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

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1. Kort 1

   Fråga

   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. Kort 2

   Fråga

   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. Kort 3

   Fråga

   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. Kort 4

   Fråga

   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. Kort 5

   Fråga

   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. Kort 6

   Fråga

   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. Kort 7

   Fråga

   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. Kort 8

   Fråga

   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. Kort 9

   Fråga

   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. Kort 10

    Fråga

    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. Kort 11

    Fråga

    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. Kort 12

    Fråga

    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. Kort 13

    Fråga

    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. Kort 14

    Fråga

    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. Kort 15

    Fråga

    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. Kort 16

    Fråga

    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. Kort 17

    Fråga

    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. Kort 18

    Fråga

    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. Kort 19

    Fråga

    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. Kort 20

    Fråga

    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. Kort 21

    Fråga

    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. Kort 22

    Fråga

    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. Kort 23

    Fråga

    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. Kort 24

    Fråga

    What makes a discontinuity infinite?

    Svar

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

25. Kort 25

    Fråga

    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. Kort 26

    Fråga

    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. Kort 27

    Fråga

    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. Kort 28

    Fråga

    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. Kort 29

    Fråga

    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. Kort 30

    Fråga

    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. Kort 31

    Fråga

    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. Kort 32

    Fråga

    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. Kort 33

    Fråga

    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. Kort 34

    Fråga

    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. Kort 35

    Fråga

    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. Kort 36

    Fråga

    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. Kort 37

    Fråga

    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. Kort 38

    Fråga

    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. Kort 39

    Fråga

    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. Kort 40

    Fråga

    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. Kort 41

    Fråga

    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. Kort 42

    Fråga

    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. Kort 43

    Fråga

    Derivative of a constant?

    Svar

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

    A constant function has zero rate of change.

44. Kort 44

    Fråga

    Derivative of $\sin x$?

    Svar

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

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

45. Kort 45

    Fråga

    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. Kort 46

    Fråga

    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. Kort 47

    Fråga

    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. Kort 48

    Fråga

    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. Kort 49

    Fråga

    Derivative of a sum or difference?

    Svar

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

50. Kort 50

    Fråga

    Derivative of $\cos x$?

    Svar

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

    The standard formula assumes radians.

51. Kort 51

    Fråga

    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. Kort 52

    Fråga

    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. Kort 53

    Fråga

    Derivative of $e^x$?

    Svar

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

54. Kort 54

    Fråga

    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. Kort 55

    Fråga

    Derivative of $\tan x$?

    Svar

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kort 56

    Fråga

    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. Kort 57

    Fråga

    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. Kort 58

    Fråga

    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. Kort 59

    Fråga

    Derivative of $\csc x$?

    Svar

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kort 60

    Fråga

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

    Svar

    $f$ is increasing on that interval.

61. Kort 61

    Fråga

    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. Kort 62

    Fråga

    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. Kort 63

    Fråga

    Derivative of $\sec x$?

    Svar

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kort 64

    Fråga

    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. Kort 65

    Fråga

    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. Kort 66

    Fråga

    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. Kort 67

    Fråga

    Derivative of $\cot x$?

    Svar

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kort 68

    Fråga

    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. Kort 69

    Fråga

    Constant-multiple rule?

    Svar

    For a constant $c$,

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

70. Kort 70

    Fråga

    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. Kort 71

    Fråga

    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. Kort 72

    Fråga

    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. Kort 73

    Fråga

    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. Kort 74

    Fråga

    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. Kort 75

    Fråga

    Derivative of $\arcsin x$?

    Svar

    For $|x|<1$,

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

76. Kort 76

    Fråga

    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. Kort 77

    Fråga

    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. Kort 78

    Fråga

    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. Kort 79

    Fråga

    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. Kort 80

    Fråga

    Derivative of $\arctan x$?

    Svar

    For every real $x$,

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

81. Kort 81

    Fråga

    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. Kort 82

    Fråga

    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. Kort 83

    Fråga

    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. Kort 84

    Fråga

    Derivative of $\arccos x$?

    Svar

    For $|x|<1$,

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

85. Kort 85

    Fråga

    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. Kort 86

    Fråga

    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. Kort 87

    Fråga

    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. Kort 88

    Fråga

    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. Kort 89

    Fråga

    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. Kort 90

    Fråga

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

    Svar

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

91. Kort 91

    Fråga

    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. Kort 92

    Fråga

    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. Kort 93

    Fråga

    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. Kort 94

    Fråga

    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. Kort 95

    Fråga

    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. Kort 96

    Fråga

    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. Kort 97

    Fråga

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

    Svar

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

98. Kort 98

    Fråga

    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. Kort 99

    Fråga

    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. Kort 100

     Fråga

     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. Kort 101

     Fråga

     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. Kort 102

     Fråga

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

     Svar

     For $a>0$,

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

103. Kort 103

     Fråga

     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. Kort 104

     Fråga

     Position, velocity, and acceleration relationships?

     Svar

     For position $s(t)$,

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

105. Kort 105

     Fråga

     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. Kort 106

     Fråga

     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. Kort 107

     Fråga

     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. Kort 108

     Fråga

     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. Kort 109

     Fråga

     Speed in terms of velocity?

     Svar

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kort 110

     Fråga

     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. Kort 111

     Fråga

     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. Kort 112

     Fråga

     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. Kort 113

     Fråga

     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. Kort 114

     Fråga

     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. Kort 115

     Fråga

     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. Kort 116

     Fråga

     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. Kort 117

     Fråga

     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. Kort 118

     Fråga

     When is a particle moving in the positive direction?

     Svar

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

119. Kort 119

     Fråga

     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. Kort 120

     Fråga

     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. Kort 121

     Fråga

     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. Kort 122

     Fråga

     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. Kort 123

     Fråga

     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. Kort 124

     Fråga

     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. Kort 125

     Fråga

     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. Kort 126

     Fråga

     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. Kort 127

     Fråga

     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. Kort 128

     Fråga

     When is speed increasing?

     Svar

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

129. Kort 129

     Fråga

     Volume changes with time: notation for its rate?

     Svar

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

130. Kort 130

     Fråga

     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. Kort 131

     Fråga

     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. Kort 132

     Fråga

     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. Kort 133

     Fråga

     When is speed decreasing?

     Svar

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

134. Kort 134

     Fråga

     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. Kort 135

     Fråga

     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. Kort 136

     Fråga

     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. Kort 137

     Fråga

     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. Kort 138

     Fråga

     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. Kort 139

     Fråga

     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. Kort 140

     Fråga

     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. Kort 141

     Fråga

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

     Svar

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

142. Kort 142

     Fråga

     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. Kort 143

     Fråga

     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. Kort 144

     Fråga

     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. Kort 145

     Fråga

     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. Kort 146

     Fråga

     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. Kort 147

     Fråga

     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. Kort 148

     Fråga

     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. Kort 149

     Fråga

     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. Kort 150

     Fråga

     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. Kort 151

     Fråga

     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. Kort 152

     Fråga

     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. Kort 153

     Fråga

     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. Kort 154

     Fråga

     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. Kort 155

     Fråga

     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. Kort 156

     Fråga

     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. Kort 157

     Fråga

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

     Svar

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

158. Kort 158

     Fråga

     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. Kort 159

     Fråga

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

     Svar

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

160. Kort 160

     Fråga

     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. Kort 161

     Fråga

     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. Kort 162

     Fråga

     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. Kort 163

     Fråga

     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. Kort 164

     Fråga

     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. Kort 165

     Fråga

     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. Kort 166

     Fråga

     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. Kort 167

     Fråga

     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. Kort 168

     Fråga

     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. Kort 169

     Fråga

     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. Kort 170

     Fråga

     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. Kort 171

     Fråga

     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. Kort 172

     Fråga

     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. Kort 173

     Fråga

     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. Kort 174

     Fråga

     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. Kort 175

     Fråga

     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. Kort 176

     Fråga

     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. Kort 177

     Fråga

     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. Kort 178

     Fråga

     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. Kort 179

     Fråga

     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. Kort 180

     Fråga

     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. Kort 181

     Fråga

     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. Kort 182

     Fråga

     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. Kort 183

     Fråga

     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. Kort 184

     Fråga

     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. Kort 185

     Fråga

     Power rule for antiderivatives?

     Svar

     For $n\ne-1$,

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

186. Kort 186

     Fråga

     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. Kort 187

     Fråga

     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. Kort 188

     Fråga

     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. Kort 189

     Fråga

     Antiderivative of $1/x$?

     Svar

     On any interval not crossing zero,

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

190. Kort 190

     Fråga

     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. Kort 191

     Fråga

     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. Kort 192

     Fråga

     Basic antiderivatives of sine and cosine?

     Svar

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

193. Kort 193

     Fråga

     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. Kort 194

     Fråga

     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. Kort 195

     Fråga

     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. Kort 196

     Fråga

     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. Kort 197

     Fråga

     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. Kort 198

     Fråga

     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. Kort 199

     Fråga

     Basic antiderivative of $e^x$?

     Svar

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

200. Kort 200

     Fråga

     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. Kort 201

     Fråga

     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. Kort 202

     Fråga

     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. Kort 203

     Fråga

     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. Kort 204

     Fråga

     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. Kort 205

     Fråga

     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. Kort 206

     Fråga

     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. Kort 207

     Fråga

     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. Kort 208

     Fråga

     Why does an indefinite integral include $+C$?

     Svar

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

209. Kort 209

     Fråga

     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. Kort 210

     Fråga

     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. Kort 211

     Fråga

     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. Kort 212

     Fråga

     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. Kort 213

     Fråga

     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. Kort 214

     Fråga

     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. Kort 215

     Fråga

     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. Kort 216

     Fråga

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

     Svar

     Where $g(x)\ne0$,

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

217. Kort 217

     Fråga

     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. Kort 218

     Fråga

     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. Kort 219

     Fråga

     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. Kort 220

     Fråga

     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. Kort 221

     Fråga

     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. Kort 222

     Fråga

     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. Kort 223

     Fråga

     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. Kort 224

     Fråga

     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. Kort 225

     Fråga

     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. Kort 226

     Fråga

     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. Kort 227

     Fråga

     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. Kort 228

     Fråga

     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. Kort 229

     Fråga

     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. Kort 230

     Fråga

     General solution of $dy/dt=ky$?

     Svar

     $$y=Ce^{kt}$$

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

231. Kort 231

     Fråga

     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. Kort 232

     Fråga

     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. Kort 233

     Fråga

     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. Kort 234

     Fråga

     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. Kort 235

     Fråga

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

     Svar

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

236. Kort 236

     Fråga

     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. Kort 237

     Fråga

     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. Kort 238

     Fråga

     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. Kort 239

     Fråga

     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. Kort 240

     Fråga

     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. Kort 241

     Fråga

     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. Kort 242

     Fråga

     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. Kort 243

     Fråga

     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. Kort 244

     Fråga

     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. Kort 245

     Fråga

     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. Kort 246

     Fråga

     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. Kort 247

     Fråga

     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. Kort 248

     Fråga

     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. Kort 249

     Fråga

     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. Kort 250

     Fråga

     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. Kort 251

     Fråga

     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. Kort 252

     Fråga

     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. Kort 253

     Fråga

     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. Kort 254

     Fråga

     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. Kort 255

     Fråga

     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. Kort 256

     Fråga

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

     Svar

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

257. Kort 257

     Fråga

     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. Kort 258

     Fråga

     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. Kort 259

     Fråga

     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. Kort 260

     Fråga

     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. Kort 261

     Fråga

     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. Kort 262

     Fråga

     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. Kort 263

     Fråga

     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. Kort 264

     Fråga

     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. Kort 265

     Fråga

     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. Kort 266

     Fråga

     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. Kort 267

     Fråga

     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. Kort 268

     Fråga

     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. Kort 269

     Fråga

     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. Kort 270

     Fråga

     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. Kort 271

     Fråga

     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. Kort 272

     Fråga

     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. Kort 273

     Fråga

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

     Svar

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

274. Kort 274

     Fråga

     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. Kort 275

     Fråga

     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. Kort 276

     Fråga

     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. Kort 277

     Fråga

     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. Kort 278

     Fråga

     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. Kort 279

     Fråga

     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. Kort 280

     Fråga

     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. Kort 281

     Fråga

     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. Kort 282

     Fråga

     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. Kort 283

     Fråga

     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. Kort 284

     Fråga

     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. Kort 285

     Fråga

     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. Kort 286

     Fråga

     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. Kort 287

     Fråga

     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. Kort 288

     Fråga

     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.