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

Cards in this deck

1. Card 1

   Question

   What does

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

   say?

   Answer

   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. Card 2

   Question

   How can a table estimate

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

   ?

   Answer

   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. Card 3

   Question

   When does direct substitution evaluate a limit?

   Answer

   When the function is continuous at the target input. Then

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

4. Card 4

   Question

   Three conditions for continuity at

   $$x=a$$

   ?

   Answer

   $$f(a)$$

   exists,

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

   exists, and

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

5. Card 5

   Question

   Intermediate Value Theorem: hypotheses and conclusion?

   Answer

   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. Card 6

   Question

   When does a two-sided limit equal

   $$L$$

   ?

   Answer

   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. Card 7

   Question

   How do you read a finite limit from a graph?

   Answer

   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. Card 8

   Question

   Limit law for a sum or difference?

   Answer

   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. Card 9

   Question

   What makes a discontinuity removable?

   Answer

   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. Card 10

    Question

    Squeeze Theorem: usable form?

    Answer

    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. Card 11

    Question

    What does

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

    mean?

    Answer

    $$f(x)$$

    grows without bound above as

    $$x$$

    approaches

    $$a$$

    . It describes unbounded behavior, not a finite limit value.

12. Card 12

    Question

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

    Answer

    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. Card 13

    Question

    Limit law for a product?

    Answer

    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. Card 14

    Question

    Graph signature of a jump discontinuity?

    Answer

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

15. Card 15

    Question

    Which theorem can guarantee a root on

    $$[a,b]$$

    ?

    Answer

    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. Card 16

    Question

    Horizontal asymptote from a limit at infinity?

    Answer

    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. Card 17

    Question

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

    Answer

    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. Card 18

    Question

    Limit law for a quotient—and its condition?

    Answer

    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. Card 19

    Question

    What does continuity on

    $$[a,b]$$

    require at the endpoints?

    Answer

    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. Card 20

    Question

    When is the Squeeze Theorem a natural choice?

    Answer

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

21. Card 21

    Question

    Vertical asymptote from one-sided behavior?

    Answer

    If at least one one-sided limit at

    $$x=a$$

    is

    $$+\infty$$

    or

    $$-\infty$$

    , then

    $$x=a$$

    is a vertical asymptote.

22. Card 22

    Question

    Limit at infinity of equal-degree rational functions?

    Answer

    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. Card 23

    Question

    When can a limit pass through a continuous outer function?

    Answer

    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. Card 24

    Question

    What makes a discontinuity infinite?

    Answer

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

25. Card 25

    Question

    Left limit

    $$=2$$

    and right limit

    $$=5$$

    : two-sided limit?

    Answer

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

26. Card 26

    Question

    Standard trigonometric limit behind

    $$\frac{\sin x}{x}$$

    ?

    Answer

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Card 27

    Question

    Continuity of a composition?

    Answer

    If

    $$g$$

    is continuous at

    $$a$$

    and

    $$f$$

    is continuous at

    $$g(a)$$

    , then

    $$f\circ g$$

    is continuous at

    $$a$$

    .

28. Card 28

    Question

    Limit at infinity when a rational numerator has lower degree?

    Answer

    $$0$$

    . If the numerator's degree is less than the denominator's, the denominator dominates as

    $$x\to\pm\infty$$

    .

29. Card 29

    Question

    What does the indeterminate form

    $$0/0$$

    tell you?

    Answer

    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. Card 30

    Question

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

    Answer

    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. Card 31

    Question

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

    Answer

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

32. Card 32

    Question

    Value of

    $$\lim_{x\to0}\frac{1-\cos x}{x}$$

    ?

    Answer

    $$0$$

    . Rationalizing gives a product involving

    $$\sin x/x$$

    and a factor that approaches

    $$0$$

    .

33. Card 33

    Question

    Can

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

    exist when

    $$f(a)$$

    doesn't?

    Answer

    Yes. A limit uses nearby values, so a hole at

    $$x=a$$

    can coexist with a finite two-sided limit.

34. Card 34

    Question

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

    Answer

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

35. Card 35

    Question

    Average rate of change of

    $$f$$

    on

    $$[a,b]$$

    ?

    Answer

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

    It is the slope of the secant line through

    $$(a,f(a))$$

    and

    $$(b,f(b))$$

    .

36. Card 36

    Question

    Derivative at

    $$x=a$$

    using an increment

    $$h$$

    ?

    Answer

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

    The derivative exists only if this finite limit exists.

37. Card 37

    Question

    Tangent-line equation to

    $$y=f(x)$$

    at

    $$x=a$$

    ?

    Answer

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

    This requires

    $$f'(a)$$

    to exist.

38. Card 38

    Question

    What does differentiability imply about continuity?

    Answer

    If

    $$f$$

    is differentiable at

    $$a$$

    , then

    $$f$$

    is continuous at

    $$a$$

    . The converse is false: continuity alone doesn't guarantee differentiability.

39. Card 39

    Question

    Power rule for derivatives?

    Answer

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

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

40. Card 40

    Question

    Units of

    $$f'(x)$$

    ?

    Answer

    Output units of

    $$f$$

    per input unit of

    $$x$$

    . A derivative is a rate of change, so its units are a quotient.

41. Card 41

    Question

    Derivative at

    $$x=a$$

    using

    $$x\to a$$

    ?

    Answer

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

    This is equivalent to the

    $$h$$

    -form after setting

    $$h=x-a$$

    .

42. Card 42

    Question

    How does a graph of

    $$f$$

    show the sign of

    $$f'$$

    ?

    Answer

    $$f'(x)>0$$

    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. Card 43

    Question

    Derivative of a constant?

    Answer

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

    A constant function has zero rate of change.

44. Card 44

    Question

    Derivative of

    $$\sin x$$

    ?

    Answer

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

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

45. Card 45

    Question

    Product rule?

    Answer

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

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

46. Card 46

    Question

    How can nearby table values estimate

    $$f'(a)$$

    ?

    Answer

    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. Card 47

    Question

    What does

    $$f''(x)$$

    measure?

    Answer

    The rate of change of

    $$f'(x)$$

    with respect to

    $$x$$

    . Its units are the units of

    $$f$$

    per square input unit.

48. Card 48

    Question

    Instantaneous rate of change of

    $$f$$

    at

    $$a$$

    ?

    Answer

    $$f'(a)$$

    . It is the limit of average rates over intervals shrinking to

    $$a$$

    , and geometrically it is the tangent-line slope.

49. Card 49

    Question

    Derivative of a sum or difference?

    Answer

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

50. Card 50

    Question

    Derivative of

    $$\cos x$$

    ?

    Answer

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

    The standard formula assumes radians.

51. Card 51

    Question

    Quotient rule?

    Answer

    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. Card 52

    Question

    Common notations for the first derivative?

    Answer

    $$f'(x)$$

    ,

    $$y'$$

    ,

    $$\dfrac{dy}{dx}$$

    , and

    $$\dfrac{d}{dx}f(x)$$

    . They describe the same derivative in different contexts.

53. Card 53

    Question

    Derivative of

    $$e^x$$

    ?

    Answer

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

54. Card 54

    Question

    What graph features can make

    $$f$$

    nondifferentiable?

    Answer

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

55. Card 55

    Question

    Derivative of

    $$\tan x$$

    ?

    Answer

    Where

    $$\tan x$$

    is defined,

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

    Angles are in radians.

56. Card 56

    Question

    What does the derivative function

    $$f'$$

    assign to each input?

    Answer

    The instantaneous rate of change—or tangent slope—of

    $$f$$

    at that input, wherever the derivative exists.

57. Card 57

    Question

    Derivative of

    $$\ln x$$

    ?

    Answer

    For

    $$x>0$$

    ,

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

    More generally,

    $$d(\ln|x|)/dx=1/x$$

    for

    $$x\ne0$$

    .

58. Card 58

    Question

    How does the power rule handle roots or negative powers?

    Answer

    Rewrite them as

    $$x^n$$

    and apply

    $$nx^{n-1}$$

    on intervals where the real-valued expression is defined. Domain restrictions still matter.

59. Card 59

    Question

    Derivative of

    $$\csc x$$

    ?

    Answer

    Where

    $$\csc x$$

    is defined,

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

    Angles are in radians.

60. Card 60

    Question

    If

    $$f'(x)>0$$

    throughout an interval, what does

    $$f$$

    do there?

    Answer

    $$f$$

    is increasing on that interval.

61. Card 61

    Question

    Derivative of

    $$a^x$$

    for a constant base?

    Answer

    For

    $$a>0$$

    ,

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

    When

    $$a=1$$

    , the derivative is

    $$0$$

    .

62. Card 62

    Question

    How can a graph estimate

    $$f'(a)$$

    ?

    Answer

    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. Card 63

    Question

    Derivative of

    $$\sec x$$

    ?

    Answer

    Where

    $$\sec x$$

    is defined,

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

    Angles are in radians.

64. Card 64

    Question

    If

    $$f''(x)>0$$

    , how is

    $$f'$$

    changing?

    Answer

    $$f'$$

    is increasing. This is also the derivative condition associated with

    $$f$$

    being concave up.

65. Card 65

    Question

    Derivative of

    $$\log_a x$$

    ?

    Answer

    For

    $$x>0$$

    ,

    $$a>0$$

    , and

    $$a\ne1$$

    ,

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

66. Card 66

    Question

    Product rule from a table at

    $$x=a$$

    ?

    Answer

    For

    $$h=fg$$

    ,

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

    Use the four table entries at the same input.

67. Card 67

    Question

    Derivative of

    $$\cot x$$

    ?

    Answer

    Where

    $$\cot x$$

    is defined,

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

    Angles are in radians.

68. Card 68

    Question

    Why isn't

    $$|x|$$

    differentiable at

    $$x=0$$

    ?

    Answer

    Its left-hand slope is

    $$-1$$

    and right-hand slope is

    $$1$$

    . The one-sided derivative limits disagree, creating a corner.

69. Card 69

    Question

    Constant-multiple rule?

    Answer

    For a constant

    $$c$$

    ,

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

70. Card 70

    Question

    Quotient rule from a table at

    $$x=a$$

    ?

    Answer

    For

    $$h=f/g$$

    with

    $$g(a)\ne0$$

    ,

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

71. Card 71

    Question

    Chain rule for

    $$f(g(x))$$

    ?

    Answer

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

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

72. Card 72

    Question

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

    Answer

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

73. Card 73

    Question

    Core rule when differentiating an implicit equation in

    $$x$$

    and

    $$y$$

    ?

    Answer

    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. Card 74

    Question

    Derivative of an inverse function at

    $$x$$

    ?

    Answer

    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. Card 75

    Question

    Derivative of

    $$\arcsin x$$

    ?

    Answer

    For

    $$|x|<1$$

    ,

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

76. Card 76

    Question

    Notation for the third derivative of

    $$f$$

    ?

    Answer

    $$f'''(x)$$

    or

    $$\dfrac{d^3f}{dx^3}$$

    . The exponent on

    $$d$$

    indicates derivative order; it is not an ordinary power.

77. Card 77

    Question

    If

    $$h(x)=f(g(x))$$

    , what table entries give

    $$h'(a)$$

    ?

    Answer

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

    Use

    $$g(a)$$

    to find the input needed for the table entry of

    $$f'$$

    .

78. Card 78

    Question

    For

    $$x^2+y^2=r^2$$

    , what is

    $$dy/dx$$

    ?

    Answer

    Where

    $$y\ne0$$

    ,

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

    Differentiate to get

    $$2x+2y(dy/dx)=0$$

    .

79. Card 79

    Question

    If

    $$f(a)=b$$

    , how do you find

    $$[f^{-1}]'(b)$$

    ?

    Answer

    Provided

    $$f'(a)\ne0$$

    ,

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

    The inverse swaps the input-output pair

    $$(a,b)$$

    .

80. Card 80

    Question

    Derivative of

    $$\arctan x$$

    ?

    Answer

    For every real

    $$x$$

    ,

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

81. Card 81

    Question

    Derivative of

    $$e^{g(x)}$$

    ?

    Answer

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

    The extra factor is the chain rule.

82. Card 82

    Question

    Slope of a tangent to an implicit curve

    $$F(x,y)=0$$

    ?

    Answer

    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. Card 83

    Question

    Why must

    $$f'(a)\ne0$$

    to use

    $$[f^{-1}]'(f(a))=1/f'(a)$$

    ?

    Answer

    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. Card 84

    Question

    Derivative of

    $$\arccos x$$

    ?

    Answer

    For

    $$|x|<1$$

    ,

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

85. Card 85

    Question

    How do you find

    $$d^2y/dx^2$$

    for an implicit relation?

    Answer

    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. Card 86

    Question

    Derivative of

    $$\ln(g(x))$$

    ?

    Answer

    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. Card 87

    Question

    Derivative of

    $$y^n$$

    when

    $$y=y(x)$$

    ?

    Answer

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

    The

    $$dy/dx$$

    factor comes from the chain rule.

88. Card 88

    Question

    How are tangent slopes of inverse graphs related?

    Answer

    At reflected points

    $$(a,b)$$

    and

    $$(b,a)$$

    , the slopes are reciprocals when both are defined and nonzero.

89. Card 89

    Question

    Derivative of

    $$\arcsin(g(x))$$

    ?

    Answer

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

90. Card 90

    Question

    Derivative of

    $$\sin(g(x))$$

    ?

    Answer

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

91. Card 91

    Question

    Horizontal tangent on an implicit curve: derivative condition?

    Answer

    $$dy/dx=0$$

    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. Card 92

    Question

    How do you differentiate

    $$f^{-1}(x)$$

    without solving for the inverse?

    Answer

    Use the reciprocal derivative formula and the matching original input: find

    $$a$$

    with

    $$f(a)=x$$

    , then compute

    $$1/f'(a)$$

    .

93. Card 93

    Question

    Difference between

    $$f''(x)$$

    and

    $$[f'(x)]^2$$

    ?

    Answer

    $$f''(x)$$

    is the derivative of

    $$f'(x)$$

    . The expression

    $$[f'(x)]^2$$

    is the square of the first derivative; they are generally unrelated.

94. Card 94

    Question

    Derivative of

    $$[g(x)]^n$$

    ?

    Answer

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

    This combines the power rule with the chain rule.

95. Card 95

    Question

    Vertical tangent on an implicit curve: derivative clue?

    Answer

    $$dy/dx$$

    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. Card 96

    Question

    Table formula for an inverse derivative at

    $$x=b$$

    ?

    Answer

    Find

    $$a$$

    in the table with

    $$f(a)=b$$

    . If

    $$f'(a)\ne0$$

    , then

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

97. Card 97

    Question

    Derivative of

    $$\arctan(g(x))$$

    ?

    Answer

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

98. Card 98

    Question

    How do product and chain rules combine in

    $$f(x)g(h(x))$$

    ?

    Answer

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

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

99. Card 99

    Question

    Why can

    $$d^2y/dx^2$$

    depend on both

    $$x$$

    and

    $$y$$

    ?

    Answer

    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. Card 100

     Question

     A quantity

     $$y$$

     changes through

     $$u$$

     , which changes with

     $$x$$

     . How are the rates connected?

     Answer

     When the functions are differentiable, the chain rule gives

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

101. Card 101

     Question

     What local property lets a function have an inverse derivative?

     Answer

     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. Card 102

     Question

     Derivative of

     $$a^{g(x)}$$

     ?

     Answer

     For

     $$a>0$$

     ,

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

103. Card 103

     Question

     How should

     $$Q'(t)$$

     be interpreted in context?

     Answer

     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. Card 104

     Question

     Position, velocity, and acceleration relationships?

     Answer

     For position

     $$s(t)$$

     ,

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

105. Card 105

     Question

     Central idea of a related-rates problem?

     Answer

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

106. Card 106

     Question

     Linearization of

     $$f$$

     near

     $$x=a$$

     ?

     Answer

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

     For

     $$x$$

     close to

     $$a$$

     ,

     $$f(x)\approx L(x)$$

     .

107. Card 107

     Question

     L’Hospital’s Rule: basic conditions?

     Answer

     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. Card 108

     Question

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

     Answer

     Meters per second squared,

     $$\text{m}/\text{s}^2$$

     . Acceleration is the rate of change of velocity with respect to time.

109. Card 109

     Question

     Speed in terms of velocity?

     Answer

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Card 110

     Question

     Why do

     $$x$$

     and

     $$y$$

     gain

     $$dx/dt$$

     and

     $$dy/dt$$

     in related rates?

     Answer

     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. Card 111

     Question

     Differential approximation connecting

     $$dx$$

     and

     $$dy$$

     ?

     Answer

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

     For a small change

     $$\Delta x$$

     , the actual change satisfies

     $$\Delta y\approx f'(x)\Delta x$$

     .

112. Card 112

     Question

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

     Answer

     $$0/0$$

     and

     $$\infty/\infty$$

     . Other indeterminate forms must first be rewritten as an appropriate quotient.

113. Card 113

     Question

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

     Answer

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

114. Card 114

     Question

     What does positive acceleration say about velocity?

     Answer

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

115. Card 115

     Question

     Related rates: when should numerical values be substituted?

     Answer

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

116. Card 116

     Question

     How does concavity predict linearization error?

     Answer

     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. Card 117

     Question

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

     Answer

     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. Card 118

     Question

     When is a particle moving in the positive direction?

     Answer

     When

     $$v(t)>0$$

     . Position then increases as time increases.

119. Card 119

     Question

     How can velocity show a change of direction?

     Answer

     Velocity changes sign. A time with

     $$v(t)=0$$

     is only a candidate; confirm the sign differs on the two sides.

120. Card 120

     Question

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

     Answer

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

121. Card 121

     Question

     Tangent-line approximation of

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

     ?

     Answer

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

     It is most reliable for small

     $$|\Delta x|$$

     where the function is well approximated by its tangent.

122. Card 122

     Question

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

     Answer

     When the derivative quotient still has

     $$0/0$$

     or

     $$\infty/\infty$$

     form and the rule's conditions continue to hold.

123. Card 123

     Question

     What must a contextual derivative sentence include?

     Answer

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

124. Card 124

     Question

     Velocity negative and acceleration positive: what happens?

     Answer

     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. Card 125

     Question

     How should a negative related rate be interpreted?

     Answer

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

126. Card 126

     Question

     When is local linearity a sound approximation tool?

     Answer

     When

     $$f$$

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

127. Card 127

     Question

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

     Answer

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

128. Card 128

     Question

     When is speed increasing?

     Answer

     When velocity and acceleration have the same sign, so

     $$v(t)a(t)>0$$

     .

129. Card 129

     Question

     Volume changes with time: notation for its rate?

     Answer

     $$dV/dt$$

     . Its units are cubic length units per time unit.

130. Card 130

     Question

     Why are similar triangles useful in related rates?

     Answer

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

131. Card 131

     Question

     Meaning of

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

     in approximation?

     Answer

     $$dy$$

     is the tangent-line estimate of the actual output change

     $$\Delta y$$

     caused by an input change

     $$dx$$

     .

132. Card 132

     Question

     What conclusion does L’Hospital’s Rule permit?

     Answer

     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. Card 133

     Question

     When is speed decreasing?

     Answer

     When velocity and acceleration have opposite signs, so

     $$v(t)a(t)<0$$

     .

134. Card 134

     Question

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

     Answer

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

135. Card 135

     Question

     Does

     $$v(t)=0$$

     guarantee a particle changes direction?

     Answer

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

136. Card 136

     Question

     How do you translate “

     $$Q$$

     increases by 3 units per minute” into derivative notation?

     Answer

     $$dQ/dt=3$$

     in the stated time interval or at the stated instant. “Decreases by 3” would give

     $$dQ/dt=-3$$

     .

137. Card 137

     Question

     Extreme Value Theorem: hypothesis and conclusion?

     Answer

     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. Card 138

     Question

     What is a critical number of

     $$f$$

     ?

     Answer

     A number

     $$c$$

     in the domain of

     $$f$$

     where

     $$f'(c)=0$$

     or

     $$f'(c)$$

     doesn't exist.

139. Card 139

     Question

     First derivative test for a local maximum?

     Answer

     $$f'$$

     changes from positive to negative at the critical point, so

     $$f$$

     changes from increasing to decreasing.

140. Card 140

     Question

     Second-derivative sign for concave up?

     Answer

     If

     $$f''(x)>0$$

     on an interval, then

     $$f$$

     is concave up there and

     $$f'$$

     is increasing.

141. Card 141

     Question

     If the graph of

     $$f'$$

     is above the

     $$x$$

     -axis, what does

     $$f$$

     do?

     Answer

     $$f$$

     is increasing because

     $$f'(x)>0$$

     .

142. Card 142

     Question

     First step in an optimization model?

     Answer

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

143. Card 143

     Question

     Mean Value Theorem: hypotheses and conclusion?

     Answer

     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. Card 144

     Question

     Candidates test for absolute extrema on

     $$[a,b]$$

     ?

     Answer

     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. Card 145

     Question

     First derivative test for a local minimum?

     Answer

     $$f'$$

     changes from negative to positive at the critical point, so

     $$f$$

     changes from decreasing to increasing.

146. Card 146

     Question

     What must happen at an inflection point?

     Answer

     The graph's concavity changes. A zero or undefined value of

     $$f''$$

     is only a candidate; verify a concavity change.

147. Card 147

     Question

     If

     $$f'$$

     has a local maximum, what can that say about

     $$f$$

     ?

     Answer

     $$f''$$

     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. Card 148

     Question

     How do you confirm an optimization answer is absolute?

     Answer

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

149. Card 149

     Question

     Rolle’s Theorem: hypotheses and conclusion?

     Answer

     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. Card 150

     Question

     Difference between absolute and relative extrema?

     Answer

     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. Card 151

     Question

     If

     $$f$$

     is continuous at a critical number

     $$c$$

     and

     $$f'$$

     is positive on both sides, is there a local extremum?

     Answer

     No. The function is increasing through

     $$c$$

     , so it has no local extremum there.

152. Card 152

     Question

     Second derivative test for a local minimum?

     Answer

     If

     $$f'(c)=0$$

     and

     $$f''(c)>0$$

     , then

     $$f$$

     has a local minimum at

     $$c$$

     .

153. Card 153

     Question

     Zeros of

     $$f'$$

     correspond to what features of

     $$f$$

     ?

     Answer

     Horizontal tangents where

     $$f'$$

     exists. They are critical numbers and possible extrema, but each needs further sign or value analysis.

154. Card 154

     Question

     Implicit relation: how can

     $$dy/dx$$

     reveal local behavior?

     Answer

     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. Card 155

     Question

     Which theorem links an average slope to an instantaneous slope?

     Answer

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

156. Card 156

     Question

     How can an implicit derivative locate a horizontal tangent?

     Answer

     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. Card 157

     Question

     Derivative-sign chart: where is

     $$f$$

     decreasing?

     Answer

     On intervals where

     $$f'(x)<0$$

     .

158. Card 158

     Question

     Second derivative test for a local maximum?

     Answer

     If

     $$f'(c)=0$$

     and

     $$f''(c)<0$$

     , then

     $$f$$

     has a local maximum at

     $$c$$

     .

159. Card 159

     Question

     If

     $$f'$$

     is increasing, what is the concavity of

     $$f$$

     ?

     Answer

     $$f$$

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

160. Card 160

     Question

     Why must an optimization domain be stated?

     Answer

     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. Card 161

     Question

     Which theorem guarantees absolute extrema, not where they occur?

     Answer

     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. Card 162

     Question

     Can

     $$f'(c)$$

     fail to exist at a local extremum?

     Answer

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

     $$f'$$

     is undefined.

163. Card 163

     Question

     For a function continuous at

     $$c$$

     , what same-sign pattern in

     $$f'$$

     rules out a local extremum there?

     Answer

     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. Card 164

     Question

     If

     $$f'(c)=0$$

     and

     $$f''(c)=0$$

     , what does the second derivative test conclude?

     Answer

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

165. Card 165

     Question

     If the graph of

     $$f'$$

     crosses from negative to positive, what feature does

     $$f$$

     have?

     Answer

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

     $$f$$

     .

166. Card 166

     Question

     How can an implicit derivative locate a vertical tangent?

     Answer

     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. Card 167

     Question

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

     Answer

     $$f(a)=f(b)$$

     . The average slope becomes zero, so the guaranteed instantaneous slope is also zero.

168. Card 168

     Question

     Why are endpoints included in the candidates test?

     Answer

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

169. Card 169

     Question

     If

     $$f'(x)=0$$

     throughout an interval, what is

     $$f$$

     there?

     Answer

     $$f$$

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

170. Card 170

     Question

     Second-derivative sign for concave down?

     Answer

     If

     $$f''(x)<0$$

     on an interval, then

     $$f$$

     is concave down there and

     $$f'$$

     is decreasing.

171. Card 171

     Question

     Graph of

     $$f'$$

     has a local minimum: possible effect on

     $$f$$

     ?

     Answer

     $$f''$$

     may change from negative to positive, so

     $$f$$

     may change from concave down to concave up. Verify the sign change.

172. Card 172

     Question

     What should the final line of an optimization solution state?

     Answer

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

173. Card 173

     Question

     Can Rolle’s Theorem be used if

     $$f$$

     has a corner inside

     $$(a,b)$$

     ?

     Answer

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

174. Card 174

     Question

     How do

     $$f''$$

     zeros help analyze a graph?

     Answer

     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. Card 175

     Question

     What does the accumulation function

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

     measure?

     Answer

     The signed net accumulation of

     $$f$$

     from

     $$a$$

     to

     $$x$$

     . Contributions above the axis are positive; contributions below it are negative.

176. Card 176

     Question

     Left Riemann sum on equal subintervals?

     Answer

     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. Card 177

     Question

     What does

     $$\int_a^b f(x)\,dx$$

     represent geometrically?

     Answer

     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. Card 178

     Question

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Answer

     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. Card 179

     Question

     Derivative of

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

     ?

     Answer

     If

     $$f$$

     is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Card 180

     Question

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

     Answer

     If

     $$F'=f$$

     and

     $$G'=f$$

     on an interval, then

     $$(F-G)'=0$$

     , so

     $$F-G=C$$

     on that interval.

181. Card 181

     Question

     Right Riemann sum on equal subintervals?

     Answer

     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. Card 182

     Question

     How does reversing integral bounds change the value?

     Answer

     It changes the sign:

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

183. Card 183

     Question

     Net Change Theorem?

     Answer

     If

     $$Q'(t)$$

     is the rate of change of a quantity, then

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

184. Card 184

     Question

     Derivative of

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

     ?

     Answer

     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. Card 185

     Question

     Power rule for antiderivatives?

     Answer

     For

     $$n\ne-1$$

     ,

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

186. Card 186

     Question

     Midpoint Riemann sum on equal subintervals?

     Answer

     With midpoint

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

     ,

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

187. Card 187

     Question

     How can an integral be split at an interior point

     $$c$$

     ?

     Answer

     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. Card 188

     Question

     Derivative of

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

     ?

     Answer

     If

     $$f$$

     is continuous, then

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

     The variable lower bound produces the negative sign.

189. Card 189

     Question

     Antiderivative of

     $$1/x$$

     ?

     Answer

     On any interval not crossing zero,

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

190. Card 190

     Question

     Trapezoidal approximation on equal subintervals?

     Answer

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

191. Card 191

     Question

     How do geometric regions help evaluate a definite integral?

     Answer

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

192. Card 192

     Question

     Basic antiderivatives of sine and cosine?

     Answer

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

193. Card 193

     Question

     Definite integral as a limit of Riemann sums?

     Answer

     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. Card 194

     Question

     Constant-multiple rule for integrals?

     Answer

     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. Card 195

     Question

     What pattern suggests

     $$u$$

     -substitution?

     Answer

     A composite expression paired with its derivative, such as

     $$f(g(x))g'(x)$$

     . Set

     $$u=g(x)$$

     so

     $$du=g'(x)\,dx$$

     .

196. Card 196

     Question

     How should bounds change in a definite

     $$u$$

     -substitution?

     Answer

     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. Card 197

     Question

     What condition makes

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

     differentiable with

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

     ?

     Answer

     Continuity of

     $$f$$

     on an interval containing

     $$a$$

     and

     $$x$$

     is the standard AP Calculus condition.

198. Card 198

     Question

     Sum-and-difference rule for definite integrals?

     Answer

     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. Card 199

     Question

     Basic antiderivative of

     $$e^x$$

     ?

     Answer

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

200. Card 200

     Question

     Basic antiderivatives of

     $$\sec^2x$$

     and

     $$\csc^2x$$

     ?

     Answer

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

201. Card 201

     Question

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

     Answer

     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. Card 202

     Question

     How does concavity predict trapezoidal and midpoint error?

     Answer

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

203. Card 203

     Question

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

     Answer

     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. Card 204

     Question

     What denominator pattern suggests an arctangent antiderivative?

     Answer

     After completing the square and scaling, a form like

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

205. Card 205

     Question

     Basic antiderivatives of

     $$\sec x\tan x$$

     and

     $$\csc x\cot x$$

     ?

     Answer

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

206. Card 206

     Question

     How does an initial condition determine an antiderivative?

     Answer

     First find the family

     $$F(x)+C$$

     . Substitute the given point, such as

     $$y(a)=b$$

     , and solve for

     $$C$$

     .

207. Card 207

     Question

     Should a definite-integral answer include

     $$+C$$

     ?

     Answer

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

208. Card 208

     Question

     Why does an indefinite integral include

     $$+C$$

     ?

     Answer

     Differentiation loses additive constants. The

     $$+C$$

     represents every function with the stated derivative.

209. Card 209

     Question

     When is

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

     increasing?

     Answer

     Where

     $$F'(x)=f(x)>0$$

     . It is decreasing where

     $$f(x)<0$$

     .

210. Card 210

     Question

     How is the concavity of

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

     determined?

     Answer

     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. Card 211

     Question

     How is

     $$\int_a^b |f(x)|\,dx$$

     interpreted?

     Answer

     As total geometric area between

     $$f$$

     and the

     $$x$$

     -axis. Split at zeros of

     $$f$$

     and make every regional contribution nonnegative.

212. Card 212

     Question

     What constant-factor check completes many

     $$u$$

     -substitutions?

     Answer

     Compare

     $$du=g'(x)\,dx$$

     with the remaining factor. Multiply or divide by a constant so the integrand contains exactly the needed differential.

213. Card 213

     Question

     How do you recover

     $$\Delta x$$

     from a sigma-form Riemann sum on

     $$[a,b]$$

     ?

     Answer

     Identify the factor multiplying each function value. For

     $$n$$

     equal subintervals, it should be

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

214. Card 214

     Question

     Riemann sum for unequal subinterval widths?

     Answer

     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. Card 215

     Question

     Does continuity guarantee integrability on a closed interval?

     Answer

     Yes. A function continuous on

     $$[a,b]$$

     is integrable there. Some discontinuous functions are also integrable, so continuity is sufficient, not necessary.

216. Card 216

     Question

     Antiderivative pattern for

     $$g'(x)/g(x)$$

     ?

     Answer

     Where

     $$g(x)\ne0$$

     ,

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

217. Card 217

     Question

     What algebraic rewrites often reveal a basic antiderivative?

     Answer

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

218. Card 218

     Question

     What units does

     $$\int_a^b r(t)\,dt$$

     have?

     Answer

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

219. Card 219

     Question

     What is a differential equation?

     Answer

     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. Card 220

     Question

     How does a verbal rate statement become a differential equation?

     Answer

     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. Card 221

     Question

     How do you verify that

     $$y=f(x)$$

     solves a differential equation?

     Answer

     Differentiate

     $$f$$

     as needed, substitute

     $$y$$

     and its derivatives into the equation, and confirm both sides agree on the claimed interval.

222. Card 222

     Question

     General solution versus particular solution?

     Answer

     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. Card 223

     Question

     What does one segment in a slope field show?

     Answer

     At

     $$(x,y)$$

     , its slope equals the value of

     $$dy/dx$$

     given by the differential equation at that point.

224. Card 224

     Question

     What units does the constant

     $$k$$

     have in

     $$dy/dt=ky$$

     ?

     Answer

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

     $$kt$$

     dimensionless.

225. Card 225

     Question

     Euler's method update formula?

     Answer

     With step size

     $$\Delta x$$

     ,

     $$y_{n+1}=y_n+F(x_n,y_n)\Delta x$$

     for

     $$dy/dx=F(x,y)$$

     .

226. Card 226

     Question

     What makes a first-order differential equation separable?

     Answer

     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. Card 227

     Question

     What is an initial value problem?

     Answer

     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. Card 228

     Question

     What is an isocline in a slope field?

     Answer

     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. Card 229

     Question

     What does step size mean in Euler's method?

     Answer

     It is the horizontal change

     $$\Delta x$$

     used in each tangent-line step. The sign of

     $$\Delta x$$

     determines whether the approximation moves right or left.

230. Card 230

     Question

     General solution of

     $$dy/dt=ky$$

     ?

     Answer

     $$y=Ce^{kt}$$

     for a constant

     $$C$$

     . The zero solution is included by

     $$C=0$$

     .

231. Card 231

     Question

     Core method for solving a separable differential equation?

     Answer

     Separate the variables, integrate both sides, include a constant of integration, and solve for

     $$y$$

     when practical. Then apply any initial condition.

232. Card 232

     Question

     How should a solution curve follow a slope field?

     Answer

     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. Card 233

     Question

     How is Euler's method repeated?

     Answer

     At each new point, recompute the slope from the differential equation, multiply by

     $$\Delta x$$

     , and add that change to the current

     $$y$$

     -value.

234. Card 234

     Question

     Why is one integration constant enough after integrating both sides?

     Answer

     Two constants can be combined:

     $$C_2-C_1$$

     is still an arbitrary constant. Write a single

     $$C$$

     .

235. Card 235

     Question

     Solution of

     $$dy/dt=ky$$

     with

     $$y(0)=y_0$$

     ?

     Answer

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

236. Card 236

     Question

     Can one differential equation have infinitely many solutions?

     Answer

     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. Card 237

     Question

     What is an equilibrium solution of

     $$dy/dx=F(y)$$

     ?

     Answer

     A constant solution

     $$y=c$$

     where

     $$F(c)=0$$

     . In the slope field, the segments along that horizontal line have zero slope.

238. Card 238

     Question

     When does a forward Euler estimate tend to lie below the true solution?

     Answer

     When the solution is concave up over the step, its tangent line lies below the curve. For concave down, the tangent-line estimate tends to lie above.

239. Card 239

     Question

     What can be lost when dividing to separate variables?

     Answer

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

240. Card 240

     Question

     In

     $$dy/dt=ky$$

     , what do the signs of

     $$k$$

     mean?

     Answer

     For a positive quantity,

     $$k>0$$

     produces exponential growth,

     $$k<0$$

     produces exponential decay, and

     $$k=0$$

     keeps the quantity constant.

241. Card 241

     Question

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

     Answer

     Test representative

     $$(x,y)$$

     entries in each candidate equation. Match zeros, signs, and repeated slope patterns before checking exact numerical values.

242. Card 242

     Question

     How does the sign of

     $$dy/dx$$

     describe a solution?

     Answer

     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. Card 243

     Question

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

     Answer

     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. Card 244

     Question

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

     Answer

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

245. Card 245

     Question

     Doubling time for exponential growth

     $$y=y_0e^{kt}$$

     ?

     Answer

     For

     $$k>0$$

     ,

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

     It is independent of the initial amount.

246. Card 246

     Question

     How can a slope field reveal whether

     $$dy/dx$$

     depends only on

     $$y$$

     ?

     Answer

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

247. Card 247

     Question

     How is an initial condition used after separation?

     Answer

     Substitute the given

     $$x$$

     and

     $$y$$

     values into the integrated relationship to solve for the arbitrary constant, then state the resulting particular solution.

248. Card 248

     Question

     How do units check a model

     $$dy/dt=F(t,y)$$

     ?

     Answer

     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. Card 249

     Question

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

     Answer

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

250. Card 250

     Question

     Half-life for exponential decay

     $$y=y_0e^{kt}$$

     ?

     Answer

     For

     $$k<0$$

     ,

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

251. Card 251

     Question

     Average value of

     $$f$$

     on

     $$[a,b]$$

     ?

     Answer

     For integrable

     $$f$$

     and

     $$a<b$$

     ,

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

252. Card 252

     Question

     Displacement from velocity

     $$v(t)$$

     on

     $$[a,b]$$

     ?

     Answer

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

     Velocity below zero contributes negative displacement.

253. Card 253

     Question

     Area between vertical curves

     $$y=f(x)$$

     and

     $$y=g(x)$$

     ?

     Answer

     On intervals where

     $$f(x)\ge g(x)$$

     ,

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

     Think top minus bottom.

254. Card 254

     Question

     Volume from known cross-sectional area

     $$A(x)$$

     ?

     Answer

     If slices are perpendicular to the

     $$x$$

     -axis,

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

255. Card 255

     Question

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Answer

     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. Card 256

     Question

     Velocity and acceleration from position

     $$s(t)$$

     ?

     Answer

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

257. Card 257

     Question

     Cross-sectional area when each slice is a square?

     Answer

     If the base segment has length

     $$s(x)$$

     , then

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

258. Card 258

     Question

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

     Answer

     Integrate the net rate:

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

259. Card 259

     Question

     Area between horizontal curves written as

     $$x=R(y)$$

     and

     $$x=L(y)$$

     ?

     Answer

     On intervals where

     $$R(y)\ge L(y)$$

     ,

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

     Think right minus left.

260. Card 260

     Question

     Disc-method volume formula?

     Answer

     For radius

     $$R(x)$$

     and slices perpendicular to the

     $$x$$

     -axis,

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

261. Card 261

     Question

     What units does average value have?

     Answer

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

262. Card 262

     Question

     Total distance traveled from velocity

     $$v(t)$$

     ?

     Answer

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

     Split the interval wherever

     $$v(t)=0$$

     and its sign changes.

263. Card 263

     Question

     Cross-sectional area when each slice is a rectangle?

     Answer

     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. Card 264

     Question

     How do you determine bounds for area between curves?

     Answer

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

265. Card 265

     Question

     How is a rotation radius measured from a horizontal axis

     $$y=k$$

     ?

     Answer

     As vertical distance:

     $$|y-k|$$

     . For washers, identify which boundary stays farther from the axis over the interval.

266. Card 266

     Question

     When is a particle moving to the right or left?

     Answer

     It moves right where

     $$v(t)>0$$

     and left where

     $$v(t)<0$$

     . Position alone does not determine direction.

267. Card 267

     Question

     Cross-sectional area when the diameter of a semicircle is

     $$d(x)$$

     ?

     Answer

     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. Card 268

     Question

     Why must an area integral be split where curves intersect?

     Answer

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

269. Card 269

     Question

     How can a velocity table approximate displacement?

     Answer

     Use a left, right, midpoint, or trapezoidal sum for

     $$\int v(t)\,dt$$

     . Each term is velocity times a time width.

270. Card 270

     Question

     Washer-method volume formula?

     Answer

     For outer radius

     $$R(x)$$

     and inner radius

     $$r(x)$$

     ,

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

271. Card 271

     Question

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

     Answer

     If

     $$s(a)$$

     is known,

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

272. Card 272

     Question

     How do you choose between vertical and horizontal area slices?

     Answer

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

273. Card 273

     Question

     Cross-sectional area of an equilateral triangle with side

     $$s(x)$$

     ?

     Answer

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

274. Card 274

     Question

     How can a table approximate the average value of

     $$f$$

     on

     $$[a,b]$$

     ?

     Answer

     First approximate

     $$\int_a^b f(x)\,dx$$

     with an appropriate Riemann or trapezoidal sum, then divide by

     $$b-a$$

     .

275. Card 275

     Question

     Single expression for area between two curves?

     Answer

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Card 276

     Question

     How is a rotation radius measured from a vertical axis

     $$x=k$$

     ?

     Answer

     As horizontal distance:

     $$|x-k|$$

     . With

     $$dy$$

     slices, write the relevant boundaries as

     $$x$$

     -functions of

     $$y$$

     .

277. Card 277

     Question

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

     Answer

     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. Card 278

     Question

     When should a volume integral use

     $$dy$$

     ?

     Answer

     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. Card 279

     Question

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

     Answer

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

280. Card 280

     Question

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

     Answer

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

281. Card 281

     Question

     Why must total distance split at velocity sign changes?

     Answer

     Distance accumulates speed

     $$|v|$$

     , not signed velocity. A single integral of

     $$v$$

     would cancel motion in opposite directions.

282. Card 282

     Question

     When does an accumulated quantity reach a local maximum?

     Answer

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

283. Card 283

     Question

     What distinguishes area from a definite integral?

     Answer

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

284. Card 284

     Question

     How do position, velocity, and acceleration graphs correspond?

     Answer

     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. Card 285

     Question

     How do you interpret

     $$\int_a^b [r(t)-c(t)]\,dt$$

     in context?

     Answer

     As net change: total amount added by

     $$r$$

     minus total amount removed by

     $$c$$

     over the interval. State the resulting quantity and units.

286. Card 286

     Question

     What units does a volume integral

     $$\int A(x)\,dx$$

     have?

     Answer

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

287. Card 287

     Question

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

     Answer

     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. Card 288

     Question

     Why should a contextual integral answer include a sentence?

     Answer

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