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

Kadi za fungu hili

1. Kadi namba 1

   Swali

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

   Jibu

   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. Kadi namba 2

   Swali

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

   Jibu

   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. Kadi namba 3

   Swali

   When does direct substitution evaluate a limit?

   Jibu

   When the function is continuous at the target input. Then

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

4. Kadi namba 4

   Swali

   Three conditions for continuity at $x=a$?

   Jibu

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

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

5. Kadi namba 5

   Swali

   Intermediate Value Theorem: hypotheses and conclusion?

   Jibu

   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. Kadi namba 6

   Swali

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

   Jibu

   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. Kadi namba 7

   Swali

   How do you read a finite limit from a graph?

   Jibu

   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. Kadi namba 8

   Swali

   Limit law for a sum or difference?

   Jibu

   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. Kadi namba 9

   Swali

   What makes a discontinuity removable?

   Jibu

   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. Kadi namba 10

    Swali

    Squeeze Theorem: usable form?

    Jibu

    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. Kadi namba 11

    Swali

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

    Jibu

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

12. Kadi namba 12

    Swali

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

    Jibu

    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. Kadi namba 13

    Swali

    Limit law for a product?

    Jibu

    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. Kadi namba 14

    Swali

    Graph signature of a jump discontinuity?

    Jibu

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

15. Kadi namba 15

    Swali

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

    Jibu

    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. Kadi namba 16

    Swali

    Horizontal asymptote from a limit at infinity?

    Jibu

    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. Kadi namba 17

    Swali

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

    Jibu

    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. Kadi namba 18

    Swali

    Limit law for a quotient—and its condition?

    Jibu

    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. Kadi namba 19

    Swali

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

    Jibu

    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. Kadi namba 20

    Swali

    When is the Squeeze Theorem a natural choice?

    Jibu

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

21. Kadi namba 21

    Swali

    Vertical asymptote from one-sided behavior?

    Jibu

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

22. Kadi namba 22

    Swali

    Limit at infinity of equal-degree rational functions?

    Jibu

    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. Kadi namba 23

    Swali

    When can a limit pass through a continuous outer function?

    Jibu

    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. Kadi namba 24

    Swali

    What makes a discontinuity infinite?

    Jibu

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

25. Kadi namba 25

    Swali

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

    Jibu

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

26. Kadi namba 26

    Swali

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

    Jibu

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kadi namba 27

    Swali

    Continuity of a composition?

    Jibu

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

28. Kadi namba 28

    Swali

    Limit at infinity when a rational numerator has lower degree?

    Jibu

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

29. Kadi namba 29

    Swali

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

    Jibu

    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. Kadi namba 30

    Swali

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

    Jibu

    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. Kadi namba 31

    Swali

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

    Jibu

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

32. Kadi namba 32

    Swali

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

    Jibu

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

33. Kadi namba 33

    Swali

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

    Jibu

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

34. Kadi namba 34

    Swali

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

    Jibu

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

35. Kadi namba 35

    Swali

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

    Jibu

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

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

36. Kadi namba 36

    Swali

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

    Jibu

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

    The derivative exists only if this finite limit exists.

37. Kadi namba 37

    Swali

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

    Jibu

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

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

38. Kadi namba 38

    Swali

    What does differentiability imply about continuity?

    Jibu

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

39. Kadi namba 39

    Swali

    Power rule for derivatives?

    Jibu

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

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

40. Kadi namba 40

    Swali

    Units of $f'(x)$?

    Jibu

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

41. Kadi namba 41

    Swali

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

    Jibu

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

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

42. Kadi namba 42

    Swali

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

    Jibu

    $f'(x)>0$ 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. Kadi namba 43

    Swali

    Derivative of a constant?

    Jibu

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

    A constant function has zero rate of change.

44. Kadi namba 44

    Swali

    Derivative of $\sin x$?

    Jibu

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

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

45. Kadi namba 45

    Swali

    Product rule?

    Jibu

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

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

46. Kadi namba 46

    Swali

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

    Jibu

    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. Kadi namba 47

    Swali

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

    Jibu

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

48. Kadi namba 48

    Swali

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

    Jibu

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

49. Kadi namba 49

    Swali

    Derivative of a sum or difference?

    Jibu

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

50. Kadi namba 50

    Swali

    Derivative of $\cos x$?

    Jibu

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

    The standard formula assumes radians.

51. Kadi namba 51

    Swali

    Quotient rule?

    Jibu

    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. Kadi namba 52

    Swali

    Common notations for the first derivative?

    Jibu

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

53. Kadi namba 53

    Swali

    Derivative of $e^x$?

    Jibu

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

54. Kadi namba 54

    Swali

    What graph features can make $f$ nondifferentiable?

    Jibu

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

55. Kadi namba 55

    Swali

    Derivative of $\tan x$?

    Jibu

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kadi namba 56

    Swali

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

    Jibu

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

57. Kadi namba 57

    Swali

    Derivative of $\ln x$?

    Jibu

    For $x>0$,

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

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

58. Kadi namba 58

    Swali

    How does the power rule handle roots or negative powers?

    Jibu

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

59. Kadi namba 59

    Swali

    Derivative of $\csc x$?

    Jibu

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kadi namba 60

    Swali

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

    Jibu

    $f$ is increasing on that interval.

61. Kadi namba 61

    Swali

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

    Jibu

    For $a>0$,

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

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

62. Kadi namba 62

    Swali

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

    Jibu

    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. Kadi namba 63

    Swali

    Derivative of $\sec x$?

    Jibu

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kadi namba 64

    Swali

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

    Jibu

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

65. Kadi namba 65

    Swali

    Derivative of $\log_a x$?

    Jibu

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

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

66. Kadi namba 66

    Swali

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

    Jibu

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kadi namba 67

    Swali

    Derivative of $\cot x$?

    Jibu

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kadi namba 68

    Swali

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

    Jibu

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

69. Kadi namba 69

    Swali

    Constant-multiple rule?

    Jibu

    For a constant $c$,

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

70. Kadi namba 70

    Swali

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

    Jibu

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

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

71. Kadi namba 71

    Swali

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

    Jibu

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

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

72. Kadi namba 72

    Swali

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

    Jibu

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

73. Kadi namba 73

    Swali

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

    Jibu

    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. Kadi namba 74

    Swali

    Derivative of an inverse function at $x$?

    Jibu

    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. Kadi namba 75

    Swali

    Derivative of $\arcsin x$?

    Jibu

    For $|x|<1$,

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

76. Kadi namba 76

    Swali

    Notation for the third derivative of $f$?

    Jibu

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

77. Kadi namba 77

    Swali

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

    Jibu

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

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

78. Kadi namba 78

    Swali

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

    Jibu

    Where $y\ne0$,

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

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

79. Kadi namba 79

    Swali

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

    Jibu

    Provided $f'(a)\ne0$,

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

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

80. Kadi namba 80

    Swali

    Derivative of $\arctan x$?

    Jibu

    For every real $x$,

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

81. Kadi namba 81

    Swali

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

    Jibu

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

    The extra factor is the chain rule.

82. Kadi namba 82

    Swali

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

    Jibu

    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. Kadi namba 83

    Swali

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

    Jibu

    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. Kadi namba 84

    Swali

    Derivative of $\arccos x$?

    Jibu

    For $|x|<1$,

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

85. Kadi namba 85

    Swali

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

    Jibu

    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. Kadi namba 86

    Swali

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

    Jibu

    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. Kadi namba 87

    Swali

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

    Jibu

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

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

88. Kadi namba 88

    Swali

    How are tangent slopes of inverse graphs related?

    Jibu

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

89. Kadi namba 89

    Swali

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

    Jibu

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

90. Kadi namba 90

    Swali

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

    Jibu

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

91. Kadi namba 91

    Swali

    Horizontal tangent on an implicit curve: derivative condition?

    Jibu

    $dy/dx=0$ 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. Kadi namba 92

    Swali

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

    Jibu

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

93. Kadi namba 93

    Swali

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

    Jibu

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

94. Kadi namba 94

    Swali

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

    Jibu

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

    This combines the power rule with the chain rule.

95. Kadi namba 95

    Swali

    Vertical tangent on an implicit curve: derivative clue?

    Jibu

    $dy/dx$ 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. Kadi namba 96

    Swali

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

    Jibu

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

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

97. Kadi namba 97

    Swali

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

    Jibu

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

98. Kadi namba 98

    Swali

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

    Jibu

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

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

99. Kadi namba 99

    Swali

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

    Jibu

    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. Kadi namba 100

     Swali

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

     Jibu

     When the functions are differentiable, the chain rule gives

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

101. Kadi namba 101

     Swali

     What local property lets a function have an inverse derivative?

     Jibu

     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. Kadi namba 102

     Swali

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

     Jibu

     For $a>0$,

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

103. Kadi namba 103

     Swali

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

     Jibu

     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. Kadi namba 104

     Swali

     Position, velocity, and acceleration relationships?

     Jibu

     For position $s(t)$,

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

105. Kadi namba 105

     Swali

     Central idea of a related-rates problem?

     Jibu

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

106. Kadi namba 106

     Swali

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

     Jibu

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

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

107. Kadi namba 107

     Swali

     L’Hospital’s Rule: basic conditions?

     Jibu

     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. Kadi namba 108

     Swali

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

     Jibu

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

109. Kadi namba 109

     Swali

     Speed in terms of velocity?

     Jibu

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kadi namba 110

     Swali

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

     Jibu

     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. Kadi namba 111

     Swali

     Differential approximation connecting $dx$ and $dy$?

     Jibu

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

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

112. Kadi namba 112

     Swali

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

     Jibu

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

113. Kadi namba 113

     Swali

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

     Jibu

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

114. Kadi namba 114

     Swali

     What does positive acceleration say about velocity?

     Jibu

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

115. Kadi namba 115

     Swali

     Related rates: when should numerical values be substituted?

     Jibu

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

116. Kadi namba 116

     Swali

     How does concavity predict linearization error?

     Jibu

     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. Kadi namba 117

     Swali

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

     Jibu

     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. Kadi namba 118

     Swali

     When is a particle moving in the positive direction?

     Jibu

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

119. Kadi namba 119

     Swali

     How can velocity show a change of direction?

     Jibu

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

120. Kadi namba 120

     Swali

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

     Jibu

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

121. Kadi namba 121

     Swali

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

     Jibu

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

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

122. Kadi namba 122

     Swali

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

     Jibu

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

123. Kadi namba 123

     Swali

     What must a contextual derivative sentence include?

     Jibu

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

124. Kadi namba 124

     Swali

     Velocity negative and acceleration positive: what happens?

     Jibu

     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. Kadi namba 125

     Swali

     How should a negative related rate be interpreted?

     Jibu

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

126. Kadi namba 126

     Swali

     When is local linearity a sound approximation tool?

     Jibu

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

127. Kadi namba 127

     Swali

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

     Jibu

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

128. Kadi namba 128

     Swali

     When is speed increasing?

     Jibu

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

129. Kadi namba 129

     Swali

     Volume changes with time: notation for its rate?

     Jibu

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

130. Kadi namba 130

     Swali

     Why are similar triangles useful in related rates?

     Jibu

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

131. Kadi namba 131

     Swali

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

     Jibu

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

132. Kadi namba 132

     Swali

     What conclusion does L’Hospital’s Rule permit?

     Jibu

     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. Kadi namba 133

     Swali

     When is speed decreasing?

     Jibu

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

134. Kadi namba 134

     Swali

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

     Jibu

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

135. Kadi namba 135

     Swali

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

     Jibu

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

136. Kadi namba 136

     Swali

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

     Jibu

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

137. Kadi namba 137

     Swali

     Extreme Value Theorem: hypothesis and conclusion?

     Jibu

     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. Kadi namba 138

     Swali

     What is a critical number of $f$?

     Jibu

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

139. Kadi namba 139

     Swali

     First derivative test for a local maximum?

     Jibu

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

140. Kadi namba 140

     Swali

     Second-derivative sign for concave up?

     Jibu

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

141. Kadi namba 141

     Swali

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

     Jibu

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

142. Kadi namba 142

     Swali

     First step in an optimization model?

     Jibu

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

143. Kadi namba 143

     Swali

     Mean Value Theorem: hypotheses and conclusion?

     Jibu

     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. Kadi namba 144

     Swali

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

     Jibu

     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.

     

     kadi 288

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

     Soma fungu hili bila malipo

     Nibomo itafunguka ili uanze kusoma.

145. Kadi namba 145

     Swali

     First derivative test for a local minimum?

     Jibu

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

146. Kadi namba 146

     Swali

     What must happen at an inflection point?

     Jibu

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

147. Kadi namba 147

     Swali

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

     Jibu

     $f''$ 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. Kadi namba 148

     Swali

     How do you confirm an optimization answer is absolute?

     Jibu

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

149. Kadi namba 149

     Swali

     Rolle’s Theorem: hypotheses and conclusion?

     Jibu

     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. Kadi namba 150

     Swali

     Difference between absolute and relative extrema?

     Jibu

     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. Kadi namba 151

     Swali

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

     Jibu

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

152. Kadi namba 152

     Swali

     Second derivative test for a local minimum?

     Jibu

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

153. Kadi namba 153

     Swali

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

     Jibu

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

154. Kadi namba 154

     Swali

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

     Jibu

     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. Kadi namba 155

     Swali

     Which theorem links an average slope to an instantaneous slope?

     Jibu

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

156. Kadi namba 156

     Swali

     How can an implicit derivative locate a horizontal tangent?

     Jibu

     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. Kadi namba 157

     Swali

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

     Jibu

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

158. Kadi namba 158

     Swali

     Second derivative test for a local maximum?

     Jibu

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

159. Kadi namba 159

     Swali

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

     Jibu

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

160. Kadi namba 160

     Swali

     Why must an optimization domain be stated?

     Jibu

     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. Kadi namba 161

     Swali

     Which theorem guarantees absolute extrema, not where they occur?

     Jibu

     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. Kadi namba 162

     Swali

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

     Jibu

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

163. Kadi namba 163

     Swali

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

     Jibu

     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. Kadi namba 164

     Swali

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

     Jibu

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

165. Kadi namba 165

     Swali

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

     Jibu

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

166. Kadi namba 166

     Swali

     How can an implicit derivative locate a vertical tangent?

     Jibu

     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. Kadi namba 167

     Swali

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

     Jibu

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

168. Kadi namba 168

     Swali

     Why are endpoints included in the candidates test?

     Jibu

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

169. Kadi namba 169

     Swali

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

     Jibu

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

170. Kadi namba 170

     Swali

     Second-derivative sign for concave down?

     Jibu

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

171. Kadi namba 171

     Swali

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

     Jibu

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

172. Kadi namba 172

     Swali

     What should the final line of an optimization solution state?

     Jibu

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

173. Kadi namba 173

     Swali

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

     Jibu

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

174. Kadi namba 174

     Swali

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

     Jibu

     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. Kadi namba 175

     Swali

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

     Jibu

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

176. Kadi namba 176

     Swali

     Left Riemann sum on equal subintervals?

     Jibu

     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. Kadi namba 177

     Swali

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

     Jibu

     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. Kadi namba 178

     Swali

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Jibu

     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. Kadi namba 179

     Swali

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

     Jibu

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kadi namba 180

     Swali

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

     Jibu

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

181. Kadi namba 181

     Swali

     Right Riemann sum on equal subintervals?

     Jibu

     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. Kadi namba 182

     Swali

     How does reversing integral bounds change the value?

     Jibu

     It changes the sign:

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

183. Kadi namba 183

     Swali

     Net Change Theorem?

     Jibu

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

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

184. Kadi namba 184

     Swali

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

     Jibu

     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. Kadi namba 185

     Swali

     Power rule for antiderivatives?

     Jibu

     For $n\ne-1$,

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

186. Kadi namba 186

     Swali

     Midpoint Riemann sum on equal subintervals?

     Jibu

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

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

187. Kadi namba 187

     Swali

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

     Jibu

     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. Kadi namba 188

     Swali

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

     Jibu

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kadi namba 189

     Swali

     Antiderivative of $1/x$?

     Jibu

     On any interval not crossing zero,

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

190. Kadi namba 190

     Swali

     Trapezoidal approximation on equal subintervals?

     Jibu

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

191. Kadi namba 191

     Swali

     How do geometric regions help evaluate a definite integral?

     Jibu

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

192. Kadi namba 192

     Swali

     Basic antiderivatives of sine and cosine?

     Jibu

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

193. Kadi namba 193

     Swali

     Definite integral as a limit of Riemann sums?

     Jibu

     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. Kadi namba 194

     Swali

     Constant-multiple rule for integrals?

     Jibu

     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. Kadi namba 195

     Swali

     What pattern suggests $u$-substitution?

     Jibu

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

196. Kadi namba 196

     Swali

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

     Jibu

     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. Kadi namba 197

     Swali

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

     Jibu

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

198. Kadi namba 198

     Swali

     Sum-and-difference rule for definite integrals?

     Jibu

     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. Kadi namba 199

     Swali

     Basic antiderivative of $e^x$?

     Jibu

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

200. Kadi namba 200

     Swali

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

     Jibu

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

201. Kadi namba 201

     Swali

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

     Jibu

     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. Kadi namba 202

     Swali

     How does concavity predict trapezoidal and midpoint error?

     Jibu

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

203. Kadi namba 203

     Swali

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

     Jibu

     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. Kadi namba 204

     Swali

     What denominator pattern suggests an arctangent antiderivative?

     Jibu

     After completing the square and scaling, a form like

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

205. Kadi namba 205

     Swali

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

     Jibu

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

206. Kadi namba 206

     Swali

     How does an initial condition determine an antiderivative?

     Jibu

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

207. Kadi namba 207

     Swali

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

     Jibu

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

208. Kadi namba 208

     Swali

     Why does an indefinite integral include $+C$?

     Jibu

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

209. Kadi namba 209

     Swali

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

     Jibu

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

210. Kadi namba 210

     Swali

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

     Jibu

     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. Kadi namba 211

     Swali

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

     Jibu

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

212. Kadi namba 212

     Swali

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

     Jibu

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

213. Kadi namba 213

     Swali

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

     Jibu

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

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

214. Kadi namba 214

     Swali

     Riemann sum for unequal subinterval widths?

     Jibu

     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. Kadi namba 215

     Swali

     Does continuity guarantee integrability on a closed interval?

     Jibu

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

216. Kadi namba 216

     Swali

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

     Jibu

     Where $g(x)\ne0$,

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

217. Kadi namba 217

     Swali

     What algebraic rewrites often reveal a basic antiderivative?

     Jibu

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

218. Kadi namba 218

     Swali

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

     Jibu

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

219. Kadi namba 219

     Swali

     What is a differential equation?

     Jibu

     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. Kadi namba 220

     Swali

     How does a verbal rate statement become a differential equation?

     Jibu

     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. Kadi namba 221

     Swali

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

     Jibu

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

222. Kadi namba 222

     Swali

     General solution versus particular solution?

     Jibu

     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. Kadi namba 223

     Swali

     What does one segment in a slope field show?

     Jibu

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

224. Kadi namba 224

     Swali

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

     Jibu

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

225. Kadi namba 225

     Swali

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

     Jibu

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

226. Kadi namba 226

     Swali

     What makes a first-order differential equation separable?

     Jibu

     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. Kadi namba 227

     Swali

     What is an initial value problem?

     Jibu

     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. Kadi namba 228

     Swali

     What is an isocline in a slope field?

     Jibu

     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. Kadi namba 229

     Swali

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

     Jibu

     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. Kadi namba 230

     Swali

     General solution of $dy/dt=ky$?

     Jibu

     $$y=Ce^{kt}$$

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

231. Kadi namba 231

     Swali

     Core method for solving a separable differential equation?

     Jibu

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

232. Kadi namba 232

     Swali

     How should a solution curve follow a slope field?

     Jibu

     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. Kadi namba 233

     Swali

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

     Jibu

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

234. Kadi namba 234

     Swali

     Why is one integration constant enough after integrating both sides?

     Jibu

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

235. Kadi namba 235

     Swali

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

     Jibu

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

236. Kadi namba 236

     Swali

     Can one differential equation have infinitely many solutions?

     Jibu

     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. Kadi namba 237

     Swali

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

     Jibu

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

238. Kadi namba 238

     Swali

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

     Jibu

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

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

239. Kadi namba 239

     Swali

     What can be lost when dividing to separate variables?

     Jibu

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

240. Kadi namba 240

     Swali

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

     Jibu

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

241. Kadi namba 241

     Swali

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

     Jibu

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

242. Kadi namba 242

     Swali

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

     Jibu

     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. Kadi namba 243

     Swali

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

     Jibu

     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. Kadi namba 244

     Swali

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

     Jibu

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

245. Kadi namba 245

     Swali

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

     Jibu

     For $k>0$,

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

     It is independent of the initial amount.

246. Kadi namba 246

     Swali

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

     Jibu

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

247. Kadi namba 247

     Swali

     How is an initial condition used after separation?

     Jibu

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

248. Kadi namba 248

     Swali

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

     Jibu

     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. Kadi namba 249

     Swali

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

     Jibu

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

250. Kadi namba 250

     Swali

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

     Jibu

     For $k<0$,

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

251. Kadi namba 251

     Swali

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

     Jibu

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

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

252. Kadi namba 252

     Swali

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

     Jibu

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

     Velocity below zero contributes negative displacement.

253. Kadi namba 253

     Swali

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

     Jibu

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

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

     Think top minus bottom.

254. Kadi namba 254

     Swali

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

     Jibu

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

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

255. Kadi namba 255

     Swali

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Jibu

     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. Kadi namba 256

     Swali

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

     Jibu

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

257. Kadi namba 257

     Swali

     Cross-sectional area when each slice is a square?

     Jibu

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

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

258. Kadi namba 258

     Swali

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

     Jibu

     Integrate the net rate:

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

259. Kadi namba 259

     Swali

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

     Jibu

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

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

     Think right minus left.

260. Kadi namba 260

     Swali

     Disc-method volume formula?

     Jibu

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

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

261. Kadi namba 261

     Swali

     What units does average value have?

     Jibu

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

262. Kadi namba 262

     Swali

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

     Jibu

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

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

263. Kadi namba 263

     Swali

     Cross-sectional area when each slice is a rectangle?

     Jibu

     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. Kadi namba 264

     Swali

     How do you determine bounds for area between curves?

     Jibu

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

265. Kadi namba 265

     Swali

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

     Jibu

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

266. Kadi namba 266

     Swali

     When is a particle moving to the right or left?

     Jibu

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

267. Kadi namba 267

     Swali

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

     Jibu

     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. Kadi namba 268

     Swali

     Why must an area integral be split where curves intersect?

     Jibu

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

269. Kadi namba 269

     Swali

     How can a velocity table approximate displacement?

     Jibu

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

270. Kadi namba 270

     Swali

     Washer-method volume formula?

     Jibu

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

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

271. Kadi namba 271

     Swali

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

     Jibu

     If $s(a)$ is known,

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

272. Kadi namba 272

     Swali

     How do you choose between vertical and horizontal area slices?

     Jibu

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

273. Kadi namba 273

     Swali

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

     Jibu

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

274. Kadi namba 274

     Swali

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

     Jibu

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

275. Kadi namba 275

     Swali

     Single expression for area between two curves?

     Jibu

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kadi namba 276

     Swali

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

     Jibu

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

277. Kadi namba 277

     Swali

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

     Jibu

     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. Kadi namba 278

     Swali

     When should a volume integral use $dy$?

     Jibu

     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. Kadi namba 279

     Swali

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

     Jibu

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

280. Kadi namba 280

     Swali

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

     Jibu

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

281. Kadi namba 281

     Swali

     Why must total distance split at velocity sign changes?

     Jibu

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

282. Kadi namba 282

     Swali

     When does an accumulated quantity reach a local maximum?

     Jibu

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

283. Kadi namba 283

     Swali

     What distinguishes area from a definite integral?

     Jibu

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

284. Kadi namba 284

     Swali

     How do position, velocity, and acceleration graphs correspond?

     Jibu

     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. Kadi namba 285

     Swali

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

     Jibu

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

286. Kadi namba 286

     Swali

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

     Jibu

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

287. Kadi namba 287

     Swali

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

     Jibu

     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. Kadi namba 288

     Swali

     Why should a contextual integral answer include a sentence?

     Jibu

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