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

Spjöld í þessum stokki

1. Spjald 1

   Spurning

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

   Svar

   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. Spjald 2

   Spurning

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

   Svar

   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. Spjald 3

   Spurning

   When does direct substitution evaluate a limit?

   Svar

   When the function is continuous at the target input. Then

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

4. Spjald 4

   Spurning

   Three conditions for continuity at $x=a$?

   Svar

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

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

5. Spjald 5

   Spurning

   Intermediate Value Theorem: hypotheses and conclusion?

   Svar

   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. Spjald 6

   Spurning

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

   Svar

   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. Spjald 7

   Spurning

   How do you read a finite limit from a graph?

   Svar

   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. Spjald 8

   Spurning

   Limit law for a sum or difference?

   Svar

   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. Spjald 9

   Spurning

   What makes a discontinuity removable?

   Svar

   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. Spjald 10

    Spurning

    Squeeze Theorem: usable form?

    Svar

    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. Spjald 11

    Spurning

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

    Svar

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

12. Spjald 12

    Spurning

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

    Svar

    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. Spjald 13

    Spurning

    Limit law for a product?

    Svar

    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. Spjald 14

    Spurning

    Graph signature of a jump discontinuity?

    Svar

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

15. Spjald 15

    Spurning

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

    Svar

    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. Spjald 16

    Spurning

    Horizontal asymptote from a limit at infinity?

    Svar

    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. Spjald 17

    Spurning

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

    Svar

    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. Spjald 18

    Spurning

    Limit law for a quotient—and its condition?

    Svar

    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. Spjald 19

    Spurning

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

    Svar

    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. Spjald 20

    Spurning

    When is the Squeeze Theorem a natural choice?

    Svar

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

21. Spjald 21

    Spurning

    Vertical asymptote from one-sided behavior?

    Svar

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

22. Spjald 22

    Spurning

    Limit at infinity of equal-degree rational functions?

    Svar

    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. Spjald 23

    Spurning

    When can a limit pass through a continuous outer function?

    Svar

    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. Spjald 24

    Spurning

    What makes a discontinuity infinite?

    Svar

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

25. Spjald 25

    Spurning

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

    Svar

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

26. Spjald 26

    Spurning

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

    Svar

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Spjald 27

    Spurning

    Continuity of a composition?

    Svar

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

28. Spjald 28

    Spurning

    Limit at infinity when a rational numerator has lower degree?

    Svar

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

29. Spjald 29

    Spurning

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

    Svar

    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. Spjald 30

    Spurning

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

    Svar

    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. Spjald 31

    Spurning

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

    Svar

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

32. Spjald 32

    Spurning

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

    Svar

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

33. Spjald 33

    Spurning

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

    Svar

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

34. Spjald 34

    Spurning

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

    Svar

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

35. Spjald 35

    Spurning

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

    Svar

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

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

36. Spjald 36

    Spurning

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

    Svar

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

    The derivative exists only if this finite limit exists.

37. Spjald 37

    Spurning

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

    Svar

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

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

38. Spjald 38

    Spurning

    What does differentiability imply about continuity?

    Svar

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

39. Spjald 39

    Spurning

    Power rule for derivatives?

    Svar

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

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

40. Spjald 40

    Spurning

    Units of $f'(x)$?

    Svar

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

41. Spjald 41

    Spurning

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

    Svar

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

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

42. Spjald 42

    Spurning

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

    Svar

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

    Spurning

    Derivative of a constant?

    Svar

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

    A constant function has zero rate of change.

44. Spjald 44

    Spurning

    Derivative of $\sin x$?

    Svar

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

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

45. Spjald 45

    Spurning

    Product rule?

    Svar

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

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

46. Spjald 46

    Spurning

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

    Svar

    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. Spjald 47

    Spurning

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

    Svar

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

48. Spjald 48

    Spurning

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

    Svar

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

49. Spjald 49

    Spurning

    Derivative of a sum or difference?

    Svar

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

50. Spjald 50

    Spurning

    Derivative of $\cos x$?

    Svar

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

    The standard formula assumes radians.

51. Spjald 51

    Spurning

    Quotient rule?

    Svar

    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. Spjald 52

    Spurning

    Common notations for the first derivative?

    Svar

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

53. Spjald 53

    Spurning

    Derivative of $e^x$?

    Svar

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

54. Spjald 54

    Spurning

    What graph features can make $f$ nondifferentiable?

    Svar

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

55. Spjald 55

    Spurning

    Derivative of $\tan x$?

    Svar

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Spjald 56

    Spurning

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

    Svar

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

57. Spjald 57

    Spurning

    Derivative of $\ln x$?

    Svar

    For $x>0$,

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

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

58. Spjald 58

    Spurning

    How does the power rule handle roots or negative powers?

    Svar

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

59. Spjald 59

    Spurning

    Derivative of $\csc x$?

    Svar

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Spjald 60

    Spurning

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

    Svar

    $f$ is increasing on that interval.

61. Spjald 61

    Spurning

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

    Svar

    For $a>0$,

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

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

62. Spjald 62

    Spurning

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

    Svar

    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. Spjald 63

    Spurning

    Derivative of $\sec x$?

    Svar

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Spjald 64

    Spurning

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

    Svar

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

65. Spjald 65

    Spurning

    Derivative of $\log_a x$?

    Svar

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

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

66. Spjald 66

    Spurning

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

    Svar

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Spjald 67

    Spurning

    Derivative of $\cot x$?

    Svar

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Spjald 68

    Spurning

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

    Svar

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

69. Spjald 69

    Spurning

    Constant-multiple rule?

    Svar

    For a constant $c$,

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

70. Spjald 70

    Spurning

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

    Svar

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

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

71. Spjald 71

    Spurning

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

    Svar

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

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

72. Spjald 72

    Spurning

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

    Svar

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

73. Spjald 73

    Spurning

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

    Svar

    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. Spjald 74

    Spurning

    Derivative of an inverse function at $x$?

    Svar

    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. Spjald 75

    Spurning

    Derivative of $\arcsin x$?

    Svar

    For $|x|<1$,

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

76. Spjald 76

    Spurning

    Notation for the third derivative of $f$?

    Svar

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

77. Spjald 77

    Spurning

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

    Svar

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

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

78. Spjald 78

    Spurning

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

    Svar

    Where $y\ne0$,

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

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

79. Spjald 79

    Spurning

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

    Svar

    Provided $f'(a)\ne0$,

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

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

80. Spjald 80

    Spurning

    Derivative of $\arctan x$?

    Svar

    For every real $x$,

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

81. Spjald 81

    Spurning

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

    Svar

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

    The extra factor is the chain rule.

82. Spjald 82

    Spurning

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

    Svar

    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. Spjald 83

    Spurning

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

    Svar

    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. Spjald 84

    Spurning

    Derivative of $\arccos x$?

    Svar

    For $|x|<1$,

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

85. Spjald 85

    Spurning

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

    Svar

    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. Spjald 86

    Spurning

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

    Svar

    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. Spjald 87

    Spurning

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

    Svar

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

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

88. Spjald 88

    Spurning

    How are tangent slopes of inverse graphs related?

    Svar

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

89. Spjald 89

    Spurning

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

    Svar

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

90. Spjald 90

    Spurning

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

    Svar

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

91. Spjald 91

    Spurning

    Horizontal tangent on an implicit curve: derivative condition?

    Svar

    $dy/dx=0$ 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. Spjald 92

    Spurning

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

    Svar

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

93. Spjald 93

    Spurning

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

    Svar

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

94. Spjald 94

    Spurning

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

    Svar

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

    This combines the power rule with the chain rule.

95. Spjald 95

    Spurning

    Vertical tangent on an implicit curve: derivative clue?

    Svar

    $dy/dx$ 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. Spjald 96

    Spurning

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

    Svar

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

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

97. Spjald 97

    Spurning

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

    Svar

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

98. Spjald 98

    Spurning

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

    Svar

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

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

99. Spjald 99

    Spurning

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

    Svar

    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. Spjald 100

     Spurning

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

     Svar

     When the functions are differentiable, the chain rule gives

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

101. Spjald 101

     Spurning

     What local property lets a function have an inverse derivative?

     Svar

     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. Spjald 102

     Spurning

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

     Svar

     For $a>0$,

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

103. Spjald 103

     Spurning

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

     Svar

     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. Spjald 104

     Spurning

     Position, velocity, and acceleration relationships?

     Svar

     For position $s(t)$,

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

105. Spjald 105

     Spurning

     Central idea of a related-rates problem?

     Svar

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

106. Spjald 106

     Spurning

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

     Svar

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

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

107. Spjald 107

     Spurning

     L’Hospital’s Rule: basic conditions?

     Svar

     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. Spjald 108

     Spurning

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

     Svar

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

109. Spjald 109

     Spurning

     Speed in terms of velocity?

     Svar

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Spjald 110

     Spurning

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

     Svar

     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. Spjald 111

     Spurning

     Differential approximation connecting $dx$ and $dy$?

     Svar

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

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

112. Spjald 112

     Spurning

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

     Svar

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

113. Spjald 113

     Spurning

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

     Svar

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

114. Spjald 114

     Spurning

     What does positive acceleration say about velocity?

     Svar

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

115. Spjald 115

     Spurning

     Related rates: when should numerical values be substituted?

     Svar

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

116. Spjald 116

     Spurning

     How does concavity predict linearization error?

     Svar

     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. Spjald 117

     Spurning

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

     Svar

     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. Spjald 118

     Spurning

     When is a particle moving in the positive direction?

     Svar

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

119. Spjald 119

     Spurning

     How can velocity show a change of direction?

     Svar

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

120. Spjald 120

     Spurning

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

     Svar

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

121. Spjald 121

     Spurning

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

     Svar

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

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

122. Spjald 122

     Spurning

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

     Svar

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

123. Spjald 123

     Spurning

     What must a contextual derivative sentence include?

     Svar

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

124. Spjald 124

     Spurning

     Velocity negative and acceleration positive: what happens?

     Svar

     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. Spjald 125

     Spurning

     How should a negative related rate be interpreted?

     Svar

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

126. Spjald 126

     Spurning

     When is local linearity a sound approximation tool?

     Svar

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

127. Spjald 127

     Spurning

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

     Svar

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

128. Spjald 128

     Spurning

     When is speed increasing?

     Svar

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

129. Spjald 129

     Spurning

     Volume changes with time: notation for its rate?

     Svar

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

130. Spjald 130

     Spurning

     Why are similar triangles useful in related rates?

     Svar

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

131. Spjald 131

     Spurning

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

     Svar

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

132. Spjald 132

     Spurning

     What conclusion does L’Hospital’s Rule permit?

     Svar

     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. Spjald 133

     Spurning

     When is speed decreasing?

     Svar

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

134. Spjald 134

     Spurning

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

     Svar

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

135. Spjald 135

     Spurning

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

     Svar

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

136. Spjald 136

     Spurning

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

     Svar

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

137. Spjald 137

     Spurning

     Extreme Value Theorem: hypothesis and conclusion?

     Svar

     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. Spjald 138

     Spurning

     What is a critical number of $f$?

     Svar

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

139. Spjald 139

     Spurning

     First derivative test for a local maximum?

     Svar

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

140. Spjald 140

     Spurning

     Second-derivative sign for concave up?

     Svar

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

141. Spjald 141

     Spurning

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

     Svar

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

142. Spjald 142

     Spurning

     First step in an optimization model?

     Svar

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

143. Spjald 143

     Spurning

     Mean Value Theorem: hypotheses and conclusion?

     Svar

     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. Spjald 144

     Spurning

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

     Svar

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

     

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145. Spjald 145

     Spurning

     First derivative test for a local minimum?

     Svar

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

146. Spjald 146

     Spurning

     What must happen at an inflection point?

     Svar

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

147. Spjald 147

     Spurning

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

     Svar

     $f''$ 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. Spjald 148

     Spurning

     How do you confirm an optimization answer is absolute?

     Svar

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

149. Spjald 149

     Spurning

     Rolle’s Theorem: hypotheses and conclusion?

     Svar

     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. Spjald 150

     Spurning

     Difference between absolute and relative extrema?

     Svar

     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. Spjald 151

     Spurning

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

     Svar

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

152. Spjald 152

     Spurning

     Second derivative test for a local minimum?

     Svar

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

153. Spjald 153

     Spurning

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

     Svar

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

154. Spjald 154

     Spurning

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

     Svar

     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. Spjald 155

     Spurning

     Which theorem links an average slope to an instantaneous slope?

     Svar

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

156. Spjald 156

     Spurning

     How can an implicit derivative locate a horizontal tangent?

     Svar

     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. Spjald 157

     Spurning

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

     Svar

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

158. Spjald 158

     Spurning

     Second derivative test for a local maximum?

     Svar

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

159. Spjald 159

     Spurning

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

     Svar

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

160. Spjald 160

     Spurning

     Why must an optimization domain be stated?

     Svar

     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. Spjald 161

     Spurning

     Which theorem guarantees absolute extrema, not where they occur?

     Svar

     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. Spjald 162

     Spurning

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

     Svar

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

163. Spjald 163

     Spurning

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

     Svar

     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. Spjald 164

     Spurning

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

     Svar

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

165. Spjald 165

     Spurning

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

     Svar

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

166. Spjald 166

     Spurning

     How can an implicit derivative locate a vertical tangent?

     Svar

     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. Spjald 167

     Spurning

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

     Svar

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

168. Spjald 168

     Spurning

     Why are endpoints included in the candidates test?

     Svar

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

169. Spjald 169

     Spurning

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

     Svar

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

170. Spjald 170

     Spurning

     Second-derivative sign for concave down?

     Svar

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

171. Spjald 171

     Spurning

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

     Svar

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

172. Spjald 172

     Spurning

     What should the final line of an optimization solution state?

     Svar

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

173. Spjald 173

     Spurning

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

     Svar

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

174. Spjald 174

     Spurning

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

     Svar

     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. Spjald 175

     Spurning

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

     Svar

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

176. Spjald 176

     Spurning

     Left Riemann sum on equal subintervals?

     Svar

     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. Spjald 177

     Spurning

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

     Svar

     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. Spjald 178

     Spurning

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Svar

     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. Spjald 179

     Spurning

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

     Svar

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Spjald 180

     Spurning

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

     Svar

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

181. Spjald 181

     Spurning

     Right Riemann sum on equal subintervals?

     Svar

     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. Spjald 182

     Spurning

     How does reversing integral bounds change the value?

     Svar

     It changes the sign:

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

183. Spjald 183

     Spurning

     Net Change Theorem?

     Svar

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

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

184. Spjald 184

     Spurning

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

     Svar

     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. Spjald 185

     Spurning

     Power rule for antiderivatives?

     Svar

     For $n\ne-1$,

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

186. Spjald 186

     Spurning

     Midpoint Riemann sum on equal subintervals?

     Svar

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

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

187. Spjald 187

     Spurning

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

     Svar

     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. Spjald 188

     Spurning

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

     Svar

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Spjald 189

     Spurning

     Antiderivative of $1/x$?

     Svar

     On any interval not crossing zero,

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

190. Spjald 190

     Spurning

     Trapezoidal approximation on equal subintervals?

     Svar

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

191. Spjald 191

     Spurning

     How do geometric regions help evaluate a definite integral?

     Svar

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

192. Spjald 192

     Spurning

     Basic antiderivatives of sine and cosine?

     Svar

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

193. Spjald 193

     Spurning

     Definite integral as a limit of Riemann sums?

     Svar

     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. Spjald 194

     Spurning

     Constant-multiple rule for integrals?

     Svar

     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. Spjald 195

     Spurning

     What pattern suggests $u$-substitution?

     Svar

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

196. Spjald 196

     Spurning

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

     Svar

     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. Spjald 197

     Spurning

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

     Svar

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

198. Spjald 198

     Spurning

     Sum-and-difference rule for definite integrals?

     Svar

     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. Spjald 199

     Spurning

     Basic antiderivative of $e^x$?

     Svar

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

200. Spjald 200

     Spurning

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

     Svar

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

201. Spjald 201

     Spurning

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

     Svar

     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. Spjald 202

     Spurning

     How does concavity predict trapezoidal and midpoint error?

     Svar

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

203. Spjald 203

     Spurning

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

     Svar

     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. Spjald 204

     Spurning

     What denominator pattern suggests an arctangent antiderivative?

     Svar

     After completing the square and scaling, a form like

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

205. Spjald 205

     Spurning

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

     Svar

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

206. Spjald 206

     Spurning

     How does an initial condition determine an antiderivative?

     Svar

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

207. Spjald 207

     Spurning

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

     Svar

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

208. Spjald 208

     Spurning

     Why does an indefinite integral include $+C$?

     Svar

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

209. Spjald 209

     Spurning

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

     Svar

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

210. Spjald 210

     Spurning

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

     Svar

     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. Spjald 211

     Spurning

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

     Svar

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

212. Spjald 212

     Spurning

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

     Svar

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

213. Spjald 213

     Spurning

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

     Svar

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

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

214. Spjald 214

     Spurning

     Riemann sum for unequal subinterval widths?

     Svar

     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. Spjald 215

     Spurning

     Does continuity guarantee integrability on a closed interval?

     Svar

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

216. Spjald 216

     Spurning

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

     Svar

     Where $g(x)\ne0$,

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

217. Spjald 217

     Spurning

     What algebraic rewrites often reveal a basic antiderivative?

     Svar

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

218. Spjald 218

     Spurning

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

     Svar

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

219. Spjald 219

     Spurning

     What is a differential equation?

     Svar

     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. Spjald 220

     Spurning

     How does a verbal rate statement become a differential equation?

     Svar

     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. Spjald 221

     Spurning

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

     Svar

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

222. Spjald 222

     Spurning

     General solution versus particular solution?

     Svar

     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. Spjald 223

     Spurning

     What does one segment in a slope field show?

     Svar

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

224. Spjald 224

     Spurning

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

     Svar

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

225. Spjald 225

     Spurning

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

     Svar

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

226. Spjald 226

     Spurning

     What makes a first-order differential equation separable?

     Svar

     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. Spjald 227

     Spurning

     What is an initial value problem?

     Svar

     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. Spjald 228

     Spurning

     What is an isocline in a slope field?

     Svar

     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. Spjald 229

     Spurning

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

     Svar

     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. Spjald 230

     Spurning

     General solution of $dy/dt=ky$?

     Svar

     $$y=Ce^{kt}$$

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

231. Spjald 231

     Spurning

     Core method for solving a separable differential equation?

     Svar

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

232. Spjald 232

     Spurning

     How should a solution curve follow a slope field?

     Svar

     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. Spjald 233

     Spurning

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

     Svar

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

234. Spjald 234

     Spurning

     Why is one integration constant enough after integrating both sides?

     Svar

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

235. Spjald 235

     Spurning

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

     Svar

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

236. Spjald 236

     Spurning

     Can one differential equation have infinitely many solutions?

     Svar

     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. Spjald 237

     Spurning

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

     Svar

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

238. Spjald 238

     Spurning

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

     Svar

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

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

239. Spjald 239

     Spurning

     What can be lost when dividing to separate variables?

     Svar

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

240. Spjald 240

     Spurning

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

     Svar

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

241. Spjald 241

     Spurning

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

     Svar

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

242. Spjald 242

     Spurning

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

     Svar

     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. Spjald 243

     Spurning

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

     Svar

     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. Spjald 244

     Spurning

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

     Svar

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

245. Spjald 245

     Spurning

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

     Svar

     For $k>0$,

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

     It is independent of the initial amount.

246. Spjald 246

     Spurning

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

     Svar

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

247. Spjald 247

     Spurning

     How is an initial condition used after separation?

     Svar

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

248. Spjald 248

     Spurning

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

     Svar

     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. Spjald 249

     Spurning

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

     Svar

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

250. Spjald 250

     Spurning

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

     Svar

     For $k<0$,

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

251. Spjald 251

     Spurning

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

     Svar

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

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

252. Spjald 252

     Spurning

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

     Svar

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

     Velocity below zero contributes negative displacement.

253. Spjald 253

     Spurning

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

     Svar

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

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

     Think top minus bottom.

254. Spjald 254

     Spurning

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

     Svar

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

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

255. Spjald 255

     Spurning

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Svar

     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. Spjald 256

     Spurning

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

     Svar

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

257. Spjald 257

     Spurning

     Cross-sectional area when each slice is a square?

     Svar

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

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

258. Spjald 258

     Spurning

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

     Svar

     Integrate the net rate:

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

259. Spjald 259

     Spurning

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

     Svar

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

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

     Think right minus left.

260. Spjald 260

     Spurning

     Disc-method volume formula?

     Svar

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

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

261. Spjald 261

     Spurning

     What units does average value have?

     Svar

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

262. Spjald 262

     Spurning

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

     Svar

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

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

263. Spjald 263

     Spurning

     Cross-sectional area when each slice is a rectangle?

     Svar

     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. Spjald 264

     Spurning

     How do you determine bounds for area between curves?

     Svar

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

265. Spjald 265

     Spurning

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

     Svar

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

266. Spjald 266

     Spurning

     When is a particle moving to the right or left?

     Svar

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

267. Spjald 267

     Spurning

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

     Svar

     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. Spjald 268

     Spurning

     Why must an area integral be split where curves intersect?

     Svar

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

269. Spjald 269

     Spurning

     How can a velocity table approximate displacement?

     Svar

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

270. Spjald 270

     Spurning

     Washer-method volume formula?

     Svar

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

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

271. Spjald 271

     Spurning

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

     Svar

     If $s(a)$ is known,

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

272. Spjald 272

     Spurning

     How do you choose between vertical and horizontal area slices?

     Svar

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

273. Spjald 273

     Spurning

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

     Svar

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

274. Spjald 274

     Spurning

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

     Svar

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

275. Spjald 275

     Spurning

     Single expression for area between two curves?

     Svar

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Spjald 276

     Spurning

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

     Svar

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

277. Spjald 277

     Spurning

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

     Svar

     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. Spjald 278

     Spurning

     When should a volume integral use $dy$?

     Svar

     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. Spjald 279

     Spurning

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

     Svar

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

280. Spjald 280

     Spurning

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

     Svar

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

281. Spjald 281

     Spurning

     Why must total distance split at velocity sign changes?

     Svar

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

282. Spjald 282

     Spurning

     When does an accumulated quantity reach a local maximum?

     Svar

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

283. Spjald 283

     Spurning

     What distinguishes area from a definite integral?

     Svar

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

284. Spjald 284

     Spurning

     How do position, velocity, and acceleration graphs correspond?

     Svar

     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. Spjald 285

     Spurning

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

     Svar

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

286. Spjald 286

     Spurning

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

     Svar

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

287. Spjald 287

     Spurning

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

     Svar

     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. Spjald 288

     Spurning

     Why should a contextual integral answer include a sentence?

     Svar

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