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

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1. การ์ด 1

   คำถาม

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

   คำตอบ

   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. การ์ด 2

   คำถาม

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

   คำตอบ

   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. การ์ด 3

   คำถาม

   When does direct substitution evaluate a limit?

   คำตอบ

   When the function is continuous at the target input. Then

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

4. การ์ด 4

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   Three conditions for continuity at $x=a$?

   คำตอบ

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

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

5. การ์ด 5

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   Intermediate Value Theorem: hypotheses and conclusion?

   คำตอบ

   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. การ์ด 6

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

   คำตอบ

   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. การ์ด 7

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   How do you read a finite limit from a graph?

   คำตอบ

   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. การ์ด 8

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   Limit law for a sum or difference?

   คำตอบ

   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. การ์ด 9

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   What makes a discontinuity removable?

   คำตอบ

   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. การ์ด 10

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    Squeeze Theorem: usable form?

    คำตอบ

    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. การ์ด 11

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

    คำตอบ

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

12. การ์ด 12

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

    คำตอบ

    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. การ์ด 13

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    Limit law for a product?

    คำตอบ

    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. การ์ด 14

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    Graph signature of a jump discontinuity?

    คำตอบ

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

15. การ์ด 15

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

    คำตอบ

    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. การ์ด 16

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    Horizontal asymptote from a limit at infinity?

    คำตอบ

    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. การ์ด 17

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

    คำตอบ

    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. การ์ด 18

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    Limit law for a quotient—and its condition?

    คำตอบ

    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. การ์ด 19

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

    คำตอบ

    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. การ์ด 20

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    When is the Squeeze Theorem a natural choice?

    คำตอบ

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

21. การ์ด 21

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    Vertical asymptote from one-sided behavior?

    คำตอบ

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

22. การ์ด 22

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    Limit at infinity of equal-degree rational functions?

    คำตอบ

    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. การ์ด 23

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    When can a limit pass through a continuous outer function?

    คำตอบ

    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. การ์ด 24

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    What makes a discontinuity infinite?

    คำตอบ

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

25. การ์ด 25

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

    คำตอบ

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

26. การ์ด 26

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

    คำตอบ

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. การ์ด 27

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    Continuity of a composition?

    คำตอบ

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

28. การ์ด 28

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    Limit at infinity when a rational numerator has lower degree?

    คำตอบ

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

29. การ์ด 29

    คำถาม

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

    คำตอบ

    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. การ์ด 30

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

    คำตอบ

    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. การ์ด 31

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

    คำตอบ

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

32. การ์ด 32

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

    คำตอบ

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

33. การ์ด 33

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

    คำตอบ

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

34. การ์ด 34

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

    คำตอบ

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

35. การ์ด 35

    คำถาม

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

    คำตอบ

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

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

36. การ์ด 36

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

    คำตอบ

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

    The derivative exists only if this finite limit exists.

37. การ์ด 37

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

    คำตอบ

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

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

38. การ์ด 38

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    What does differentiability imply about continuity?

    คำตอบ

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

39. การ์ด 39

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    Power rule for derivatives?

    คำตอบ

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

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

40. การ์ด 40

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    Units of $f'(x)$?

    คำตอบ

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

41. การ์ด 41

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

    คำตอบ

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

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

42. การ์ด 42

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

    คำตอบ

    $f'(x)>0$ 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. การ์ด 43

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    Derivative of a constant?

    คำตอบ

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

    A constant function has zero rate of change.

44. การ์ด 44

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    Derivative of $\sin x$?

    คำตอบ

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

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

45. การ์ด 45

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    Product rule?

    คำตอบ

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

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

46. การ์ด 46

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

    คำตอบ

    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. การ์ด 47

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

    คำตอบ

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

48. การ์ด 48

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

    คำตอบ

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

49. การ์ด 49

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    Derivative of a sum or difference?

    คำตอบ

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

50. การ์ด 50

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    Derivative of $\cos x$?

    คำตอบ

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

    The standard formula assumes radians.

51. การ์ด 51

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    Quotient rule?

    คำตอบ

    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. การ์ด 52

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    Common notations for the first derivative?

    คำตอบ

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

53. การ์ด 53

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    Derivative of $e^x$?

    คำตอบ

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

54. การ์ด 54

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    What graph features can make $f$ nondifferentiable?

    คำตอบ

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

55. การ์ด 55

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    Derivative of $\tan x$?

    คำตอบ

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. การ์ด 56

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

    คำตอบ

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

57. การ์ด 57

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    Derivative of $\ln x$?

    คำตอบ

    For $x>0$,

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

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

58. การ์ด 58

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    How does the power rule handle roots or negative powers?

    คำตอบ

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

59. การ์ด 59

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    Derivative of $\csc x$?

    คำตอบ

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. การ์ด 60

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

    คำตอบ

    $f$ is increasing on that interval.

61. การ์ด 61

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

    คำตอบ

    For $a>0$,

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

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

62. การ์ด 62

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

    คำตอบ

    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. การ์ด 63

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    Derivative of $\sec x$?

    คำตอบ

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. การ์ด 64

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

    คำตอบ

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

65. การ์ด 65

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    Derivative of $\log_a x$?

    คำตอบ

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

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

66. การ์ด 66

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

    คำตอบ

    For $h=fg$,

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

    Use the four table entries at the same input.

67. การ์ด 67

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    Derivative of $\cot x$?

    คำตอบ

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. การ์ด 68

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

    คำตอบ

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

69. การ์ด 69

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    Constant-multiple rule?

    คำตอบ

    For a constant $c$,

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

70. การ์ด 70

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

    คำตอบ

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

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

71. การ์ด 71

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

    คำตอบ

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

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

72. การ์ด 72

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

    คำตอบ

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

73. การ์ด 73

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

    คำตอบ

    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. การ์ด 74

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    Derivative of an inverse function at $x$?

    คำตอบ

    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. การ์ด 75

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    Derivative of $\arcsin x$?

    คำตอบ

    For $|x|<1$,

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

76. การ์ด 76

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    Notation for the third derivative of $f$?

    คำตอบ

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

77. การ์ด 77

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

    คำตอบ

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

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

78. การ์ด 78

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

    คำตอบ

    Where $y\ne0$,

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

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

79. การ์ด 79

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

    คำตอบ

    Provided $f'(a)\ne0$,

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

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

80. การ์ด 80

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    Derivative of $\arctan x$?

    คำตอบ

    For every real $x$,

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

81. การ์ด 81

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

    คำตอบ

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

    The extra factor is the chain rule.

82. การ์ด 82

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

    คำตอบ

    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. การ์ด 83

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

    คำตอบ

    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. การ์ด 84

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    Derivative of $\arccos x$?

    คำตอบ

    For $|x|<1$,

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

85. การ์ด 85

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

    คำตอบ

    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. การ์ด 86

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

    คำตอบ

    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. การ์ด 87

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

    คำตอบ

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

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

88. การ์ด 88

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    How are tangent slopes of inverse graphs related?

    คำตอบ

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

89. การ์ด 89

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

    คำตอบ

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

90. การ์ด 90

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

    คำตอบ

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

91. การ์ด 91

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    Horizontal tangent on an implicit curve: derivative condition?

    คำตอบ

    $dy/dx=0$ 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. การ์ด 92

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

    คำตอบ

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

93. การ์ด 93

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

    คำตอบ

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

94. การ์ด 94

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

    คำตอบ

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

    This combines the power rule with the chain rule.

95. การ์ด 95

    คำถาม

    Vertical tangent on an implicit curve: derivative clue?

    คำตอบ

    $dy/dx$ 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. การ์ด 96

    คำถาม

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

    คำตอบ

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

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

97. การ์ด 97

    คำถาม

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

    คำตอบ

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

98. การ์ด 98

    คำถาม

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

    คำตอบ

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

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

99. การ์ด 99

    คำถาม

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

    คำตอบ

    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. การ์ด 100

     คำถาม

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

     คำตอบ

     When the functions are differentiable, the chain rule gives

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

101. การ์ด 101

     คำถาม

     What local property lets a function have an inverse derivative?

     คำตอบ

     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. การ์ด 102

     คำถาม

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

     คำตอบ

     For $a>0$,

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

103. การ์ด 103

     คำถาม

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

     คำตอบ

     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. การ์ด 104

     คำถาม

     Position, velocity, and acceleration relationships?

     คำตอบ

     For position $s(t)$,

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

105. การ์ด 105

     คำถาม

     Central idea of a related-rates problem?

     คำตอบ

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

106. การ์ด 106

     คำถาม

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

     คำตอบ

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

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

107. การ์ด 107

     คำถาม

     L’Hospital’s Rule: basic conditions?

     คำตอบ

     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. การ์ด 108

     คำถาม

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

     คำตอบ

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

109. การ์ด 109

     คำถาม

     Speed in terms of velocity?

     คำตอบ

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

     Velocity includes direction; speed is nonnegative magnitude.

110. การ์ด 110

     คำถาม

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

     คำตอบ

     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. การ์ด 111

     คำถาม

     Differential approximation connecting $dx$ and $dy$?

     คำตอบ

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

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

112. การ์ด 112

     คำถาม

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

     คำตอบ

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

113. การ์ด 113

     คำถาม

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

     คำตอบ

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

114. การ์ด 114

     คำถาม

     What does positive acceleration say about velocity?

     คำตอบ

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

115. การ์ด 115

     คำถาม

     Related rates: when should numerical values be substituted?

     คำตอบ

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

116. การ์ด 116

     คำถาม

     How does concavity predict linearization error?

     คำตอบ

     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. การ์ด 117

     คำถาม

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

     คำตอบ

     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. การ์ด 118

     คำถาม

     When is a particle moving in the positive direction?

     คำตอบ

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

119. การ์ด 119

     คำถาม

     How can velocity show a change of direction?

     คำตอบ

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

120. การ์ด 120

     คำถาม

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

     คำตอบ

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

121. การ์ด 121

     คำถาม

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

     คำตอบ

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

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

122. การ์ด 122

     คำถาม

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

     คำตอบ

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

123. การ์ด 123

     คำถาม

     What must a contextual derivative sentence include?

     คำตอบ

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

124. การ์ด 124

     คำถาม

     Velocity negative and acceleration positive: what happens?

     คำตอบ

     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. การ์ด 125

     คำถาม

     How should a negative related rate be interpreted?

     คำตอบ

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

126. การ์ด 126

     คำถาม

     When is local linearity a sound approximation tool?

     คำตอบ

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

127. การ์ด 127

     คำถาม

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

     คำตอบ

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

128. การ์ด 128

     คำถาม

     When is speed increasing?

     คำตอบ

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

129. การ์ด 129

     คำถาม

     Volume changes with time: notation for its rate?

     คำตอบ

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

130. การ์ด 130

     คำถาม

     Why are similar triangles useful in related rates?

     คำตอบ

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

131. การ์ด 131

     คำถาม

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

     คำตอบ

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

132. การ์ด 132

     คำถาม

     What conclusion does L’Hospital’s Rule permit?

     คำตอบ

     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. การ์ด 133

     คำถาม

     When is speed decreasing?

     คำตอบ

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

134. การ์ด 134

     คำถาม

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

     คำตอบ

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

135. การ์ด 135

     คำถาม

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

     คำตอบ

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

136. การ์ด 136

     คำถาม

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

     คำตอบ

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

137. การ์ด 137

     คำถาม

     Extreme Value Theorem: hypothesis and conclusion?

     คำตอบ

     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. การ์ด 138

     คำถาม

     What is a critical number of $f$?

     คำตอบ

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

139. การ์ด 139

     คำถาม

     First derivative test for a local maximum?

     คำตอบ

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

140. การ์ด 140

     คำถาม

     Second-derivative sign for concave up?

     คำตอบ

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

141. การ์ด 141

     คำถาม

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

     คำตอบ

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

142. การ์ด 142

     คำถาม

     First step in an optimization model?

     คำตอบ

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

143. การ์ด 143

     คำถาม

     Mean Value Theorem: hypotheses and conclusion?

     คำตอบ

     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. การ์ด 144

     คำถาม

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

     คำตอบ

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

     

     288 การ์ด

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

     เรียนเด็คนี้ฟรี

     Nibomo จะเปิดขึ้นเพื่อให้คุณเริ่มเรียนได้ทันที

145. การ์ด 145

     คำถาม

     First derivative test for a local minimum?

     คำตอบ

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

146. การ์ด 146

     คำถาม

     What must happen at an inflection point?

     คำตอบ

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

147. การ์ด 147

     คำถาม

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

     คำตอบ

     $f''$ 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. การ์ด 148

     คำถาม

     How do you confirm an optimization answer is absolute?

     คำตอบ

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

149. การ์ด 149

     คำถาม

     Rolle’s Theorem: hypotheses and conclusion?

     คำตอบ

     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. การ์ด 150

     คำถาม

     Difference between absolute and relative extrema?

     คำตอบ

     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. การ์ด 151

     คำถาม

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

     คำตอบ

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

152. การ์ด 152

     คำถาม

     Second derivative test for a local minimum?

     คำตอบ

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

153. การ์ด 153

     คำถาม

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

     คำตอบ

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

154. การ์ด 154

     คำถาม

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

     คำตอบ

     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. การ์ด 155

     คำถาม

     Which theorem links an average slope to an instantaneous slope?

     คำตอบ

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

156. การ์ด 156

     คำถาม

     How can an implicit derivative locate a horizontal tangent?

     คำตอบ

     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. การ์ด 157

     คำถาม

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

     คำตอบ

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

158. การ์ด 158

     คำถาม

     Second derivative test for a local maximum?

     คำตอบ

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

159. การ์ด 159

     คำถาม

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

     คำตอบ

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

160. การ์ด 160

     คำถาม

     Why must an optimization domain be stated?

     คำตอบ

     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. การ์ด 161

     คำถาม

     Which theorem guarantees absolute extrema, not where they occur?

     คำตอบ

     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. การ์ด 162

     คำถาม

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

     คำตอบ

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

163. การ์ด 163

     คำถาม

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

     คำตอบ

     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. การ์ด 164

     คำถาม

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

     คำตอบ

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

165. การ์ด 165

     คำถาม

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

     คำตอบ

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

166. การ์ด 166

     คำถาม

     How can an implicit derivative locate a vertical tangent?

     คำตอบ

     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. การ์ด 167

     คำถาม

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

     คำตอบ

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

168. การ์ด 168

     คำถาม

     Why are endpoints included in the candidates test?

     คำตอบ

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

169. การ์ด 169

     คำถาม

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

     คำตอบ

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

170. การ์ด 170

     คำถาม

     Second-derivative sign for concave down?

     คำตอบ

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

171. การ์ด 171

     คำถาม

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

     คำตอบ

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

172. การ์ด 172

     คำถาม

     What should the final line of an optimization solution state?

     คำตอบ

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

173. การ์ด 173

     คำถาม

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

     คำตอบ

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

174. การ์ด 174

     คำถาม

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

     คำตอบ

     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. การ์ด 175

     คำถาม

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

     คำตอบ

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

176. การ์ด 176

     คำถาม

     Left Riemann sum on equal subintervals?

     คำตอบ

     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. การ์ด 177

     คำถาม

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

     คำตอบ

     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. การ์ด 178

     คำถาม

     Fundamental Theorem of Calculus: evaluate a definite integral?

     คำตอบ

     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. การ์ด 179

     คำถาม

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

     คำตอบ

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. การ์ด 180

     คำถาม

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

     คำตอบ

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

181. การ์ด 181

     คำถาม

     Right Riemann sum on equal subintervals?

     คำตอบ

     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. การ์ด 182

     คำถาม

     How does reversing integral bounds change the value?

     คำตอบ

     It changes the sign:

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

183. การ์ด 183

     คำถาม

     Net Change Theorem?

     คำตอบ

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

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

184. การ์ด 184

     คำถาม

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

     คำตอบ

     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. การ์ด 185

     คำถาม

     Power rule for antiderivatives?

     คำตอบ

     For $n\ne-1$,

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

186. การ์ด 186

     คำถาม

     Midpoint Riemann sum on equal subintervals?

     คำตอบ

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

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

187. การ์ด 187

     คำถาม

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

     คำตอบ

     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. การ์ด 188

     คำถาม

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

     คำตอบ

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. การ์ด 189

     คำถาม

     Antiderivative of $1/x$?

     คำตอบ

     On any interval not crossing zero,

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

190. การ์ด 190

     คำถาม

     Trapezoidal approximation on equal subintervals?

     คำตอบ

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

191. การ์ด 191

     คำถาม

     How do geometric regions help evaluate a definite integral?

     คำตอบ

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

192. การ์ด 192

     คำถาม

     Basic antiderivatives of sine and cosine?

     คำตอบ

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

193. การ์ด 193

     คำถาม

     Definite integral as a limit of Riemann sums?

     คำตอบ

     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. การ์ด 194

     คำถาม

     Constant-multiple rule for integrals?

     คำตอบ

     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. การ์ด 195

     คำถาม

     What pattern suggests $u$-substitution?

     คำตอบ

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

196. การ์ด 196

     คำถาม

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

     คำตอบ

     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. การ์ด 197

     คำถาม

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

     คำตอบ

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

198. การ์ด 198

     คำถาม

     Sum-and-difference rule for definite integrals?

     คำตอบ

     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. การ์ด 199

     คำถาม

     Basic antiderivative of $e^x$?

     คำตอบ

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

200. การ์ด 200

     คำถาม

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

     คำตอบ

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

201. การ์ด 201

     คำถาม

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

     คำตอบ

     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. การ์ด 202

     คำถาม

     How does concavity predict trapezoidal and midpoint error?

     คำตอบ

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

203. การ์ด 203

     คำถาม

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

     คำตอบ

     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. การ์ด 204

     คำถาม

     What denominator pattern suggests an arctangent antiderivative?

     คำตอบ

     After completing the square and scaling, a form like

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

205. การ์ด 205

     คำถาม

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

     คำตอบ

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

206. การ์ด 206

     คำถาม

     How does an initial condition determine an antiderivative?

     คำตอบ

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

207. การ์ด 207

     คำถาม

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

     คำตอบ

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

208. การ์ด 208

     คำถาม

     Why does an indefinite integral include $+C$?

     คำตอบ

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

209. การ์ด 209

     คำถาม

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

     คำตอบ

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

210. การ์ด 210

     คำถาม

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

     คำตอบ

     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. การ์ด 211

     คำถาม

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

     คำตอบ

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

212. การ์ด 212

     คำถาม

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

     คำตอบ

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

213. การ์ด 213

     คำถาม

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

     คำตอบ

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

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

214. การ์ด 214

     คำถาม

     Riemann sum for unequal subinterval widths?

     คำตอบ

     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. การ์ด 215

     คำถาม

     Does continuity guarantee integrability on a closed interval?

     คำตอบ

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

216. การ์ด 216

     คำถาม

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

     คำตอบ

     Where $g(x)\ne0$,

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

217. การ์ด 217

     คำถาม

     What algebraic rewrites often reveal a basic antiderivative?

     คำตอบ

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

218. การ์ด 218

     คำถาม

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

     คำตอบ

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

219. การ์ด 219

     คำถาม

     What is a differential equation?

     คำตอบ

     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. การ์ด 220

     คำถาม

     How does a verbal rate statement become a differential equation?

     คำตอบ

     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. การ์ด 221

     คำถาม

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

     คำตอบ

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

222. การ์ด 222

     คำถาม

     General solution versus particular solution?

     คำตอบ

     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. การ์ด 223

     คำถาม

     What does one segment in a slope field show?

     คำตอบ

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

224. การ์ด 224

     คำถาม

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

     คำตอบ

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

225. การ์ด 225

     คำถาม

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

     คำตอบ

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

226. การ์ด 226

     คำถาม

     What makes a first-order differential equation separable?

     คำตอบ

     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. การ์ด 227

     คำถาม

     What is an initial value problem?

     คำตอบ

     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. การ์ด 228

     คำถาม

     What is an isocline in a slope field?

     คำตอบ

     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. การ์ด 229

     คำถาม

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

     คำตอบ

     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. การ์ด 230

     คำถาม

     General solution of $dy/dt=ky$?

     คำตอบ

     $$y=Ce^{kt}$$

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

231. การ์ด 231

     คำถาม

     Core method for solving a separable differential equation?

     คำตอบ

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

232. การ์ด 232

     คำถาม

     How should a solution curve follow a slope field?

     คำตอบ

     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. การ์ด 233

     คำถาม

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

     คำตอบ

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

234. การ์ด 234

     คำถาม

     Why is one integration constant enough after integrating both sides?

     คำตอบ

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

235. การ์ด 235

     คำถาม

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

     คำตอบ

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

236. การ์ด 236

     คำถาม

     Can one differential equation have infinitely many solutions?

     คำตอบ

     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. การ์ด 237

     คำถาม

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

     คำตอบ

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

238. การ์ด 238

     คำถาม

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

     คำตอบ

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

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

239. การ์ด 239

     คำถาม

     What can be lost when dividing to separate variables?

     คำตอบ

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

240. การ์ด 240

     คำถาม

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

     คำตอบ

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

241. การ์ด 241

     คำถาม

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

     คำตอบ

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

242. การ์ด 242

     คำถาม

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

     คำตอบ

     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. การ์ด 243

     คำถาม

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

     คำตอบ

     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. การ์ด 244

     คำถาม

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

     คำตอบ

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

245. การ์ด 245

     คำถาม

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

     คำตอบ

     For $k>0$,

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

     It is independent of the initial amount.

246. การ์ด 246

     คำถาม

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

     คำตอบ

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

247. การ์ด 247

     คำถาม

     How is an initial condition used after separation?

     คำตอบ

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

248. การ์ด 248

     คำถาม

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

     คำตอบ

     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. การ์ด 249

     คำถาม

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

     คำตอบ

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

250. การ์ด 250

     คำถาม

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

     คำตอบ

     For $k<0$,

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

251. การ์ด 251

     คำถาม

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

     คำตอบ

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

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

252. การ์ด 252

     คำถาม

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

     คำตอบ

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

     Velocity below zero contributes negative displacement.

253. การ์ด 253

     คำถาม

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

     คำตอบ

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

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

     Think top minus bottom.

254. การ์ด 254

     คำถาม

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

     คำตอบ

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

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

255. การ์ด 255

     คำถาม

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     คำตอบ

     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. การ์ด 256

     คำถาม

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

     คำตอบ

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

257. การ์ด 257

     คำถาม

     Cross-sectional area when each slice is a square?

     คำตอบ

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

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

258. การ์ด 258

     คำถาม

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

     คำตอบ

     Integrate the net rate:

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

259. การ์ด 259

     คำถาม

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

     คำตอบ

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

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

     Think right minus left.

260. การ์ด 260

     คำถาม

     Disc-method volume formula?

     คำตอบ

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

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

261. การ์ด 261

     คำถาม

     What units does average value have?

     คำตอบ

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

262. การ์ด 262

     คำถาม

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

     คำตอบ

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

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

263. การ์ด 263

     คำถาม

     Cross-sectional area when each slice is a rectangle?

     คำตอบ

     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. การ์ด 264

     คำถาม

     How do you determine bounds for area between curves?

     คำตอบ

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

265. การ์ด 265

     คำถาม

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

     คำตอบ

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

266. การ์ด 266

     คำถาม

     When is a particle moving to the right or left?

     คำตอบ

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

267. การ์ด 267

     คำถาม

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

     คำตอบ

     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. การ์ด 268

     คำถาม

     Why must an area integral be split where curves intersect?

     คำตอบ

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

269. การ์ด 269

     คำถาม

     How can a velocity table approximate displacement?

     คำตอบ

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

270. การ์ด 270

     คำถาม

     Washer-method volume formula?

     คำตอบ

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

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

271. การ์ด 271

     คำถาม

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

     คำตอบ

     If $s(a)$ is known,

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

272. การ์ด 272

     คำถาม

     How do you choose between vertical and horizontal area slices?

     คำตอบ

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

273. การ์ด 273

     คำถาม

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

     คำตอบ

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

274. การ์ด 274

     คำถาม

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

     คำตอบ

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

275. การ์ด 275

     คำถาม

     Single expression for area between two curves?

     คำตอบ

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. การ์ด 276

     คำถาม

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

     คำตอบ

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

277. การ์ด 277

     คำถาม

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

     คำตอบ

     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. การ์ด 278

     คำถาม

     When should a volume integral use $dy$?

     คำตอบ

     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. การ์ด 279

     คำถาม

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

     คำตอบ

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

280. การ์ด 280

     คำถาม

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

     คำตอบ

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

281. การ์ด 281

     คำถาม

     Why must total distance split at velocity sign changes?

     คำตอบ

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

282. การ์ด 282

     คำถาม

     When does an accumulated quantity reach a local maximum?

     คำตอบ

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

283. การ์ด 283

     คำถาม

     What distinguishes area from a definite integral?

     คำตอบ

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

284. การ์ด 284

     คำถาม

     How do position, velocity, and acceleration graphs correspond?

     คำตอบ

     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. การ์ด 285

     คำถาม

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

     คำตอบ

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

286. การ์ด 286

     คำถาม

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

     คำตอบ

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

287. การ์ด 287

     คำถาม

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

     คำตอบ

     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. การ์ด 288

     คำถาม

     Why should a contextual integral answer include a sentence?

     คำตอบ

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