x$?

    Antwoord

    For every real $x$,

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

81. Kaart 81

    Vraag

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

    Antwoord

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

    The extra factor is the chain rule.

82. Kaart 82

    Vraag

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

    Antwoord

    Differentiate the relation with respect to $x$, solve for $dy/dx$, then substitute the point. Confirm the point lies on the curve and the resulting slope is defined.

83. Kaart 83

    Vraag

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

    Antwoord

    Because the reciprocal slope would be undefined when $f'(a)=0$. The inverse may have a vertical tangent or fail to be differentiable there.

84. Kaart 84

    Vraag

    Derivative of $\arccos x$?

    Antwoord

    For $|x|<1$,

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

85. Kaart 85

    Vraag

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

    Antwoord

    Differentiate the first-derivative equation again with respect to $x$, include $dy/dx$ factors, then substitute the known expression for $dy/dx$ if needed.

86. Kaart 86

    Vraag

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

    Antwoord

    Where $g(x)>0$,

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

    For $\ln|g(x)|$, the same derivative holds where $g(x)\ne0$.

87. Kaart 87

    Vraag

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

    Antwoord

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

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

88. Kaart 88

    Vraag

    How are tangent slopes of inverse graphs related?

    Antwoord

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

89. Kaart 89

    Vraag

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

    Antwoord

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

90. Kaart 90

    Vraag

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

    Antwoord

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

91. Kaart 91

    Vraag

    Horizontal tangent on an implicit curve: derivative condition?

    Antwoord

    $dy/dx=0$ at a valid point, with the derivative defined there. In a fraction for $dy/dx$, the numerator is typically zero while the denominator is nonzero.

92. Kaart 92

    Vraag

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

    Antwoord

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

93. Kaart 93

    Vraag

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

    Antwoord

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

94. Kaart 94

    Vraag

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

    Antwoord

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

    This combines the power rule with the chain rule.

95. Kaart 95

    Vraag

    Vertical tangent on an implicit curve: derivative clue?

    Antwoord

    $dy/dx$ becomes unbounded or undefined while the curve still has a vertical tangent. In a simplified derivative fraction, the denominator is often zero and the numerator nonzero.

96. Kaart 96

    Vraag

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

    Antwoord

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

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

97. Kaart 97

    Vraag

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

    Antwoord

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

98. Kaart 98

    Vraag

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

    Antwoord

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

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

99. Kaart 99

    Vraag

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

    Antwoord

    An implicit relation may never solve explicitly for $y$. After differentiating twice and replacing $dy/dx$, the result can naturally remain a function of both coordinates.

100. Kaart 100

     Vraag

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

     Antwoord

     When the functions are differentiable, the chain rule gives

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

101. Kaart 101

     Vraag

     What local property lets a function have an inverse derivative?

     Antwoord

     The function must be one-to-one near the point, differentiable there, and have a nonzero derivative. Then the inverse is locally differentiable with reciprocal slope.

102. Kaart 102

     Vraag

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

     Antwoord

     For $a>0$,

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

103. Kaart 103

     Vraag

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

     Antwoord

     At time $t$, the quantity $Q$ changes at an instantaneous rate of $Q'(t)$ output units per unit of time. Include the quantity, time, direction or sign, and units.

104. Kaart 104

     Vraag

     Position, velocity, and acceleration relationships?

     Antwoord

     For position $s(t)$,

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

105. Kaart 105

     Vraag

     Central idea of a related-rates problem?

     Antwoord

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

106. Kaart 106

     Vraag

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

     Antwoord

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

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

107. Kaart 107

     Vraag

     L’Hospital’s Rule: basic conditions?

     Antwoord

     For a quotient with differentiable numerator and denominator near the target, denominator derivative nonzero nearby, and indeterminate form $0/0$ or $\infty/\infty$, the original limit equals the limit of the derivative quotient when that latter limit exists or is infinite.

108. Kaart 108

     Vraag

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

     Antwoord

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

109. Kaart 109

     Vraag

     Speed in terms of velocity?

     Antwoord

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kaart 110

     Vraag

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

     Antwoord

     They are functions of time. Differentiating an expression such as $x^2$ with respect to $t$ gives $2x(dx/dt)$ by the chain rule.

111. Kaart 111

     Vraag

     Differential approximation connecting $dx$ and $dy$?

     Antwoord

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

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

112. Kaart 112

     Vraag

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

     Antwoord

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

113. Kaart 113

     Vraag

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

     Antwoord

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

114. Kaart 114

     Vraag

     What does positive acceleration say about velocity?

     Antwoord

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

115. Kaart 115

     Vraag

     Related rates: when should numerical values be substituted?

     Antwoord

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

116. Kaart 116

     Vraag

     How does concavity predict linearization error?

     Antwoord

     Near the tangency point, a concave-up graph generally lies above its tangent line, so the linearization underestimates. A concave-down graph generally lies below, so it overestimates.

117. Kaart 117

     Vraag

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

     Antwoord

     The rule applies to quotients with $0/0$ or $\infty/\infty$ form. Rewrite an indeterminate product such as $0\cdot\infty$ as a quotient first.

118. Kaart 118

     Vraag

     When is a particle moving in the positive direction?

     Antwoord

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

119. Kaart 119

     Vraag

     How can velocity show a change of direction?

     Antwoord

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

120. Kaart 120

     Vraag

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

     Antwoord

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

121. Kaart 121

     Vraag

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

     Antwoord

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

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

122. Kaart 122

     Vraag

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

     Antwoord

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

123. Kaart 123

     Vraag

     What must a contextual derivative sentence include?

     Antwoord

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

124. Kaart 124

     Vraag

     Velocity negative and acceleration positive: what happens?

     Antwoord

     The particle moves in the negative direction while its velocity increases toward zero. Its speed decreases as long as velocity and acceleration have opposite signs.

125. Kaart 125

     Vraag

     How should a negative related rate be interpreted?

     Antwoord

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

126. Kaart 126

     Vraag

     When is local linearity a sound approximation tool?

     Antwoord

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

127. Kaart 127

     Vraag

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

     Antwoord

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

128. Kaart 128

     Vraag

     When is speed increasing?

     Antwoord

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

129. Kaart 129

     Vraag

     Volume changes with time: notation for its rate?

     Antwoord

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

130. Kaart 130

     Vraag

     Why are similar triangles useful in related rates?

     Antwoord

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

131. Kaart 131

     Vraag

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

     Antwoord

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

132. Kaart 132

     Vraag

     What conclusion does L’Hospital’s Rule permit?

     Antwoord

     Under its conditions,

     $$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}.$$

     It does not say the two quotients are equal as functions.

133. Kaart 133

     Vraag

     When is speed decreasing?

     Antwoord

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

134. Kaart 134

     Vraag

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

     Antwoord

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

135. Kaart 135

     Vraag

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

     Antwoord

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

136. Kaart 136

     Vraag

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

     Antwoord

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

137. Kaart 137

     Vraag

     Extreme Value Theorem: hypothesis and conclusion?

     Antwoord

     If $f$ is continuous on the closed interval $[a,b]$, then $f$ has at least one absolute minimum value and at least one absolute maximum value on $[a,b]$.

138. Kaart 138

     Vraag

     What is a critical number of $f$?

     Antwoord

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

139. Kaart 139

     Vraag

     First derivative test for a local maximum?

     Antwoord

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

140. Kaart 140

     Vraag

     Second-derivative sign for concave up?

     Antwoord

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

141. Kaart 141

     Vraag

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

     Antwoord

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

142. Kaart 142

     Vraag

     First step in an optimization model?

     Antwoord

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

143. Kaart 143

     Vraag

     Mean Value Theorem: hypotheses and conclusion?

     Antwoord

     If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c$ in $(a,b)$ satisfies

     $$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

144. Kaart 144

     Vraag

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

     Antwoord

     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. Kaart 145

     Vraag

     First derivative test for a local minimum?

     Antwoord

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

146. Kaart 146

     Vraag

     What must happen at an inflection point?

     Antwoord

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

147. Kaart 147

     Vraag

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

     Antwoord

     $f''$ may change from positive to negative there, so $f$ may change from concave up to concave down. Confirm the sign change rather than relying only on the point.

148. Kaart 148

     Vraag

     How do you confirm an optimization answer is absolute?

     Antwoord

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

149. Kaart 149

     Vraag

     Rolle’s Theorem: hypotheses and conclusion?

     Antwoord

     If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then some $c$ in $(a,b)$ satisfies $f'(c)=0$.

150. Kaart 150

     Vraag

     Difference between absolute and relative extrema?

     Antwoord

     An absolute maximum is at least every other value on the stated domain, and an absolute minimum is at most every other value. A relative extremum makes the corresponding comparison only with nearby values.

151. Kaart 151

     Vraag

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

     Antwoord

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

152. Kaart 152

     Vraag

     Second derivative test for a local minimum?

     Antwoord

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

153. Kaart 153

     Vraag

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

     Antwoord

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

154. Kaart 154

     Vraag

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

     Antwoord

     Its sign shows whether the relation's local branch rises or falls as $x$ increases; zeros and undefined values mark possible horizontal or vertical tangents.

155. Kaart 155

     Vraag

     Which theorem links an average slope to an instantaneous slope?

     Antwoord

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

156. Kaart 156

     Vraag

     How can an implicit derivative locate a horizontal tangent?

     Antwoord

     At a valid point on the relation, find where the simplified numerator of $dy/dx$ is zero while its denominator is nonzero, then confirm the local branch has the expected behavior.

157. Kaart 157

     Vraag

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

     Antwoord

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

158. Kaart 158

     Vraag

     Second derivative test for a local maximum?

     Antwoord

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

159. Kaart 159

     Vraag

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

     Antwoord

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

160. Kaart 160

     Vraag

     Why must an optimization domain be stated?

     Antwoord

     The context can exclude algebraic candidates and add boundary cases. Absolute extrema depend on the feasible inputs, not just on where the derivative is zero.

161. Kaart 161

     Vraag

     Which theorem guarantees absolute extrema, not where they occur?

     Antwoord

     The Extreme Value Theorem. Continuity on a closed interval guarantees at least one maximum value and one minimum value, but finding them requires analysis.

162. Kaart 162

     Vraag

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

     Antwoord

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

163. Kaart 163

     Vraag

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

     Antwoord

     If $f'$ is positive on both sides of $c$, or negative on both sides, then $f$ keeps the same monotonic direction through $c$ and has no local extremum there.

164. Kaart 164

     Vraag

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

     Antwoord

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

165. Kaart 165

     Vraag

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

     Antwoord

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

166. Kaart 166

     Vraag

     How can an implicit derivative locate a vertical tangent?

     Antwoord

     Find valid points where the simplified $dy/dx$ denominator is zero and numerator is nonzero, then confirm the curve has the expected local behavior.

167. Kaart 167

     Vraag

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

     Antwoord

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

168. Kaart 168

     Vraag

     Why are endpoints included in the candidates test?

     Antwoord

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

169. Kaart 169

     Vraag

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

     Antwoord

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

170. Kaart 170

     Vraag

     Second-derivative sign for concave down?

     Antwoord

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

171. Kaart 171

     Vraag

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

     Antwoord

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

172. Kaart 172

     Vraag

     What should the final line of an optimization solution state?

     Antwoord

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

173. Kaart 173

     Vraag

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

     Antwoord

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

174. Kaart 174

     Vraag

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

     Antwoord

     They are candidates for changes in concavity. Test the sign of $f''$ on both sides; a zero alone doesn't guarantee an inflection point.

175. Kaart 175

     Vraag

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

     Antwoord

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

176. Kaart 176

     Vraag

     Left Riemann sum on equal subintervals?

     Antwoord

     If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$, then

     $$L_n=\sum_{i=1}^{n} f(x_{i-1})\Delta x.$$

177. Kaart 177

     Vraag

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

     Antwoord

     Signed area between the graph and the $x$-axis from $a$ to $b$, when $f$ is integrable. Regions below the axis subtract from regions above it.

178. Kaart 178

     Vraag

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Antwoord

     If $f$ is continuous on $[a,b]$ and $F$ is an antiderivative of $f$, then

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

179. Kaart 179

     Vraag

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

     Antwoord

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kaart 180

     Vraag

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

     Antwoord

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

181. Kaart 181

     Vraag

     Right Riemann sum on equal subintervals?

     Antwoord

     If $\Delta x=(b-a)/n$ and $x_i=a+i\Delta x$, then

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

182. Kaart 182

     Vraag

     How does reversing integral bounds change the value?

     Antwoord

     It changes the sign:

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

183. Kaart 183

     Vraag

     Net Change Theorem?

     Antwoord

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

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

184. Kaart 184

     Vraag

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

     Antwoord

     If $f$ is continuous on an interval containing $a$ and the range of $g$, and $g$ is differentiable, then

     $$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x).$$

185. Kaart 185

     Vraag

     Power rule for antiderivatives?

     Antwoord

     For $n\ne-1$,

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

186. Kaart 186

     Vraag

     Midpoint Riemann sum on equal subintervals?

     Antwoord

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

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

187. Kaart 187

     Vraag

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

     Antwoord

     For $a\le c\le b$,

     $$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.$$

188. Kaart 188

     Vraag

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

     Antwoord

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kaart 189

     Vraag

     Antiderivative of $1/x$?

     Antwoord

     On any interval not crossing zero,

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

190. Kaart 190

     Vraag

     Trapezoidal approximation on equal subintervals?

     Antwoord

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

191. Kaart 191

     Vraag

     How do geometric regions help evaluate a definite integral?

     Antwoord

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

192. Kaart 192

     Vraag

     Basic antiderivatives of sine and cosine?

     Antwoord

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

193. Kaart 193

     Vraag

     Definite integral as a limit of Riemann sums?

     Antwoord

     For an integrable function and sample points $x_i^*$,

     $$\int_a^b f(x)\,dx=\lim_{\max\Delta x_i\to0}\sum_i f(x_i^*)\Delta x_i.$$

194. Kaart 194

     Vraag

     Constant-multiple rule for integrals?

     Antwoord

     For a constant $k$,

     $$\int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx.$$

     The analogous rule holds for indefinite integrals.

195. Kaart 195

     Vraag

     What pattern suggests $u$-substitution?

     Antwoord

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

196. Kaart 196

     Vraag

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

     Antwoord

     If $u=g(x)$, replace the $x$-bounds $a,b$ with $u$-bounds $g(a),g(b)$. Then finish entirely in $u$, or return to $x$ before applying the original bounds.

197. Kaart 197

     Vraag

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

     Antwoord

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

198. Kaart 198

     Vraag

     Sum-and-difference rule for definite integrals?

     Antwoord

     For integrable $f$ and $g$,

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

199. Kaart 199

     Vraag

     Basic antiderivative of $e^x$?

     Antwoord

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

200. Kaart 200

     Vraag

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

     Antwoord

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

201. Kaart 201

     Vraag

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

     Antwoord

     On the same partition, the left sum is an underestimate and the right sum is an overestimate. Reverse those conclusions when the function is decreasing.

202. Kaart 202

     Vraag

     How does concavity predict trapezoidal and midpoint error?

     Antwoord

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

203. Kaart 203

     Vraag

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

     Antwoord

     When the numerator's degree is at least the denominator's, division rewrites the expression as a polynomial plus a simpler proper rational function.

204. Kaart 204

     Vraag

     What denominator pattern suggests an arctangent antiderivative?

     Antwoord

     After completing the square and scaling, a form like

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

205. Kaart 205

     Vraag

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

     Antwoord

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

206. Kaart 206

     Vraag

     How does an initial condition determine an antiderivative?

     Antwoord

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

207. Kaart 207

     Vraag

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

     Antwoord

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

208. Kaart 208

     Vraag

     Why does an indefinite integral include $+C$?

     Antwoord

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

209. Kaart 209

     Vraag

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

     Antwoord

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

210. Kaart 210

     Vraag

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

     Antwoord

     Since $F'(x)=f(x)$, $F$ is concave up where $f$ is increasing and concave down where $f$ is decreasing, assuming the needed derivatives exist.

211. Kaart 211

     Vraag

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

     Antwoord

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

212. Kaart 212

     Vraag

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

     Antwoord

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

213. Kaart 213

     Vraag

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

     Antwoord

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

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

214. Kaart 214

     Vraag

     Riemann sum for unequal subinterval widths?

     Antwoord

     If $x_{i-1}$ to $x_i$ has width $\Delta x_i$ and sample point $x_i^*$, use

     $$\sum_i f(x_i^*)\Delta x_i.$$

215. Kaart 215

     Vraag

     Does continuity guarantee integrability on a closed interval?

     Antwoord

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

216. Kaart 216

     Vraag

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

     Antwoord

     Where $g(x)\ne0$,

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

217. Kaart 217

     Vraag

     What algebraic rewrites often reveal a basic antiderivative?

     Antwoord

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

218. Kaart 218

     Vraag

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

     Antwoord

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

219. Kaart 219

     Vraag

     What is a differential equation?

     Antwoord

     An equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true on an interval.

220. Kaart 220

     Vraag

     How does a verbal rate statement become a differential equation?

     Antwoord

     Name the changing quantity and its independent variable, translate its rate as a derivative, and express that derivative using the stated relationship. For example, “rate proportional to $y$” becomes $dy/dt=ky$.

221. Kaart 221

     Vraag

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

     Antwoord

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

222. Kaart 222

     Vraag

     General solution versus particular solution?

     Antwoord

     A general solution contains an arbitrary constant and represents a family of solution curves. A particular solution uses an initial condition to determine that constant.

223. Kaart 223

     Vraag

     What does one segment in a slope field show?

     Antwoord

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

224. Kaart 224

     Vraag

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

     Antwoord

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

225. Kaart 225

     Vraag

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

     Antwoord

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

226. Kaart 226

     Vraag

     What makes a first-order differential equation separable?

     Antwoord

     It can be rearranged so all $y$ factors accompany $dy$ and all $x$ factors accompany $dx$, such as

     $$g(y)\,dy=f(x)\,dx.$$

227. Kaart 227

     Vraag

     What is an initial value problem?

     Antwoord

     A differential equation paired with a value such as $y(x_0)=y_0$. It asks for a solution through $(x_0,y_0)$; depending on the equation, there may be zero, one, or multiple such solutions.

228. Kaart 228

     Vraag

     What is an isocline in a slope field?

     Antwoord

     A curve along which the differential equation gives the same slope. For $dy/dx=F(x,y)$, an isocline satisfies $F(x,y)=k$ for a constant $k$.

229. Kaart 229

     Vraag

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

     Antwoord

     Evaluate the differential equation at $(x_0,y_0)$ to find the slope, then draw a short segment through that point with the resulting slope.

230. Kaart 230

     Vraag

     General solution of $dy/dt=ky$?

     Antwoord

     $$y=Ce^{kt}$$

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

231. Kaart 231

     Vraag

     Core method for solving a separable differential equation?

     Antwoord

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

232. Kaart 232

     Vraag

     How should a solution curve follow a slope field?

     Antwoord

     It must pass through its initial point and remain tangent to the short field segments it crosses. It should not be drawn by connecting segment endpoints.

233. Kaart 233

     Vraag

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

     Antwoord

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

234. Kaart 234

     Vraag

     Why is one integration constant enough after integrating both sides?

     Antwoord

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

235. Kaart 235

     Vraag

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

     Antwoord

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

236. Kaart 236

     Vraag

     Can one differential equation have infinitely many solutions?

     Antwoord

     Yes. A differential equation commonly describes a family of solution functions. An initial condition may select one particular solution when the relevant conditions support uniqueness.

237. Kaart 237

     Vraag

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

     Antwoord

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

238. Kaart 238

     Vraag

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

     Antwoord

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

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

239. Kaart 239

     Vraag

     What can be lost when dividing to separate variables?

     Antwoord

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

240. Kaart 240

     Vraag

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

     Antwoord

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

241. Kaart 241

     Vraag

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

     Antwoord

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

242. Kaart 242

     Vraag

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

     Antwoord

     The solution is increasing where $dy/dx>0$ and decreasing where $dy/dx<0$. At one point, $dy/dx=0$ gives a horizontal tangent; a constant $y=c$ is an equilibrium only when the derivative equation gives zero all along that level.

243. Kaart 243

     Vraag

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

     Antwoord

     Differentiate the equation with respect to the independent variable to obtain $y''$, using the chain rule for any $y$-dependence. Then use the sign of $y''$ along the solution.

244. Kaart 244

     Vraag

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

     Antwoord

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

245. Kaart 245

     Vraag

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

     Antwoord

     For $k>0$,

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

     It is independent of the initial amount.

246. Kaart 246

     Vraag

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

     Antwoord

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

247. Kaart 247

     Vraag

     How is an initial condition used after separation?

     Antwoord

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

248. Kaart 248

     Vraag

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

     Antwoord

     The right side must have the same units as $dy/dt$: units of $y$ per unit of $t$. A mismatch signals an incorrect translation or parameter unit.

249. Kaart 249

     Vraag

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

     Antwoord

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

250. Kaart 250

     Vraag

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

     Antwoord

     For $k<0$,

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

251. Kaart 251

     Vraag

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

     Antwoord

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

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

252. Kaart 252

     Vraag

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

     Antwoord

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

     Velocity below zero contributes negative displacement.

253. Kaart 253

     Vraag

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

     Antwoord

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

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

     Think top minus bottom.

254. Kaart 254

     Vraag

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

     Antwoord

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

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

255. Kaart 255

     Vraag

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Antwoord

     If $f$ is continuous on $[a,b]$, then some $c\in[a,b]$ satisfies

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

     If $a<b$, a point can also be chosen in $(a,b)$.

256. Kaart 256

     Vraag

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

     Antwoord

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

257. Kaart 257

     Vraag

     Cross-sectional area when each slice is a square?

     Antwoord

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

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

258. Kaart 258

     Vraag

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

     Antwoord

     Integrate the net rate:

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

259. Kaart 259

     Vraag

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

     Antwoord

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

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

     Think right minus left.

260. Kaart 260

     Vraag

     Disc-method volume formula?

     Antwoord

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

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

261. Kaart 261

     Vraag

     What units does average value have?

     Antwoord

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

262. Kaart 262

     Vraag

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

     Antwoord

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

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

263. Kaart 263

     Vraag

     Cross-sectional area when each slice is a rectangle?

     Antwoord

     If the slice has base $b(x)$ and height $h(x)$, then

     $$A(x)=b(x)h(x).$$

     Use the stated relationship to express both in the integration variable.

264. Kaart 264

     Vraag

     How do you determine bounds for area between curves?

     Antwoord

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

265. Kaart 265

     Vraag

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

     Antwoord

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

266. Kaart 266

     Vraag

     When is a particle moving to the right or left?

     Antwoord

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

267. Kaart 267

     Vraag

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

     Antwoord

     The radius is $d(x)/2$, so

     $$A(x)=\frac12\pi\left(\frac{d(x)}2\right)^2=\frac{\pi}{8}[d(x)]^2.$$

268. Kaart 268

     Vraag

     Why must an area integral be split where curves intersect?

     Antwoord

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

269. Kaart 269

     Vraag

     How can a velocity table approximate displacement?

     Antwoord

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

270. Kaart 270

     Vraag

     Washer-method volume formula?

     Antwoord

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

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

271. Kaart 271

     Vraag

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

     Antwoord

     If $s(a)$ is known,

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

272. Kaart 272

     Vraag

     How do you choose between vertical and horizontal area slices?

     Antwoord

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

273. Kaart 273

     Vraag

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

     Antwoord

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

274. Kaart 274

     Vraag

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

     Antwoord

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

275. Kaart 275

     Vraag

     Single expression for area between two curves?

     Antwoord

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kaart 276

     Vraag

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

     Antwoord

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

277. Kaart 277

     Vraag

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

     Antwoord

     Multiply a representative rate on each interval by that interval's actual width, then sum. Do not assume a common $\Delta t$ when the table is uneven.

278. Kaart 278

     Vraag

     When should a volume integral use $dy$?

     Antwoord

     When slices perpendicular to the $y$-axis make the cross-sectional area easiest to express as $A(y)$. Then use $V=\int_c^d A(y)\,dy$.

279. Kaart 279

     Vraag

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

     Antwoord

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

280. Kaart 280

     Vraag

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

     Antwoord

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

281. Kaart 281

     Vraag

     Why must total distance split at velocity sign changes?

     Antwoord

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

282. Kaart 282

     Vraag

     When does an accumulated quantity reach a local maximum?

     Antwoord

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

283. Kaart 283

     Vraag

     What distinguishes area from a definite integral?

     Antwoord

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

284. Kaart 284

     Vraag

     How do position, velocity, and acceleration graphs correspond?

     Antwoord

     Velocity is the slope of position, and acceleration is the slope of velocity. Conversely, signed area under velocity gives position change, and signed area under acceleration gives velocity change.

285. Kaart 285

     Vraag

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

     Antwoord

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

286. Kaart 286

     Vraag

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

     Antwoord

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

287. Kaart 287

     Vraag

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

     Antwoord

     Compare accumulation at endpoints and at times where the rate is zero or undefined. Compute signed areas to track the quantity's changes from its initial value.

288. Kaart 288

     Vraag

     Why should a contextual integral answer include a sentence?

     Antwoord

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