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

Šio rinkinio kortelės

1. 1 kortelė

   Klausimas

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

   Atsakymas

   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. 2 kortelė

   Klausimas

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

   Atsakymas

   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. 3 kortelė

   Klausimas

   When does direct substitution evaluate a limit?

   Atsakymas

   When the function is continuous at the target input. Then

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

4. 4 kortelė

   Klausimas

   Three conditions for continuity at $x=a$?

   Atsakymas

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

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

5. 5 kortelė

   Klausimas

   Intermediate Value Theorem: hypotheses and conclusion?

   Atsakymas

   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. 6 kortelė

   Klausimas

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

   Atsakymas

   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. 7 kortelė

   Klausimas

   How do you read a finite limit from a graph?

   Atsakymas

   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. 8 kortelė

   Klausimas

   Limit law for a sum or difference?

   Atsakymas

   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. 9 kortelė

   Klausimas

   What makes a discontinuity removable?

   Atsakymas

   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. 10 kortelė

    Klausimas

    Squeeze Theorem: usable form?

    Atsakymas

    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. 11 kortelė

    Klausimas

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

    Atsakymas

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

12. 12 kortelė

    Klausimas

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

    Atsakymas

    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. 13 kortelė

    Klausimas

    Limit law for a product?

    Atsakymas

    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. 14 kortelė

    Klausimas

    Graph signature of a jump discontinuity?

    Atsakymas

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

15. 15 kortelė

    Klausimas

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

    Atsakymas

    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. 16 kortelė

    Klausimas

    Horizontal asymptote from a limit at infinity?

    Atsakymas

    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. 17 kortelė

    Klausimas

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

    Atsakymas

    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. 18 kortelė

    Klausimas

    Limit law for a quotient—and its condition?

    Atsakymas

    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. 19 kortelė

    Klausimas

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

    Atsakymas

    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. 20 kortelė

    Klausimas

    When is the Squeeze Theorem a natural choice?

    Atsakymas

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

21. 21 kortelė

    Klausimas

    Vertical asymptote from one-sided behavior?

    Atsakymas

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

22. 22 kortelė

    Klausimas

    Limit at infinity of equal-degree rational functions?

    Atsakymas

    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. 23 kortelė

    Klausimas

    When can a limit pass through a continuous outer function?

    Atsakymas

    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. 24 kortelė

    Klausimas

    What makes a discontinuity infinite?

    Atsakymas

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

25. 25 kortelė

    Klausimas

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

    Atsakymas

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

26. 26 kortelė

    Klausimas

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

    Atsakymas

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. 27 kortelė

    Klausimas

    Continuity of a composition?

    Atsakymas

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

28. 28 kortelė

    Klausimas

    Limit at infinity when a rational numerator has lower degree?

    Atsakymas

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

29. 29 kortelė

    Klausimas

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

    Atsakymas

    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. 30 kortelė

    Klausimas

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

    Atsakymas

    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. 31 kortelė

    Klausimas

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

    Atsakymas

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

32. 32 kortelė

    Klausimas

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

    Atsakymas

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

33. 33 kortelė

    Klausimas

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

    Atsakymas

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

34. 34 kortelė

    Klausimas

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

    Atsakymas

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

35. 35 kortelė

    Klausimas

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

    Atsakymas

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

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

36. 36 kortelė

    Klausimas

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

    Atsakymas

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

    The derivative exists only if this finite limit exists.

37. 37 kortelė

    Klausimas

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

    Atsakymas

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

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

38. 38 kortelė

    Klausimas

    What does differentiability imply about continuity?

    Atsakymas

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

39. 39 kortelė

    Klausimas

    Power rule for derivatives?

    Atsakymas

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

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

40. 40 kortelė

    Klausimas

    Units of $f'(x)$?

    Atsakymas

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

41. 41 kortelė

    Klausimas

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

    Atsakymas

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

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

42. 42 kortelė

    Klausimas

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

    Atsakymas

    $f'(x)>0$ 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. 43 kortelė

    Klausimas

    Derivative of a constant?

    Atsakymas

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

    A constant function has zero rate of change.

44. 44 kortelė

    Klausimas

    Derivative of $\sin x$?

    Atsakymas

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

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

45. 45 kortelė

    Klausimas

    Product rule?

    Atsakymas

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

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

46. 46 kortelė

    Klausimas

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

    Atsakymas

    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. 47 kortelė

    Klausimas

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

    Atsakymas

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

48. 48 kortelė

    Klausimas

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

    Atsakymas

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

49. 49 kortelė

    Klausimas

    Derivative of a sum or difference?

    Atsakymas

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

50. 50 kortelė

    Klausimas

    Derivative of $\cos x$?

    Atsakymas

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

    The standard formula assumes radians.

51. 51 kortelė

    Klausimas

    Quotient rule?

    Atsakymas

    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. 52 kortelė

    Klausimas

    Common notations for the first derivative?

    Atsakymas

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

53. 53 kortelė

    Klausimas

    Derivative of $e^x$?

    Atsakymas

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

54. 54 kortelė

    Klausimas

    What graph features can make $f$ nondifferentiable?

    Atsakymas

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

55. 55 kortelė

    Klausimas

    Derivative of $\tan x$?

    Atsakymas

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. 56 kortelė

    Klausimas

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

    Atsakymas

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

57. 57 kortelė

    Klausimas

    Derivative of $\ln x$?

    Atsakymas

    For $x>0$,

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

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

58. 58 kortelė

    Klausimas

    How does the power rule handle roots or negative powers?

    Atsakymas

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

59. 59 kortelė

    Klausimas

    Derivative of $\csc x$?

    Atsakymas

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. 60 kortelė

    Klausimas

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

    Atsakymas

    $f$ is increasing on that interval.

61. 61 kortelė

    Klausimas

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

    Atsakymas

    For $a>0$,

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

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

62. 62 kortelė

    Klausimas

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

    Atsakymas

    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. 63 kortelė

    Klausimas

    Derivative of $\sec x$?

    Atsakymas

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. 64 kortelė

    Klausimas

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

    Atsakymas

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

65. 65 kortelė

    Klausimas

    Derivative of $\log_a x$?

    Atsakymas

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

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

66. 66 kortelė

    Klausimas

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

    Atsakymas

    For $h=fg$,

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

    Use the four table entries at the same input.

67. 67 kortelė

    Klausimas

    Derivative of $\cot x$?

    Atsakymas

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. 68 kortelė

    Klausimas

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

    Atsakymas

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

69. 69 kortelė

    Klausimas

    Constant-multiple rule?

    Atsakymas

    For a constant $c$,

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

70. 70 kortelė

    Klausimas

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

    Atsakymas

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

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

71. 71 kortelė

    Klausimas

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

    Atsakymas

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

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

72. 72 kortelė

    Klausimas

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

    Atsakymas

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

73. 73 kortelė

    Klausimas

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

    Atsakymas

    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. 74 kortelė

    Klausimas

    Derivative of an inverse function at $x$?

    Atsakymas

    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. 75 kortelė

    Klausimas

    Derivative of $\arcsin x$?

    Atsakymas

    For $|x|<1$,

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

76. 76 kortelė

    Klausimas

    Notation for the third derivative of $f$?

    Atsakymas

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

77. 77 kortelė

    Klausimas

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

    Atsakymas

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

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

78. 78 kortelė

    Klausimas

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

    Atsakymas

    Where $y\ne0$,

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

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

79. 79 kortelė

    Klausimas

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

    Atsakymas

    Provided $f'(a)\ne0$,

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

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

80. 80 kortelė

    Klausimas

    Derivative of $\arctan x$?

    Atsakymas

    For every real $x$,

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

81. 81 kortelė

    Klausimas

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

    Atsakymas

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

    The extra factor is the chain rule.

82. 82 kortelė

    Klausimas

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

    Atsakymas

    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. 83 kortelė

    Klausimas

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

    Atsakymas

    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. 84 kortelė

    Klausimas

    Derivative of $\arccos x$?

    Atsakymas

    For $|x|<1$,

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

85. 85 kortelė

    Klausimas

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

    Atsakymas

    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. 86 kortelė

    Klausimas

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

    Atsakymas

    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. 87 kortelė

    Klausimas

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

    Atsakymas

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

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

88. 88 kortelė

    Klausimas

    How are tangent slopes of inverse graphs related?

    Atsakymas

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

89. 89 kortelė

    Klausimas

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

    Atsakymas

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

90. 90 kortelė

    Klausimas

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

    Atsakymas

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

91. 91 kortelė

    Klausimas

    Horizontal tangent on an implicit curve: derivative condition?

    Atsakymas

    $dy/dx=0$ 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. 92 kortelė

    Klausimas

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

    Atsakymas

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

93. 93 kortelė

    Klausimas

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

    Atsakymas

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

94. 94 kortelė

    Klausimas

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

    Atsakymas

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

    This combines the power rule with the chain rule.

95. 95 kortelė

    Klausimas

    Vertical tangent on an implicit curve: derivative clue?

    Atsakymas

    $dy/dx$ 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. 96 kortelė

    Klausimas

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

    Atsakymas

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

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

97. 97 kortelė

    Klausimas

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

    Atsakymas

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

98. 98 kortelė

    Klausimas

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

    Atsakymas

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

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

99. 99 kortelė

    Klausimas

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

    Atsakymas

    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. 100 kortelė

     Klausimas

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

     Atsakymas

     When the functions are differentiable, the chain rule gives

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

101. 101 kortelė

     Klausimas

     What local property lets a function have an inverse derivative?

     Atsakymas

     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. 102 kortelė

     Klausimas

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

     Atsakymas

     For $a>0$,

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

103. 103 kortelė

     Klausimas

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

     Atsakymas

     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. 104 kortelė

     Klausimas

     Position, velocity, and acceleration relationships?

     Atsakymas

     For position $s(t)$,

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

105. 105 kortelė

     Klausimas

     Central idea of a related-rates problem?

     Atsakymas

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

106. 106 kortelė

     Klausimas

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

     Atsakymas

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

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

107. 107 kortelė

     Klausimas

     L’Hospital’s Rule: basic conditions?

     Atsakymas

     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. 108 kortelė

     Klausimas

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

     Atsakymas

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

109. 109 kortelė

     Klausimas

     Speed in terms of velocity?

     Atsakymas

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

     Velocity includes direction; speed is nonnegative magnitude.

110. 110 kortelė

     Klausimas

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

     Atsakymas

     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. 111 kortelė

     Klausimas

     Differential approximation connecting $dx$ and $dy$?

     Atsakymas

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

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

112. 112 kortelė

     Klausimas

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

     Atsakymas

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

113. 113 kortelė

     Klausimas

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

     Atsakymas

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

114. 114 kortelė

     Klausimas

     What does positive acceleration say about velocity?

     Atsakymas

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

115. 115 kortelė

     Klausimas

     Related rates: when should numerical values be substituted?

     Atsakymas

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

116. 116 kortelė

     Klausimas

     How does concavity predict linearization error?

     Atsakymas

     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. 117 kortelė

     Klausimas

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

     Atsakymas

     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. 118 kortelė

     Klausimas

     When is a particle moving in the positive direction?

     Atsakymas

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

119. 119 kortelė

     Klausimas

     How can velocity show a change of direction?

     Atsakymas

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

120. 120 kortelė

     Klausimas

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

     Atsakymas

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

121. 121 kortelė

     Klausimas

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

     Atsakymas

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

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

122. 122 kortelė

     Klausimas

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

     Atsakymas

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

123. 123 kortelė

     Klausimas

     What must a contextual derivative sentence include?

     Atsakymas

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

124. 124 kortelė

     Klausimas

     Velocity negative and acceleration positive: what happens?

     Atsakymas

     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. 125 kortelė

     Klausimas

     How should a negative related rate be interpreted?

     Atsakymas

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

126. 126 kortelė

     Klausimas

     When is local linearity a sound approximation tool?

     Atsakymas

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

127. 127 kortelė

     Klausimas

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

     Atsakymas

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

128. 128 kortelė

     Klausimas

     When is speed increasing?

     Atsakymas

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

129. 129 kortelė

     Klausimas

     Volume changes with time: notation for its rate?

     Atsakymas

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

130. 130 kortelė

     Klausimas

     Why are similar triangles useful in related rates?

     Atsakymas

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

131. 131 kortelė

     Klausimas

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

     Atsakymas

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

132. 132 kortelė

     Klausimas

     What conclusion does L’Hospital’s Rule permit?

     Atsakymas

     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. 133 kortelė

     Klausimas

     When is speed decreasing?

     Atsakymas

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

134. 134 kortelė

     Klausimas

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

     Atsakymas

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

135. 135 kortelė

     Klausimas

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

     Atsakymas

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

136. 136 kortelė

     Klausimas

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

     Atsakymas

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

137. 137 kortelė

     Klausimas

     Extreme Value Theorem: hypothesis and conclusion?

     Atsakymas

     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. 138 kortelė

     Klausimas

     What is a critical number of $f$?

     Atsakymas

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

139. 139 kortelė

     Klausimas

     First derivative test for a local maximum?

     Atsakymas

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

140. 140 kortelė

     Klausimas

     Second-derivative sign for concave up?

     Atsakymas

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

141. 141 kortelė

     Klausimas

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

     Atsakymas

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

142. 142 kortelė

     Klausimas

     First step in an optimization model?

     Atsakymas

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

143. 143 kortelė

     Klausimas

     Mean Value Theorem: hypotheses and conclusion?

     Atsakymas

     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. 144 kortelė

     Klausimas

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

     Atsakymas

     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.

     

     288 kortelės

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

     Mokytis iš šio rinkinio nemokamai

     Atsidarys Nibomo ir galėsite pradėti mokytis.

145. 145 kortelė

     Klausimas

     First derivative test for a local minimum?

     Atsakymas

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

146. 146 kortelė

     Klausimas

     What must happen at an inflection point?

     Atsakymas

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

147. 147 kortelė

     Klausimas

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

     Atsakymas

     $f''$ 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. 148 kortelė

     Klausimas

     How do you confirm an optimization answer is absolute?

     Atsakymas

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

149. 149 kortelė

     Klausimas

     Rolle’s Theorem: hypotheses and conclusion?

     Atsakymas

     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. 150 kortelė

     Klausimas

     Difference between absolute and relative extrema?

     Atsakymas

     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. 151 kortelė

     Klausimas

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

     Atsakymas

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

152. 152 kortelė

     Klausimas

     Second derivative test for a local minimum?

     Atsakymas

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

153. 153 kortelė

     Klausimas

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

     Atsakymas

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

154. 154 kortelė

     Klausimas

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

     Atsakymas

     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. 155 kortelė

     Klausimas

     Which theorem links an average slope to an instantaneous slope?

     Atsakymas

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

156. 156 kortelė

     Klausimas

     How can an implicit derivative locate a horizontal tangent?

     Atsakymas

     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. 157 kortelė

     Klausimas

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

     Atsakymas

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

158. 158 kortelė

     Klausimas

     Second derivative test for a local maximum?

     Atsakymas

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

159. 159 kortelė

     Klausimas

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

     Atsakymas

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

160. 160 kortelė

     Klausimas

     Why must an optimization domain be stated?

     Atsakymas

     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. 161 kortelė

     Klausimas

     Which theorem guarantees absolute extrema, not where they occur?

     Atsakymas

     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. 162 kortelė

     Klausimas

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

     Atsakymas

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

163. 163 kortelė

     Klausimas

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

     Atsakymas

     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. 164 kortelė

     Klausimas

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

     Atsakymas

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

165. 165 kortelė

     Klausimas

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

     Atsakymas

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

166. 166 kortelė

     Klausimas

     How can an implicit derivative locate a vertical tangent?

     Atsakymas

     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. 167 kortelė

     Klausimas

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

     Atsakymas

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

168. 168 kortelė

     Klausimas

     Why are endpoints included in the candidates test?

     Atsakymas

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

169. 169 kortelė

     Klausimas

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

     Atsakymas

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

170. 170 kortelė

     Klausimas

     Second-derivative sign for concave down?

     Atsakymas

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

171. 171 kortelė

     Klausimas

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

     Atsakymas

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

172. 172 kortelė

     Klausimas

     What should the final line of an optimization solution state?

     Atsakymas

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

173. 173 kortelė

     Klausimas

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

     Atsakymas

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

174. 174 kortelė

     Klausimas

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

     Atsakymas

     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. 175 kortelė

     Klausimas

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

     Atsakymas

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

176. 176 kortelė

     Klausimas

     Left Riemann sum on equal subintervals?

     Atsakymas

     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. 177 kortelė

     Klausimas

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

     Atsakymas

     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. 178 kortelė

     Klausimas

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Atsakymas

     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. 179 kortelė

     Klausimas

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

     Atsakymas

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. 180 kortelė

     Klausimas

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

     Atsakymas

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

181. 181 kortelė

     Klausimas

     Right Riemann sum on equal subintervals?

     Atsakymas

     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. 182 kortelė

     Klausimas

     How does reversing integral bounds change the value?

     Atsakymas

     It changes the sign:

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

183. 183 kortelė

     Klausimas

     Net Change Theorem?

     Atsakymas

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

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

184. 184 kortelė

     Klausimas

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

     Atsakymas

     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. 185 kortelė

     Klausimas

     Power rule for antiderivatives?

     Atsakymas

     For $n\ne-1$,

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

186. 186 kortelė

     Klausimas

     Midpoint Riemann sum on equal subintervals?

     Atsakymas

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

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

187. 187 kortelė

     Klausimas

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

     Atsakymas

     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. 188 kortelė

     Klausimas

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

     Atsakymas

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. 189 kortelė

     Klausimas

     Antiderivative of $1/x$?

     Atsakymas

     On any interval not crossing zero,

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

190. 190 kortelė

     Klausimas

     Trapezoidal approximation on equal subintervals?

     Atsakymas

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

191. 191 kortelė

     Klausimas

     How do geometric regions help evaluate a definite integral?

     Atsakymas

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

192. 192 kortelė

     Klausimas

     Basic antiderivatives of sine and cosine?

     Atsakymas

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

193. 193 kortelė

     Klausimas

     Definite integral as a limit of Riemann sums?

     Atsakymas

     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. 194 kortelė

     Klausimas

     Constant-multiple rule for integrals?

     Atsakymas

     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. 195 kortelė

     Klausimas

     What pattern suggests $u$-substitution?

     Atsakymas

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

196. 196 kortelė

     Klausimas

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

     Atsakymas

     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. 197 kortelė

     Klausimas

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

     Atsakymas

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

198. 198 kortelė

     Klausimas

     Sum-and-difference rule for definite integrals?

     Atsakymas

     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. 199 kortelė

     Klausimas

     Basic antiderivative of $e^x$?

     Atsakymas

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

200. 200 kortelė

     Klausimas

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

     Atsakymas

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

201. 201 kortelė

     Klausimas

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

     Atsakymas

     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. 202 kortelė

     Klausimas

     How does concavity predict trapezoidal and midpoint error?

     Atsakymas

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

203. 203 kortelė

     Klausimas

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

     Atsakymas

     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. 204 kortelė

     Klausimas

     What denominator pattern suggests an arctangent antiderivative?

     Atsakymas

     After completing the square and scaling, a form like

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

205. 205 kortelė

     Klausimas

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

     Atsakymas

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

206. 206 kortelė

     Klausimas

     How does an initial condition determine an antiderivative?

     Atsakymas

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

207. 207 kortelė

     Klausimas

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

     Atsakymas

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

208. 208 kortelė

     Klausimas

     Why does an indefinite integral include $+C$?

     Atsakymas

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

209. 209 kortelė

     Klausimas

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

     Atsakymas

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

210. 210 kortelė

     Klausimas

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

     Atsakymas

     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. 211 kortelė

     Klausimas

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

     Atsakymas

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

212. 212 kortelė

     Klausimas

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

     Atsakymas

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

213. 213 kortelė

     Klausimas

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

     Atsakymas

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

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

214. 214 kortelė

     Klausimas

     Riemann sum for unequal subinterval widths?

     Atsakymas

     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. 215 kortelė

     Klausimas

     Does continuity guarantee integrability on a closed interval?

     Atsakymas

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

216. 216 kortelė

     Klausimas

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

     Atsakymas

     Where $g(x)\ne0$,

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

217. 217 kortelė

     Klausimas

     What algebraic rewrites often reveal a basic antiderivative?

     Atsakymas

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

218. 218 kortelė

     Klausimas

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

     Atsakymas

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

219. 219 kortelė

     Klausimas

     What is a differential equation?

     Atsakymas

     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. 220 kortelė

     Klausimas

     How does a verbal rate statement become a differential equation?

     Atsakymas

     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. 221 kortelė

     Klausimas

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

     Atsakymas

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

222. 222 kortelė

     Klausimas

     General solution versus particular solution?

     Atsakymas

     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. 223 kortelė

     Klausimas

     What does one segment in a slope field show?

     Atsakymas

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

224. 224 kortelė

     Klausimas

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

     Atsakymas

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

225. 225 kortelė

     Klausimas

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

     Atsakymas

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

226. 226 kortelė

     Klausimas

     What makes a first-order differential equation separable?

     Atsakymas

     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. 227 kortelė

     Klausimas

     What is an initial value problem?

     Atsakymas

     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. 228 kortelė

     Klausimas

     What is an isocline in a slope field?

     Atsakymas

     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. 229 kortelė

     Klausimas

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

     Atsakymas

     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. 230 kortelė

     Klausimas

     General solution of $dy/dt=ky$?

     Atsakymas

     $$y=Ce^{kt}$$

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

231. 231 kortelė

     Klausimas

     Core method for solving a separable differential equation?

     Atsakymas

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

232. 232 kortelė

     Klausimas

     How should a solution curve follow a slope field?

     Atsakymas

     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. 233 kortelė

     Klausimas

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

     Atsakymas

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

234. 234 kortelė

     Klausimas

     Why is one integration constant enough after integrating both sides?

     Atsakymas

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

235. 235 kortelė

     Klausimas

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

     Atsakymas

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

236. 236 kortelė

     Klausimas

     Can one differential equation have infinitely many solutions?

     Atsakymas

     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. 237 kortelė

     Klausimas

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

     Atsakymas

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

238. 238 kortelė

     Klausimas

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

     Atsakymas

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

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

239. 239 kortelė

     Klausimas

     What can be lost when dividing to separate variables?

     Atsakymas

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

240. 240 kortelė

     Klausimas

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

     Atsakymas

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

241. 241 kortelė

     Klausimas

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

     Atsakymas

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

242. 242 kortelė

     Klausimas

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

     Atsakymas

     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. 243 kortelė

     Klausimas

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

     Atsakymas

     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. 244 kortelė

     Klausimas

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

     Atsakymas

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

245. 245 kortelė

     Klausimas

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

     Atsakymas

     For $k>0$,

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

     It is independent of the initial amount.

246. 246 kortelė

     Klausimas

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

     Atsakymas

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

247. 247 kortelė

     Klausimas

     How is an initial condition used after separation?

     Atsakymas

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

248. 248 kortelė

     Klausimas

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

     Atsakymas

     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. 249 kortelė

     Klausimas

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

     Atsakymas

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

250. 250 kortelė

     Klausimas

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

     Atsakymas

     For $k<0$,

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

251. 251 kortelė

     Klausimas

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

     Atsakymas

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

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

252. 252 kortelė

     Klausimas

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

     Atsakymas

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

     Velocity below zero contributes negative displacement.

253. 253 kortelė

     Klausimas

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

     Atsakymas

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

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

     Think top minus bottom.

254. 254 kortelė

     Klausimas

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

     Atsakymas

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

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

255. 255 kortelė

     Klausimas

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Atsakymas

     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. 256 kortelė

     Klausimas

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

     Atsakymas

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

257. 257 kortelė

     Klausimas

     Cross-sectional area when each slice is a square?

     Atsakymas

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

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

258. 258 kortelė

     Klausimas

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

     Atsakymas

     Integrate the net rate:

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

259. 259 kortelė

     Klausimas

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

     Atsakymas

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

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

     Think right minus left.

260. 260 kortelė

     Klausimas

     Disc-method volume formula?

     Atsakymas

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

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

261. 261 kortelė

     Klausimas

     What units does average value have?

     Atsakymas

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

262. 262 kortelė

     Klausimas

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

     Atsakymas

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

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

263. 263 kortelė

     Klausimas

     Cross-sectional area when each slice is a rectangle?

     Atsakymas

     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. 264 kortelė

     Klausimas

     How do you determine bounds for area between curves?

     Atsakymas

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

265. 265 kortelė

     Klausimas

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

     Atsakymas

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

266. 266 kortelė

     Klausimas

     When is a particle moving to the right or left?

     Atsakymas

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

267. 267 kortelė

     Klausimas

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

     Atsakymas

     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. 268 kortelė

     Klausimas

     Why must an area integral be split where curves intersect?

     Atsakymas

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

269. 269 kortelė

     Klausimas

     How can a velocity table approximate displacement?

     Atsakymas

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

270. 270 kortelė

     Klausimas

     Washer-method volume formula?

     Atsakymas

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

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

271. 271 kortelė

     Klausimas

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

     Atsakymas

     If $s(a)$ is known,

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

272. 272 kortelė

     Klausimas

     How do you choose between vertical and horizontal area slices?

     Atsakymas

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

273. 273 kortelė

     Klausimas

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

     Atsakymas

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

274. 274 kortelė

     Klausimas

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

     Atsakymas

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

275. 275 kortelė

     Klausimas

     Single expression for area between two curves?

     Atsakymas

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. 276 kortelė

     Klausimas

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

     Atsakymas

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

277. 277 kortelė

     Klausimas

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

     Atsakymas

     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. 278 kortelė

     Klausimas

     When should a volume integral use $dy$?

     Atsakymas

     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. 279 kortelė

     Klausimas

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

     Atsakymas

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

280. 280 kortelė

     Klausimas

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

     Atsakymas

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

281. 281 kortelė

     Klausimas

     Why must total distance split at velocity sign changes?

     Atsakymas

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

282. 282 kortelė

     Klausimas

     When does an accumulated quantity reach a local maximum?

     Atsakymas

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

283. 283 kortelė

     Klausimas

     What distinguishes area from a definite integral?

     Atsakymas

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

284. 284 kortelė

     Klausimas

     How do position, velocity, and acceleration graphs correspond?

     Atsakymas

     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. 285 kortelė

     Klausimas

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

     Atsakymas

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

286. 286 kortelė

     Klausimas

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

     Atsakymas

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

287. 287 kortelė

     Klausimas

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

     Atsakymas

     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. 288 kortelė

     Klausimas

     Why should a contextual integral answer include a sentence?

     Atsakymas

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