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

Tämän pakan kortit

1. Kortti 1

   Kysymys

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

   Vastaus

   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. Kortti 2

   Kysymys

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

   Vastaus

   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. Kortti 3

   Kysymys

   When does direct substitution evaluate a limit?

   Vastaus

   When the function is continuous at the target input. Then

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

4. Kortti 4

   Kysymys

   Three conditions for continuity at $x=a$?

   Vastaus

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

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

5. Kortti 5

   Kysymys

   Intermediate Value Theorem: hypotheses and conclusion?

   Vastaus

   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. Kortti 6

   Kysymys

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

   Vastaus

   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. Kortti 7

   Kysymys

   How do you read a finite limit from a graph?

   Vastaus

   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. Kortti 8

   Kysymys

   Limit law for a sum or difference?

   Vastaus

   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. Kortti 9

   Kysymys

   What makes a discontinuity removable?

   Vastaus

   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. Kortti 10

    Kysymys

    Squeeze Theorem: usable form?

    Vastaus

    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. Kortti 11

    Kysymys

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

    Vastaus

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

12. Kortti 12

    Kysymys

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

    Vastaus

    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. Kortti 13

    Kysymys

    Limit law for a product?

    Vastaus

    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. Kortti 14

    Kysymys

    Graph signature of a jump discontinuity?

    Vastaus

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

15. Kortti 15

    Kysymys

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

    Vastaus

    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. Kortti 16

    Kysymys

    Horizontal asymptote from a limit at infinity?

    Vastaus

    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. Kortti 17

    Kysymys

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

    Vastaus

    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. Kortti 18

    Kysymys

    Limit law for a quotient—and its condition?

    Vastaus

    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. Kortti 19

    Kysymys

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

    Vastaus

    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. Kortti 20

    Kysymys

    When is the Squeeze Theorem a natural choice?

    Vastaus

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

21. Kortti 21

    Kysymys

    Vertical asymptote from one-sided behavior?

    Vastaus

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

22. Kortti 22

    Kysymys

    Limit at infinity of equal-degree rational functions?

    Vastaus

    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. Kortti 23

    Kysymys

    When can a limit pass through a continuous outer function?

    Vastaus

    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. Kortti 24

    Kysymys

    What makes a discontinuity infinite?

    Vastaus

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

25. Kortti 25

    Kysymys

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

    Vastaus

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

26. Kortti 26

    Kysymys

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

    Vastaus

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. Kortti 27

    Kysymys

    Continuity of a composition?

    Vastaus

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

28. Kortti 28

    Kysymys

    Limit at infinity when a rational numerator has lower degree?

    Vastaus

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

29. Kortti 29

    Kysymys

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

    Vastaus

    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. Kortti 30

    Kysymys

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

    Vastaus

    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. Kortti 31

    Kysymys

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

    Vastaus

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

32. Kortti 32

    Kysymys

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

    Vastaus

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

33. Kortti 33

    Kysymys

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

    Vastaus

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

34. Kortti 34

    Kysymys

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

    Vastaus

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

35. Kortti 35

    Kysymys

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

    Vastaus

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

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

36. Kortti 36

    Kysymys

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

    Vastaus

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

    The derivative exists only if this finite limit exists.

37. Kortti 37

    Kysymys

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

    Vastaus

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

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

38. Kortti 38

    Kysymys

    What does differentiability imply about continuity?

    Vastaus

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

39. Kortti 39

    Kysymys

    Power rule for derivatives?

    Vastaus

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

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

40. Kortti 40

    Kysymys

    Units of $f'(x)$?

    Vastaus

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

41. Kortti 41

    Kysymys

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

    Vastaus

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

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

42. Kortti 42

    Kysymys

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

    Vastaus

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

    Kysymys

    Derivative of a constant?

    Vastaus

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

    A constant function has zero rate of change.

44. Kortti 44

    Kysymys

    Derivative of $\sin x$?

    Vastaus

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

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

45. Kortti 45

    Kysymys

    Product rule?

    Vastaus

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

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

46. Kortti 46

    Kysymys

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

    Vastaus

    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. Kortti 47

    Kysymys

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

    Vastaus

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

48. Kortti 48

    Kysymys

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

    Vastaus

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

49. Kortti 49

    Kysymys

    Derivative of a sum or difference?

    Vastaus

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

50. Kortti 50

    Kysymys

    Derivative of $\cos x$?

    Vastaus

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

    The standard formula assumes radians.

51. Kortti 51

    Kysymys

    Quotient rule?

    Vastaus

    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. Kortti 52

    Kysymys

    Common notations for the first derivative?

    Vastaus

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

53. Kortti 53

    Kysymys

    Derivative of $e^x$?

    Vastaus

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

54. Kortti 54

    Kysymys

    What graph features can make $f$ nondifferentiable?

    Vastaus

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

55. Kortti 55

    Kysymys

    Derivative of $\tan x$?

    Vastaus

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. Kortti 56

    Kysymys

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

    Vastaus

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

57. Kortti 57

    Kysymys

    Derivative of $\ln x$?

    Vastaus

    For $x>0$,

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

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

58. Kortti 58

    Kysymys

    How does the power rule handle roots or negative powers?

    Vastaus

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

59. Kortti 59

    Kysymys

    Derivative of $\csc x$?

    Vastaus

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. Kortti 60

    Kysymys

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

    Vastaus

    $f$ is increasing on that interval.

61. Kortti 61

    Kysymys

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

    Vastaus

    For $a>0$,

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

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

62. Kortti 62

    Kysymys

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

    Vastaus

    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. Kortti 63

    Kysymys

    Derivative of $\sec x$?

    Vastaus

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. Kortti 64

    Kysymys

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

    Vastaus

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

65. Kortti 65

    Kysymys

    Derivative of $\log_a x$?

    Vastaus

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

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

66. Kortti 66

    Kysymys

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

    Vastaus

    For $h=fg$,

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

    Use the four table entries at the same input.

67. Kortti 67

    Kysymys

    Derivative of $\cot x$?

    Vastaus

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. Kortti 68

    Kysymys

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

    Vastaus

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

69. Kortti 69

    Kysymys

    Constant-multiple rule?

    Vastaus

    For a constant $c$,

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

70. Kortti 70

    Kysymys

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

    Vastaus

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

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

71. Kortti 71

    Kysymys

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

    Vastaus

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

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

72. Kortti 72

    Kysymys

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

    Vastaus

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

73. Kortti 73

    Kysymys

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

    Vastaus

    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. Kortti 74

    Kysymys

    Derivative of an inverse function at $x$?

    Vastaus

    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. Kortti 75

    Kysymys

    Derivative of $\arcsin x$?

    Vastaus

    For $|x|<1$,

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

76. Kortti 76

    Kysymys

    Notation for the third derivative of $f$?

    Vastaus

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

77. Kortti 77

    Kysymys

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

    Vastaus

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

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

78. Kortti 78

    Kysymys

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

    Vastaus

    Where $y\ne0$,

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

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

79. Kortti 79

    Kysymys

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

    Vastaus

    Provided $f'(a)\ne0$,

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

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

80. Kortti 80

    Kysymys

    Derivative of $\arctan x$?

    Vastaus

    For every real $x$,

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

81. Kortti 81

    Kysymys

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

    Vastaus

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

    The extra factor is the chain rule.

82. Kortti 82

    Kysymys

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

    Vastaus

    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. Kortti 83

    Kysymys

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

    Vastaus

    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. Kortti 84

    Kysymys

    Derivative of $\arccos x$?

    Vastaus

    For $|x|<1$,

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

85. Kortti 85

    Kysymys

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

    Vastaus

    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. Kortti 86

    Kysymys

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

    Vastaus

    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. Kortti 87

    Kysymys

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

    Vastaus

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

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

88. Kortti 88

    Kysymys

    How are tangent slopes of inverse graphs related?

    Vastaus

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

89. Kortti 89

    Kysymys

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

    Vastaus

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

90. Kortti 90

    Kysymys

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

    Vastaus

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

91. Kortti 91

    Kysymys

    Horizontal tangent on an implicit curve: derivative condition?

    Vastaus

    $dy/dx=0$ 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. Kortti 92

    Kysymys

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

    Vastaus

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

93. Kortti 93

    Kysymys

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

    Vastaus

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

94. Kortti 94

    Kysymys

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

    Vastaus

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

    This combines the power rule with the chain rule.

95. Kortti 95

    Kysymys

    Vertical tangent on an implicit curve: derivative clue?

    Vastaus

    $dy/dx$ 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. Kortti 96

    Kysymys

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

    Vastaus

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

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

97. Kortti 97

    Kysymys

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

    Vastaus

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

98. Kortti 98

    Kysymys

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

    Vastaus

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

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

99. Kortti 99

    Kysymys

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

    Vastaus

    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. Kortti 100

     Kysymys

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

     Vastaus

     When the functions are differentiable, the chain rule gives

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

101. Kortti 101

     Kysymys

     What local property lets a function have an inverse derivative?

     Vastaus

     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. Kortti 102

     Kysymys

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

     Vastaus

     For $a>0$,

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

103. Kortti 103

     Kysymys

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

     Vastaus

     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. Kortti 104

     Kysymys

     Position, velocity, and acceleration relationships?

     Vastaus

     For position $s(t)$,

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

105. Kortti 105

     Kysymys

     Central idea of a related-rates problem?

     Vastaus

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

106. Kortti 106

     Kysymys

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

     Vastaus

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

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

107. Kortti 107

     Kysymys

     L’Hospital’s Rule: basic conditions?

     Vastaus

     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. Kortti 108

     Kysymys

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

     Vastaus

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

109. Kortti 109

     Kysymys

     Speed in terms of velocity?

     Vastaus

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

     Velocity includes direction; speed is nonnegative magnitude.

110. Kortti 110

     Kysymys

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

     Vastaus

     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. Kortti 111

     Kysymys

     Differential approximation connecting $dx$ and $dy$?

     Vastaus

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

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

112. Kortti 112

     Kysymys

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

     Vastaus

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

113. Kortti 113

     Kysymys

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

     Vastaus

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

114. Kortti 114

     Kysymys

     What does positive acceleration say about velocity?

     Vastaus

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

115. Kortti 115

     Kysymys

     Related rates: when should numerical values be substituted?

     Vastaus

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

116. Kortti 116

     Kysymys

     How does concavity predict linearization error?

     Vastaus

     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. Kortti 117

     Kysymys

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

     Vastaus

     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. Kortti 118

     Kysymys

     When is a particle moving in the positive direction?

     Vastaus

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

119. Kortti 119

     Kysymys

     How can velocity show a change of direction?

     Vastaus

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

120. Kortti 120

     Kysymys

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

     Vastaus

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

121. Kortti 121

     Kysymys

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

     Vastaus

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

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

122. Kortti 122

     Kysymys

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

     Vastaus

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

123. Kortti 123

     Kysymys

     What must a contextual derivative sentence include?

     Vastaus

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

124. Kortti 124

     Kysymys

     Velocity negative and acceleration positive: what happens?

     Vastaus

     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. Kortti 125

     Kysymys

     How should a negative related rate be interpreted?

     Vastaus

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

126. Kortti 126

     Kysymys

     When is local linearity a sound approximation tool?

     Vastaus

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

127. Kortti 127

     Kysymys

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

     Vastaus

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

128. Kortti 128

     Kysymys

     When is speed increasing?

     Vastaus

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

129. Kortti 129

     Kysymys

     Volume changes with time: notation for its rate?

     Vastaus

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

130. Kortti 130

     Kysymys

     Why are similar triangles useful in related rates?

     Vastaus

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

131. Kortti 131

     Kysymys

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

     Vastaus

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

132. Kortti 132

     Kysymys

     What conclusion does L’Hospital’s Rule permit?

     Vastaus

     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. Kortti 133

     Kysymys

     When is speed decreasing?

     Vastaus

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

134. Kortti 134

     Kysymys

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

     Vastaus

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

135. Kortti 135

     Kysymys

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

     Vastaus

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

136. Kortti 136

     Kysymys

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

     Vastaus

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

137. Kortti 137

     Kysymys

     Extreme Value Theorem: hypothesis and conclusion?

     Vastaus

     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. Kortti 138

     Kysymys

     What is a critical number of $f$?

     Vastaus

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

139. Kortti 139

     Kysymys

     First derivative test for a local maximum?

     Vastaus

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

140. Kortti 140

     Kysymys

     Second-derivative sign for concave up?

     Vastaus

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

141. Kortti 141

     Kysymys

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

     Vastaus

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

142. Kortti 142

     Kysymys

     First step in an optimization model?

     Vastaus

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

143. Kortti 143

     Kysymys

     Mean Value Theorem: hypotheses and conclusion?

     Vastaus

     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. Kortti 144

     Kysymys

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

     Vastaus

     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. Kortti 145

     Kysymys

     First derivative test for a local minimum?

     Vastaus

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

146. Kortti 146

     Kysymys

     What must happen at an inflection point?

     Vastaus

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

147. Kortti 147

     Kysymys

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

     Vastaus

     $f''$ 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. Kortti 148

     Kysymys

     How do you confirm an optimization answer is absolute?

     Vastaus

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

149. Kortti 149

     Kysymys

     Rolle’s Theorem: hypotheses and conclusion?

     Vastaus

     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. Kortti 150

     Kysymys

     Difference between absolute and relative extrema?

     Vastaus

     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. Kortti 151

     Kysymys

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

     Vastaus

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

152. Kortti 152

     Kysymys

     Second derivative test for a local minimum?

     Vastaus

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

153. Kortti 153

     Kysymys

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

     Vastaus

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

154. Kortti 154

     Kysymys

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

     Vastaus

     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. Kortti 155

     Kysymys

     Which theorem links an average slope to an instantaneous slope?

     Vastaus

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

156. Kortti 156

     Kysymys

     How can an implicit derivative locate a horizontal tangent?

     Vastaus

     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. Kortti 157

     Kysymys

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

     Vastaus

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

158. Kortti 158

     Kysymys

     Second derivative test for a local maximum?

     Vastaus

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

159. Kortti 159

     Kysymys

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

     Vastaus

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

160. Kortti 160

     Kysymys

     Why must an optimization domain be stated?

     Vastaus

     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. Kortti 161

     Kysymys

     Which theorem guarantees absolute extrema, not where they occur?

     Vastaus

     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. Kortti 162

     Kysymys

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

     Vastaus

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

163. Kortti 163

     Kysymys

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

     Vastaus

     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. Kortti 164

     Kysymys

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

     Vastaus

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

165. Kortti 165

     Kysymys

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

     Vastaus

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

166. Kortti 166

     Kysymys

     How can an implicit derivative locate a vertical tangent?

     Vastaus

     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. Kortti 167

     Kysymys

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

     Vastaus

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

168. Kortti 168

     Kysymys

     Why are endpoints included in the candidates test?

     Vastaus

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

169. Kortti 169

     Kysymys

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

     Vastaus

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

170. Kortti 170

     Kysymys

     Second-derivative sign for concave down?

     Vastaus

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

171. Kortti 171

     Kysymys

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

     Vastaus

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

172. Kortti 172

     Kysymys

     What should the final line of an optimization solution state?

     Vastaus

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

173. Kortti 173

     Kysymys

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

     Vastaus

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

174. Kortti 174

     Kysymys

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

     Vastaus

     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. Kortti 175

     Kysymys

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

     Vastaus

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

176. Kortti 176

     Kysymys

     Left Riemann sum on equal subintervals?

     Vastaus

     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. Kortti 177

     Kysymys

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

     Vastaus

     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. Kortti 178

     Kysymys

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Vastaus

     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. Kortti 179

     Kysymys

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

     Vastaus

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. Kortti 180

     Kysymys

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

     Vastaus

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

181. Kortti 181

     Kysymys

     Right Riemann sum on equal subintervals?

     Vastaus

     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. Kortti 182

     Kysymys

     How does reversing integral bounds change the value?

     Vastaus

     It changes the sign:

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

183. Kortti 183

     Kysymys

     Net Change Theorem?

     Vastaus

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

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

184. Kortti 184

     Kysymys

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

     Vastaus

     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. Kortti 185

     Kysymys

     Power rule for antiderivatives?

     Vastaus

     For $n\ne-1$,

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

186. Kortti 186

     Kysymys

     Midpoint Riemann sum on equal subintervals?

     Vastaus

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

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

187. Kortti 187

     Kysymys

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

     Vastaus

     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. Kortti 188

     Kysymys

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

     Vastaus

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. Kortti 189

     Kysymys

     Antiderivative of $1/x$?

     Vastaus

     On any interval not crossing zero,

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

190. Kortti 190

     Kysymys

     Trapezoidal approximation on equal subintervals?

     Vastaus

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

191. Kortti 191

     Kysymys

     How do geometric regions help evaluate a definite integral?

     Vastaus

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

192. Kortti 192

     Kysymys

     Basic antiderivatives of sine and cosine?

     Vastaus

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

193. Kortti 193

     Kysymys

     Definite integral as a limit of Riemann sums?

     Vastaus

     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. Kortti 194

     Kysymys

     Constant-multiple rule for integrals?

     Vastaus

     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. Kortti 195

     Kysymys

     What pattern suggests $u$-substitution?

     Vastaus

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

196. Kortti 196

     Kysymys

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

     Vastaus

     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. Kortti 197

     Kysymys

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

     Vastaus

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

198. Kortti 198

     Kysymys

     Sum-and-difference rule for definite integrals?

     Vastaus

     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. Kortti 199

     Kysymys

     Basic antiderivative of $e^x$?

     Vastaus

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

200. Kortti 200

     Kysymys

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

     Vastaus

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

201. Kortti 201

     Kysymys

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

     Vastaus

     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. Kortti 202

     Kysymys

     How does concavity predict trapezoidal and midpoint error?

     Vastaus

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

203. Kortti 203

     Kysymys

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

     Vastaus

     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. Kortti 204

     Kysymys

     What denominator pattern suggests an arctangent antiderivative?

     Vastaus

     After completing the square and scaling, a form like

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

205. Kortti 205

     Kysymys

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

     Vastaus

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

206. Kortti 206

     Kysymys

     How does an initial condition determine an antiderivative?

     Vastaus

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

207. Kortti 207

     Kysymys

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

     Vastaus

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

208. Kortti 208

     Kysymys

     Why does an indefinite integral include $+C$?

     Vastaus

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

209. Kortti 209

     Kysymys

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

     Vastaus

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

210. Kortti 210

     Kysymys

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

     Vastaus

     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. Kortti 211

     Kysymys

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

     Vastaus

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

212. Kortti 212

     Kysymys

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

     Vastaus

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

213. Kortti 213

     Kysymys

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

     Vastaus

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

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

214. Kortti 214

     Kysymys

     Riemann sum for unequal subinterval widths?

     Vastaus

     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. Kortti 215

     Kysymys

     Does continuity guarantee integrability on a closed interval?

     Vastaus

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

216. Kortti 216

     Kysymys

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

     Vastaus

     Where $g(x)\ne0$,

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

217. Kortti 217

     Kysymys

     What algebraic rewrites often reveal a basic antiderivative?

     Vastaus

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

218. Kortti 218

     Kysymys

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

     Vastaus

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

219. Kortti 219

     Kysymys

     What is a differential equation?

     Vastaus

     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. Kortti 220

     Kysymys

     How does a verbal rate statement become a differential equation?

     Vastaus

     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. Kortti 221

     Kysymys

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

     Vastaus

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

222. Kortti 222

     Kysymys

     General solution versus particular solution?

     Vastaus

     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. Kortti 223

     Kysymys

     What does one segment in a slope field show?

     Vastaus

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

224. Kortti 224

     Kysymys

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

     Vastaus

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

225. Kortti 225

     Kysymys

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

     Vastaus

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

226. Kortti 226

     Kysymys

     What makes a first-order differential equation separable?

     Vastaus

     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. Kortti 227

     Kysymys

     What is an initial value problem?

     Vastaus

     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. Kortti 228

     Kysymys

     What is an isocline in a slope field?

     Vastaus

     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. Kortti 229

     Kysymys

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

     Vastaus

     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. Kortti 230

     Kysymys

     General solution of $dy/dt=ky$?

     Vastaus

     $$y=Ce^{kt}$$

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

231. Kortti 231

     Kysymys

     Core method for solving a separable differential equation?

     Vastaus

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

232. Kortti 232

     Kysymys

     How should a solution curve follow a slope field?

     Vastaus

     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. Kortti 233

     Kysymys

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

     Vastaus

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

234. Kortti 234

     Kysymys

     Why is one integration constant enough after integrating both sides?

     Vastaus

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

235. Kortti 235

     Kysymys

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

     Vastaus

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

236. Kortti 236

     Kysymys

     Can one differential equation have infinitely many solutions?

     Vastaus

     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. Kortti 237

     Kysymys

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

     Vastaus

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

238. Kortti 238

     Kysymys

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

     Vastaus

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

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

239. Kortti 239

     Kysymys

     What can be lost when dividing to separate variables?

     Vastaus

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

240. Kortti 240

     Kysymys

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

     Vastaus

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

241. Kortti 241

     Kysymys

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

     Vastaus

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

242. Kortti 242

     Kysymys

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

     Vastaus

     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. Kortti 243

     Kysymys

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

     Vastaus

     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. Kortti 244

     Kysymys

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

     Vastaus

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

245. Kortti 245

     Kysymys

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

     Vastaus

     For $k>0$,

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

     It is independent of the initial amount.

246. Kortti 246

     Kysymys

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

     Vastaus

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

247. Kortti 247

     Kysymys

     How is an initial condition used after separation?

     Vastaus

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

248. Kortti 248

     Kysymys

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

     Vastaus

     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. Kortti 249

     Kysymys

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

     Vastaus

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

250. Kortti 250

     Kysymys

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

     Vastaus

     For $k<0$,

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

251. Kortti 251

     Kysymys

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

     Vastaus

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

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

252. Kortti 252

     Kysymys

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

     Vastaus

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

     Velocity below zero contributes negative displacement.

253. Kortti 253

     Kysymys

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

     Vastaus

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

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

     Think top minus bottom.

254. Kortti 254

     Kysymys

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

     Vastaus

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

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

255. Kortti 255

     Kysymys

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Vastaus

     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. Kortti 256

     Kysymys

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

     Vastaus

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

257. Kortti 257

     Kysymys

     Cross-sectional area when each slice is a square?

     Vastaus

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

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

258. Kortti 258

     Kysymys

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

     Vastaus

     Integrate the net rate:

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

259. Kortti 259

     Kysymys

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

     Vastaus

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

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

     Think right minus left.

260. Kortti 260

     Kysymys

     Disc-method volume formula?

     Vastaus

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

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

261. Kortti 261

     Kysymys

     What units does average value have?

     Vastaus

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

262. Kortti 262

     Kysymys

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

     Vastaus

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

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

263. Kortti 263

     Kysymys

     Cross-sectional area when each slice is a rectangle?

     Vastaus

     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. Kortti 264

     Kysymys

     How do you determine bounds for area between curves?

     Vastaus

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

265. Kortti 265

     Kysymys

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

     Vastaus

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

266. Kortti 266

     Kysymys

     When is a particle moving to the right or left?

     Vastaus

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

267. Kortti 267

     Kysymys

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

     Vastaus

     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. Kortti 268

     Kysymys

     Why must an area integral be split where curves intersect?

     Vastaus

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

269. Kortti 269

     Kysymys

     How can a velocity table approximate displacement?

     Vastaus

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

270. Kortti 270

     Kysymys

     Washer-method volume formula?

     Vastaus

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

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

271. Kortti 271

     Kysymys

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

     Vastaus

     If $s(a)$ is known,

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

272. Kortti 272

     Kysymys

     How do you choose between vertical and horizontal area slices?

     Vastaus

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

273. Kortti 273

     Kysymys

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

     Vastaus

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

274. Kortti 274

     Kysymys

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

     Vastaus

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

275. Kortti 275

     Kysymys

     Single expression for area between two curves?

     Vastaus

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. Kortti 276

     Kysymys

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

     Vastaus

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

277. Kortti 277

     Kysymys

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

     Vastaus

     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. Kortti 278

     Kysymys

     When should a volume integral use $dy$?

     Vastaus

     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. Kortti 279

     Kysymys

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

     Vastaus

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

280. Kortti 280

     Kysymys

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

     Vastaus

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

281. Kortti 281

     Kysymys

     Why must total distance split at velocity sign changes?

     Vastaus

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

282. Kortti 282

     Kysymys

     When does an accumulated quantity reach a local maximum?

     Vastaus

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

283. Kortti 283

     Kysymys

     What distinguishes area from a definite integral?

     Vastaus

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

284. Kortti 284

     Kysymys

     How do position, velocity, and acceleration graphs correspond?

     Vastaus

     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. Kortti 285

     Kysymys

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

     Vastaus

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

286. Kortti 286

     Kysymys

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

     Vastaus

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

287. Kortti 287

     Kysymys

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

     Vastaus

     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. Kortti 288

     Kysymys

     Why should a contextual integral answer include a sentence?

     Vastaus

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