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

Amakhadi akuleli qoqo

1. Ikhadi 1

   Umbuzo

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

   Impendulo

   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. Ikhadi 2

   Umbuzo

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

   Impendulo

   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. Ikhadi 3

   Umbuzo

   When does direct substitution evaluate a limit?

   Impendulo

   When the function is continuous at the target input. Then

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

4. Ikhadi 4

   Umbuzo

   Three conditions for continuity at $x=a$?

   Impendulo

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

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

5. Ikhadi 5

   Umbuzo

   Intermediate Value Theorem: hypotheses and conclusion?

   Impendulo

   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. Ikhadi 6

   Umbuzo

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

   Impendulo

   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. Ikhadi 7

   Umbuzo

   How do you read a finite limit from a graph?

   Impendulo

   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. Ikhadi 8

   Umbuzo

   Limit law for a sum or difference?

   Impendulo

   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. Ikhadi 9

   Umbuzo

   What makes a discontinuity removable?

   Impendulo

   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. Ikhadi 10

    Umbuzo

    Squeeze Theorem: usable form?

    Impendulo

    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. Ikhadi 11

    Umbuzo

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

    Impendulo

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

12. Ikhadi 12

    Umbuzo

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

    Impendulo

    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. Ikhadi 13

    Umbuzo

    Limit law for a product?

    Impendulo

    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. Ikhadi 14

    Umbuzo

    Graph signature of a jump discontinuity?

    Impendulo

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

15. Ikhadi 15

    Umbuzo

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

    Impendulo

    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. Ikhadi 16

    Umbuzo

    Horizontal asymptote from a limit at infinity?

    Impendulo

    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. Ikhadi 17

    Umbuzo

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

    Impendulo

    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. Ikhadi 18

    Umbuzo

    Limit law for a quotient—and its condition?

    Impendulo

    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. Ikhadi 19

    Umbuzo

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

    Impendulo

    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. Ikhadi 20

    Umbuzo

    When is the Squeeze Theorem a natural choice?

    Impendulo

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

21. Ikhadi 21

    Umbuzo

    Vertical asymptote from one-sided behavior?

    Impendulo

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

22. Ikhadi 22

    Umbuzo

    Limit at infinity of equal-degree rational functions?

    Impendulo

    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. Ikhadi 23

    Umbuzo

    When can a limit pass through a continuous outer function?

    Impendulo

    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. Ikhadi 24

    Umbuzo

    What makes a discontinuity infinite?

    Impendulo

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

25. Ikhadi 25

    Umbuzo

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

    Impendulo

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

26. Ikhadi 26

    Umbuzo

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

    Impendulo

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Ikhadi 27

    Umbuzo

    Continuity of a composition?

    Impendulo

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

28. Ikhadi 28

    Umbuzo

    Limit at infinity when a rational numerator has lower degree?

    Impendulo

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

29. Ikhadi 29

    Umbuzo

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

    Impendulo

    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. Ikhadi 30

    Umbuzo

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

    Impendulo

    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. Ikhadi 31

    Umbuzo

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

    Impendulo

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

32. Ikhadi 32

    Umbuzo

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

    Impendulo

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

33. Ikhadi 33

    Umbuzo

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

    Impendulo

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

34. Ikhadi 34

    Umbuzo

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

    Impendulo

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

35. Ikhadi 35

    Umbuzo

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

    Impendulo

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

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

36. Ikhadi 36

    Umbuzo

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

    Impendulo

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

    The derivative exists only if this finite limit exists.

37. Ikhadi 37

    Umbuzo

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

    Impendulo

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

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

38. Ikhadi 38

    Umbuzo

    What does differentiability imply about continuity?

    Impendulo

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

39. Ikhadi 39

    Umbuzo

    Power rule for derivatives?

    Impendulo

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

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

40. Ikhadi 40

    Umbuzo

    Units of $f'(x)$?

    Impendulo

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

41. Ikhadi 41

    Umbuzo

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

    Impendulo

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

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

42. Ikhadi 42

    Umbuzo

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

    Impendulo

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

    Umbuzo

    Derivative of a constant?

    Impendulo

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

    A constant function has zero rate of change.

44. Ikhadi 44

    Umbuzo

    Derivative of $\sin x$?

    Impendulo

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

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

45. Ikhadi 45

    Umbuzo

    Product rule?

    Impendulo

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

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

46. Ikhadi 46

    Umbuzo

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

    Impendulo

    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. Ikhadi 47

    Umbuzo

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

    Impendulo

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

48. Ikhadi 48

    Umbuzo

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

    Impendulo

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

49. Ikhadi 49

    Umbuzo

    Derivative of a sum or difference?

    Impendulo

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

50. Ikhadi 50

    Umbuzo

    Derivative of $\cos x$?

    Impendulo

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

    The standard formula assumes radians.

51. Ikhadi 51

    Umbuzo

    Quotient rule?

    Impendulo

    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. Ikhadi 52

    Umbuzo

    Common notations for the first derivative?

    Impendulo

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

53. Ikhadi 53

    Umbuzo

    Derivative of $e^x$?

    Impendulo

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

54. Ikhadi 54

    Umbuzo

    What graph features can make $f$ nondifferentiable?

    Impendulo

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

55. Ikhadi 55

    Umbuzo

    Derivative of $\tan x$?

    Impendulo

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Ikhadi 56

    Umbuzo

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

    Impendulo

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

57. Ikhadi 57

    Umbuzo

    Derivative of $\ln x$?

    Impendulo

    For $x>0$,

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

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

58. Ikhadi 58

    Umbuzo

    How does the power rule handle roots or negative powers?

    Impendulo

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

59. Ikhadi 59

    Umbuzo

    Derivative of $\csc x$?

    Impendulo

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Ikhadi 60

    Umbuzo

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

    Impendulo

    $f$ is increasing on that interval.

61. Ikhadi 61

    Umbuzo

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

    Impendulo

    For $a>0$,

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

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

62. Ikhadi 62

    Umbuzo

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

    Impendulo

    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. Ikhadi 63

    Umbuzo

    Derivative of $\sec x$?

    Impendulo

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Ikhadi 64

    Umbuzo

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

    Impendulo

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

65. Ikhadi 65

    Umbuzo

    Derivative of $\log_a x$?

    Impendulo

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

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

66. Ikhadi 66

    Umbuzo

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

    Impendulo

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Ikhadi 67

    Umbuzo

    Derivative of $\cot x$?

    Impendulo

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Ikhadi 68

    Umbuzo

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

    Impendulo

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

69. Ikhadi 69

    Umbuzo

    Constant-multiple rule?

    Impendulo

    For a constant $c$,

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

70. Ikhadi 70

    Umbuzo

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

    Impendulo

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

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

71. Ikhadi 71

    Umbuzo

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

    Impendulo

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

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

72. Ikhadi 72

    Umbuzo

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

    Impendulo

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

73. Ikhadi 73

    Umbuzo

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

    Impendulo

    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. Ikhadi 74

    Umbuzo

    Derivative of an inverse function at $x$?

    Impendulo

    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. Ikhadi 75

    Umbuzo

    Derivative of $\arcsin x$?

    Impendulo

    For $|x|<1$,

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

76. Ikhadi 76

    Umbuzo

    Notation for the third derivative of $f$?

    Impendulo

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

77. Ikhadi 77

    Umbuzo

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

    Impendulo

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

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

78. Ikhadi 78

    Umbuzo

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

    Impendulo

    Where $y\ne0$,

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

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

79. Ikhadi 79

    Umbuzo

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

    Impendulo

    Provided $f'(a)\ne0$,

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

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

80. Ikhadi 80

    Umbuzo

    Derivative of $\arctan x$?

    Impendulo

    For every real $x$,

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

81. Ikhadi 81

    Umbuzo

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

    Impendulo

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

    The extra factor is the chain rule.

82. Ikhadi 82

    Umbuzo

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

    Impendulo

    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. Ikhadi 83

    Umbuzo

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

    Impendulo

    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. Ikhadi 84

    Umbuzo

    Derivative of $\arccos x$?

    Impendulo

    For $|x|<1$,

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

85. Ikhadi 85

    Umbuzo

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

    Impendulo

    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. Ikhadi 86

    Umbuzo

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

    Impendulo

    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. Ikhadi 87

    Umbuzo

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

    Impendulo

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

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

88. Ikhadi 88

    Umbuzo

    How are tangent slopes of inverse graphs related?

    Impendulo

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

89. Ikhadi 89

    Umbuzo

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

    Impendulo

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

90. Ikhadi 90

    Umbuzo

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

    Impendulo

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

91. Ikhadi 91

    Umbuzo

    Horizontal tangent on an implicit curve: derivative condition?

    Impendulo

    $dy/dx=0$ 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. Ikhadi 92

    Umbuzo

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

    Impendulo

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

93. Ikhadi 93

    Umbuzo

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

    Impendulo

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

94. Ikhadi 94

    Umbuzo

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

    Impendulo

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

    This combines the power rule with the chain rule.

95. Ikhadi 95

    Umbuzo

    Vertical tangent on an implicit curve: derivative clue?

    Impendulo

    $dy/dx$ 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. Ikhadi 96

    Umbuzo

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

    Impendulo

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

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

97. Ikhadi 97

    Umbuzo

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

    Impendulo

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

98. Ikhadi 98

    Umbuzo

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

    Impendulo

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

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

99. Ikhadi 99

    Umbuzo

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

    Impendulo

    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. Ikhadi 100

     Umbuzo

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

     Impendulo

     When the functions are differentiable, the chain rule gives

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

101. Ikhadi 101

     Umbuzo

     What local property lets a function have an inverse derivative?

     Impendulo

     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. Ikhadi 102

     Umbuzo

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

     Impendulo

     For $a>0$,

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

103. Ikhadi 103

     Umbuzo

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

     Impendulo

     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. Ikhadi 104

     Umbuzo

     Position, velocity, and acceleration relationships?

     Impendulo

     For position $s(t)$,

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

105. Ikhadi 105

     Umbuzo

     Central idea of a related-rates problem?

     Impendulo

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

106. Ikhadi 106

     Umbuzo

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

     Impendulo

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

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

107. Ikhadi 107

     Umbuzo

     L’Hospital’s Rule: basic conditions?

     Impendulo

     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. Ikhadi 108

     Umbuzo

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

     Impendulo

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

109. Ikhadi 109

     Umbuzo

     Speed in terms of velocity?

     Impendulo

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Ikhadi 110

     Umbuzo

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

     Impendulo

     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. Ikhadi 111

     Umbuzo

     Differential approximation connecting $dx$ and $dy$?

     Impendulo

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

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

112. Ikhadi 112

     Umbuzo

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

     Impendulo

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

113. Ikhadi 113

     Umbuzo

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

     Impendulo

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

114. Ikhadi 114

     Umbuzo

     What does positive acceleration say about velocity?

     Impendulo

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

115. Ikhadi 115

     Umbuzo

     Related rates: when should numerical values be substituted?

     Impendulo

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

116. Ikhadi 116

     Umbuzo

     How does concavity predict linearization error?

     Impendulo

     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. Ikhadi 117

     Umbuzo

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

     Impendulo

     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. Ikhadi 118

     Umbuzo

     When is a particle moving in the positive direction?

     Impendulo

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

119. Ikhadi 119

     Umbuzo

     How can velocity show a change of direction?

     Impendulo

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

120. Ikhadi 120

     Umbuzo

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

     Impendulo

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

121. Ikhadi 121

     Umbuzo

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

     Impendulo

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

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

122. Ikhadi 122

     Umbuzo

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

     Impendulo

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

123. Ikhadi 123

     Umbuzo

     What must a contextual derivative sentence include?

     Impendulo

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

124. Ikhadi 124

     Umbuzo

     Velocity negative and acceleration positive: what happens?

     Impendulo

     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. Ikhadi 125

     Umbuzo

     How should a negative related rate be interpreted?

     Impendulo

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

126. Ikhadi 126

     Umbuzo

     When is local linearity a sound approximation tool?

     Impendulo

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

127. Ikhadi 127

     Umbuzo

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

     Impendulo

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

128. Ikhadi 128

     Umbuzo

     When is speed increasing?

     Impendulo

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

129. Ikhadi 129

     Umbuzo

     Volume changes with time: notation for its rate?

     Impendulo

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

130. Ikhadi 130

     Umbuzo

     Why are similar triangles useful in related rates?

     Impendulo

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

131. Ikhadi 131

     Umbuzo

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

     Impendulo

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

132. Ikhadi 132

     Umbuzo

     What conclusion does L’Hospital’s Rule permit?

     Impendulo

     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. Ikhadi 133

     Umbuzo

     When is speed decreasing?

     Impendulo

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

134. Ikhadi 134

     Umbuzo

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

     Impendulo

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

135. Ikhadi 135

     Umbuzo

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

     Impendulo

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

136. Ikhadi 136

     Umbuzo

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

     Impendulo

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

137. Ikhadi 137

     Umbuzo

     Extreme Value Theorem: hypothesis and conclusion?

     Impendulo

     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. Ikhadi 138

     Umbuzo

     What is a critical number of $f$?

     Impendulo

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

139. Ikhadi 139

     Umbuzo

     First derivative test for a local maximum?

     Impendulo

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

140. Ikhadi 140

     Umbuzo

     Second-derivative sign for concave up?

     Impendulo

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

141. Ikhadi 141

     Umbuzo

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

     Impendulo

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

142. Ikhadi 142

     Umbuzo

     First step in an optimization model?

     Impendulo

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

143. Ikhadi 143

     Umbuzo

     Mean Value Theorem: hypotheses and conclusion?

     Impendulo

     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. Ikhadi 144

     Umbuzo

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

     Impendulo

     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.

     

     amakhadi angu-288

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

     Funda leli qoqo mahhala

     I-Nibomo iyavuleka ukuze uqale ukufunda.

145. Ikhadi 145

     Umbuzo

     First derivative test for a local minimum?

     Impendulo

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

146. Ikhadi 146

     Umbuzo

     What must happen at an inflection point?

     Impendulo

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

147. Ikhadi 147

     Umbuzo

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

     Impendulo

     $f''$ 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. Ikhadi 148

     Umbuzo

     How do you confirm an optimization answer is absolute?

     Impendulo

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

149. Ikhadi 149

     Umbuzo

     Rolle’s Theorem: hypotheses and conclusion?

     Impendulo

     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. Ikhadi 150

     Umbuzo

     Difference between absolute and relative extrema?

     Impendulo

     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. Ikhadi 151

     Umbuzo

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

     Impendulo

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

152. Ikhadi 152

     Umbuzo

     Second derivative test for a local minimum?

     Impendulo

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

153. Ikhadi 153

     Umbuzo

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

     Impendulo

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

154. Ikhadi 154

     Umbuzo

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

     Impendulo

     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. Ikhadi 155

     Umbuzo

     Which theorem links an average slope to an instantaneous slope?

     Impendulo

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

156. Ikhadi 156

     Umbuzo

     How can an implicit derivative locate a horizontal tangent?

     Impendulo

     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. Ikhadi 157

     Umbuzo

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

     Impendulo

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

158. Ikhadi 158

     Umbuzo

     Second derivative test for a local maximum?

     Impendulo

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

159. Ikhadi 159

     Umbuzo

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

     Impendulo

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

160. Ikhadi 160

     Umbuzo

     Why must an optimization domain be stated?

     Impendulo

     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. Ikhadi 161

     Umbuzo

     Which theorem guarantees absolute extrema, not where they occur?

     Impendulo

     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. Ikhadi 162

     Umbuzo

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

     Impendulo

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

163. Ikhadi 163

     Umbuzo

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

     Impendulo

     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. Ikhadi 164

     Umbuzo

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

     Impendulo

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

165. Ikhadi 165

     Umbuzo

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

     Impendulo

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

166. Ikhadi 166

     Umbuzo

     How can an implicit derivative locate a vertical tangent?

     Impendulo

     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. Ikhadi 167

     Umbuzo

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

     Impendulo

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

168. Ikhadi 168

     Umbuzo

     Why are endpoints included in the candidates test?

     Impendulo

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

169. Ikhadi 169

     Umbuzo

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

     Impendulo

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

170. Ikhadi 170

     Umbuzo

     Second-derivative sign for concave down?

     Impendulo

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

171. Ikhadi 171

     Umbuzo

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

     Impendulo

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

172. Ikhadi 172

     Umbuzo

     What should the final line of an optimization solution state?

     Impendulo

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

173. Ikhadi 173

     Umbuzo

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

     Impendulo

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

174. Ikhadi 174

     Umbuzo

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

     Impendulo

     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. Ikhadi 175

     Umbuzo

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

     Impendulo

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

176. Ikhadi 176

     Umbuzo

     Left Riemann sum on equal subintervals?

     Impendulo

     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. Ikhadi 177

     Umbuzo

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

     Impendulo

     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. Ikhadi 178

     Umbuzo

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Impendulo

     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. Ikhadi 179

     Umbuzo

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

     Impendulo

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Ikhadi 180

     Umbuzo

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

     Impendulo

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

181. Ikhadi 181

     Umbuzo

     Right Riemann sum on equal subintervals?

     Impendulo

     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. Ikhadi 182

     Umbuzo

     How does reversing integral bounds change the value?

     Impendulo

     It changes the sign:

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

183. Ikhadi 183

     Umbuzo

     Net Change Theorem?

     Impendulo

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

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

184. Ikhadi 184

     Umbuzo

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

     Impendulo

     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. Ikhadi 185

     Umbuzo

     Power rule for antiderivatives?

     Impendulo

     For $n\ne-1$,

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

186. Ikhadi 186

     Umbuzo

     Midpoint Riemann sum on equal subintervals?

     Impendulo

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

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

187. Ikhadi 187

     Umbuzo

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

     Impendulo

     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. Ikhadi 188

     Umbuzo

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

     Impendulo

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Ikhadi 189

     Umbuzo

     Antiderivative of $1/x$?

     Impendulo

     On any interval not crossing zero,

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

190. Ikhadi 190

     Umbuzo

     Trapezoidal approximation on equal subintervals?

     Impendulo

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

191. Ikhadi 191

     Umbuzo

     How do geometric regions help evaluate a definite integral?

     Impendulo

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

192. Ikhadi 192

     Umbuzo

     Basic antiderivatives of sine and cosine?

     Impendulo

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

193. Ikhadi 193

     Umbuzo

     Definite integral as a limit of Riemann sums?

     Impendulo

     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. Ikhadi 194

     Umbuzo

     Constant-multiple rule for integrals?

     Impendulo

     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. Ikhadi 195

     Umbuzo

     What pattern suggests $u$-substitution?

     Impendulo

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

196. Ikhadi 196

     Umbuzo

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

     Impendulo

     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. Ikhadi 197

     Umbuzo

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

     Impendulo

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

198. Ikhadi 198

     Umbuzo

     Sum-and-difference rule for definite integrals?

     Impendulo

     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. Ikhadi 199

     Umbuzo

     Basic antiderivative of $e^x$?

     Impendulo

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

200. Ikhadi 200

     Umbuzo

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

     Impendulo

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

201. Ikhadi 201

     Umbuzo

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

     Impendulo

     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. Ikhadi 202

     Umbuzo

     How does concavity predict trapezoidal and midpoint error?

     Impendulo

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

203. Ikhadi 203

     Umbuzo

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

     Impendulo

     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. Ikhadi 204

     Umbuzo

     What denominator pattern suggests an arctangent antiderivative?

     Impendulo

     After completing the square and scaling, a form like

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

205. Ikhadi 205

     Umbuzo

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

     Impendulo

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

206. Ikhadi 206

     Umbuzo

     How does an initial condition determine an antiderivative?

     Impendulo

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

207. Ikhadi 207

     Umbuzo

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

     Impendulo

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

208. Ikhadi 208

     Umbuzo

     Why does an indefinite integral include $+C$?

     Impendulo

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

209. Ikhadi 209

     Umbuzo

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

     Impendulo

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

210. Ikhadi 210

     Umbuzo

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

     Impendulo

     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. Ikhadi 211

     Umbuzo

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

     Impendulo

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

212. Ikhadi 212

     Umbuzo

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

     Impendulo

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

213. Ikhadi 213

     Umbuzo

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

     Impendulo

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

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

214. Ikhadi 214

     Umbuzo

     Riemann sum for unequal subinterval widths?

     Impendulo

     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. Ikhadi 215

     Umbuzo

     Does continuity guarantee integrability on a closed interval?

     Impendulo

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

216. Ikhadi 216

     Umbuzo

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

     Impendulo

     Where $g(x)\ne0$,

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

217. Ikhadi 217

     Umbuzo

     What algebraic rewrites often reveal a basic antiderivative?

     Impendulo

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

218. Ikhadi 218

     Umbuzo

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

     Impendulo

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

219. Ikhadi 219

     Umbuzo

     What is a differential equation?

     Impendulo

     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. Ikhadi 220

     Umbuzo

     How does a verbal rate statement become a differential equation?

     Impendulo

     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. Ikhadi 221

     Umbuzo

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

     Impendulo

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

222. Ikhadi 222

     Umbuzo

     General solution versus particular solution?

     Impendulo

     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. Ikhadi 223

     Umbuzo

     What does one segment in a slope field show?

     Impendulo

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

224. Ikhadi 224

     Umbuzo

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

     Impendulo

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

225. Ikhadi 225

     Umbuzo

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

     Impendulo

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

226. Ikhadi 226

     Umbuzo

     What makes a first-order differential equation separable?

     Impendulo

     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. Ikhadi 227

     Umbuzo

     What is an initial value problem?

     Impendulo

     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. Ikhadi 228

     Umbuzo

     What is an isocline in a slope field?

     Impendulo

     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. Ikhadi 229

     Umbuzo

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

     Impendulo

     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. Ikhadi 230

     Umbuzo

     General solution of $dy/dt=ky$?

     Impendulo

     $$y=Ce^{kt}$$

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

231. Ikhadi 231

     Umbuzo

     Core method for solving a separable differential equation?

     Impendulo

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

232. Ikhadi 232

     Umbuzo

     How should a solution curve follow a slope field?

     Impendulo

     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. Ikhadi 233

     Umbuzo

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

     Impendulo

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

234. Ikhadi 234

     Umbuzo

     Why is one integration constant enough after integrating both sides?

     Impendulo

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

235. Ikhadi 235

     Umbuzo

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

     Impendulo

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

236. Ikhadi 236

     Umbuzo

     Can one differential equation have infinitely many solutions?

     Impendulo

     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. Ikhadi 237

     Umbuzo

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

     Impendulo

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

238. Ikhadi 238

     Umbuzo

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

     Impendulo

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

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

239. Ikhadi 239

     Umbuzo

     What can be lost when dividing to separate variables?

     Impendulo

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

240. Ikhadi 240

     Umbuzo

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

     Impendulo

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

241. Ikhadi 241

     Umbuzo

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

     Impendulo

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

242. Ikhadi 242

     Umbuzo

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

     Impendulo

     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. Ikhadi 243

     Umbuzo

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

     Impendulo

     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. Ikhadi 244

     Umbuzo

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

     Impendulo

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

245. Ikhadi 245

     Umbuzo

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

     Impendulo

     For $k>0$,

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

     It is independent of the initial amount.

246. Ikhadi 246

     Umbuzo

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

     Impendulo

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

247. Ikhadi 247

     Umbuzo

     How is an initial condition used after separation?

     Impendulo

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

248. Ikhadi 248

     Umbuzo

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

     Impendulo

     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. Ikhadi 249

     Umbuzo

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

     Impendulo

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

250. Ikhadi 250

     Umbuzo

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

     Impendulo

     For $k<0$,

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

251. Ikhadi 251

     Umbuzo

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

     Impendulo

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

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

252. Ikhadi 252

     Umbuzo

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

     Impendulo

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

     Velocity below zero contributes negative displacement.

253. Ikhadi 253

     Umbuzo

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

     Impendulo

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

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

     Think top minus bottom.

254. Ikhadi 254

     Umbuzo

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

     Impendulo

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

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

255. Ikhadi 255

     Umbuzo

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Impendulo

     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. Ikhadi 256

     Umbuzo

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

     Impendulo

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

257. Ikhadi 257

     Umbuzo

     Cross-sectional area when each slice is a square?

     Impendulo

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

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

258. Ikhadi 258

     Umbuzo

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

     Impendulo

     Integrate the net rate:

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

259. Ikhadi 259

     Umbuzo

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

     Impendulo

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

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

     Think right minus left.

260. Ikhadi 260

     Umbuzo

     Disc-method volume formula?

     Impendulo

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

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

261. Ikhadi 261

     Umbuzo

     What units does average value have?

     Impendulo

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

262. Ikhadi 262

     Umbuzo

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

     Impendulo

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

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

263. Ikhadi 263

     Umbuzo

     Cross-sectional area when each slice is a rectangle?

     Impendulo

     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. Ikhadi 264

     Umbuzo

     How do you determine bounds for area between curves?

     Impendulo

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

265. Ikhadi 265

     Umbuzo

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

     Impendulo

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

266. Ikhadi 266

     Umbuzo

     When is a particle moving to the right or left?

     Impendulo

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

267. Ikhadi 267

     Umbuzo

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

     Impendulo

     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. Ikhadi 268

     Umbuzo

     Why must an area integral be split where curves intersect?

     Impendulo

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

269. Ikhadi 269

     Umbuzo

     How can a velocity table approximate displacement?

     Impendulo

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

270. Ikhadi 270

     Umbuzo

     Washer-method volume formula?

     Impendulo

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

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

271. Ikhadi 271

     Umbuzo

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

     Impendulo

     If $s(a)$ is known,

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

272. Ikhadi 272

     Umbuzo

     How do you choose between vertical and horizontal area slices?

     Impendulo

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

273. Ikhadi 273

     Umbuzo

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

     Impendulo

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

274. Ikhadi 274

     Umbuzo

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

     Impendulo

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

275. Ikhadi 275

     Umbuzo

     Single expression for area between two curves?

     Impendulo

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Ikhadi 276

     Umbuzo

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

     Impendulo

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

277. Ikhadi 277

     Umbuzo

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

     Impendulo

     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. Ikhadi 278

     Umbuzo

     When should a volume integral use $dy$?

     Impendulo

     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. Ikhadi 279

     Umbuzo

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

     Impendulo

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

280. Ikhadi 280

     Umbuzo

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

     Impendulo

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

281. Ikhadi 281

     Umbuzo

     Why must total distance split at velocity sign changes?

     Impendulo

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

282. Ikhadi 282

     Umbuzo

     When does an accumulated quantity reach a local maximum?

     Impendulo

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

283. Ikhadi 283

     Umbuzo

     What distinguishes area from a definite integral?

     Impendulo

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

284. Ikhadi 284

     Umbuzo

     How do position, velocity, and acceleration graphs correspond?

     Impendulo

     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. Ikhadi 285

     Umbuzo

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

     Impendulo

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

286. Ikhadi 286

     Umbuzo

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

     Impendulo

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

287. Ikhadi 287

     Umbuzo

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

     Impendulo

     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. Ikhadi 288

     Umbuzo

     Why should a contextual integral answer include a sentence?

     Impendulo

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