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

Kártyák ebben a pakliban

1. 1. kártya

   Kérdés

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

   Válasz

   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. kártya

   Kérdés

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

   Válasz

   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. kártya

   Kérdés

   When does direct substitution evaluate a limit?

   Válasz

   When the function is continuous at the target input. Then

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

4. 4. kártya

   Kérdés

   Three conditions for continuity at $x=a$?

   Válasz

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

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

5. 5. kártya

   Kérdés

   Intermediate Value Theorem: hypotheses and conclusion?

   Válasz

   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. kártya

   Kérdés

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

   Válasz

   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. kártya

   Kérdés

   How do you read a finite limit from a graph?

   Válasz

   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. kártya

   Kérdés

   Limit law for a sum or difference?

   Válasz

   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. kártya

   Kérdés

   What makes a discontinuity removable?

   Válasz

   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. kártya

    Kérdés

    Squeeze Theorem: usable form?

    Válasz

    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. kártya

    Kérdés

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

    Válasz

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

12. 12. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    Limit law for a product?

    Válasz

    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. kártya

    Kérdés

    Graph signature of a jump discontinuity?

    Válasz

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

15. 15. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    Horizontal asymptote from a limit at infinity?

    Válasz

    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. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    Limit law for a quotient—and its condition?

    Válasz

    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. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    When is the Squeeze Theorem a natural choice?

    Válasz

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

21. 21. kártya

    Kérdés

    Vertical asymptote from one-sided behavior?

    Válasz

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

22. 22. kártya

    Kérdés

    Limit at infinity of equal-degree rational functions?

    Válasz

    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. kártya

    Kérdés

    When can a limit pass through a continuous outer function?

    Válasz

    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. kártya

    Kérdés

    What makes a discontinuity infinite?

    Válasz

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

25. 25. kártya

    Kérdés

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

    Válasz

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

26. 26. kártya

    Kérdés

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

    Válasz

    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. 27. kártya

    Kérdés

    Continuity of a composition?

    Válasz

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

28. 28. kártya

    Kérdés

    Limit at infinity when a rational numerator has lower degree?

    Válasz

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

29. 29. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

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

32. 32. kártya

    Kérdés

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

    Válasz

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

33. 33. kártya

    Kérdés

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

    Válasz

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

34. 34. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

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

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

36. 36. kártya

    Kérdés

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

    Válasz

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

    The derivative exists only if this finite limit exists.

37. 37. kártya

    Kérdés

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

    Válasz

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

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

38. 38. kártya

    Kérdés

    What does differentiability imply about continuity?

    Válasz

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

39. 39. kártya

    Kérdés

    Power rule for derivatives?

    Válasz

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

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

40. 40. kártya

    Kérdés

    Units of $f'(x)$?

    Válasz

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

41. 41. kártya

    Kérdés

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

    Válasz

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

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

42. 42. kártya

    Kérdés

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

    Válasz

    $f'(x)>0$ 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. kártya

    Kérdés

    Derivative of a constant?

    Válasz

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

    A constant function has zero rate of change.

44. 44. kártya

    Kérdés

    Derivative of $\sin x$?

    Válasz

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

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

45. 45. kártya

    Kérdés

    Product rule?

    Válasz

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

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

46. 46. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

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

48. 48. kártya

    Kérdés

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

    Válasz

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

49. 49. kártya

    Kérdés

    Derivative of a sum or difference?

    Válasz

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

50. 50. kártya

    Kérdés

    Derivative of $\cos x$?

    Válasz

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

    The standard formula assumes radians.

51. 51. kártya

    Kérdés

    Quotient rule?

    Válasz

    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. kártya

    Kérdés

    Common notations for the first derivative?

    Válasz

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

53. 53. kártya

    Kérdés

    Derivative of $e^x$?

    Válasz

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

54. 54. kártya

    Kérdés

    What graph features can make $f$ nondifferentiable?

    Válasz

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

55. 55. kártya

    Kérdés

    Derivative of $\tan x$?

    Válasz

    Where $\tan x$ is defined,

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

    Angles are in radians.

56. 56. kártya

    Kérdés

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

    Válasz

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

57. 57. kártya

    Kérdés

    Derivative of $\ln x$?

    Válasz

    For $x>0$,

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

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

58. 58. kártya

    Kérdés

    How does the power rule handle roots or negative powers?

    Válasz

    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. kártya

    Kérdés

    Derivative of $\csc x$?

    Válasz

    Where $\csc x$ is defined,

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

    Angles are in radians.

60. 60. kártya

    Kérdés

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

    Válasz

    $f$ is increasing on that interval.

61. 61. kártya

    Kérdés

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

    Válasz

    For $a>0$,

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

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

62. 62. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    Derivative of $\sec x$?

    Válasz

    Where $\sec x$ is defined,

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

    Angles are in radians.

64. 64. kártya

    Kérdés

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

    Válasz

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

65. 65. kártya

    Kérdés

    Derivative of $\log_a x$?

    Válasz

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

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

66. 66. kártya

    Kérdés

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

    Válasz

    For $h=fg$,

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

    Use the four table entries at the same input.

67. 67. kártya

    Kérdés

    Derivative of $\cot x$?

    Válasz

    Where $\cot x$ is defined,

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

    Angles are in radians.

68. 68. kártya

    Kérdés

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

    Válasz

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

69. 69. kártya

    Kérdés

    Constant-multiple rule?

    Válasz

    For a constant $c$,

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

70. 70. kártya

    Kérdés

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

    Válasz

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

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

71. 71. kártya

    Kérdés

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

    Válasz

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

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

72. 72. kártya

    Kérdés

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

    Válasz

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

73. 73. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    Derivative of an inverse function at $x$?

    Válasz

    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. kártya

    Kérdés

    Derivative of $\arcsin x$?

    Válasz

    For $|x|<1$,

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

76. 76. kártya

    Kérdés

    Notation for the third derivative of $f$?

    Válasz

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

77. 77. kártya

    Kérdés

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

    Válasz

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

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

78. 78. kártya

    Kérdés

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

    Válasz

    Where $y\ne0$,

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

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

79. 79. kártya

    Kérdés

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

    Válasz

    Provided $f'(a)\ne0$,

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

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

80. 80. kártya

    Kérdés

    Derivative of $\arctan x$?

    Válasz

    For every real $x$,

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

81. 81. kártya

    Kérdés

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

    Válasz

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

    The extra factor is the chain rule.

82. 82. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

    Derivative of $\arccos x$?

    Válasz

    For $|x|<1$,

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

85. 85. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

    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. kártya

    Kérdés

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

    Válasz

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

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

88. 88. kártya

    Kérdés

    How are tangent slopes of inverse graphs related?

    Válasz

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

89. 89. kártya

    Kérdés

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

    Válasz

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

90. 90. kártya

    Kérdés

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

    Válasz

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

91. 91. kártya

    Kérdés

    Horizontal tangent on an implicit curve: derivative condition?

    Válasz

    $dy/dx=0$ 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. kártya

    Kérdés

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

    Válasz

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

93. 93. kártya

    Kérdés

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

    Válasz

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

94. 94. kártya

    Kérdés

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

    Válasz

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

    This combines the power rule with the chain rule.

95. 95. kártya

    Kérdés

    Vertical tangent on an implicit curve: derivative clue?

    Válasz

    $dy/dx$ 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. kártya

    Kérdés

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

    Válasz

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

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

97. 97. kártya

    Kérdés

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

    Válasz

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

98. 98. kártya

    Kérdés

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

    Válasz

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

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

99. 99. kártya

    Kérdés

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

    Válasz

    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. kártya

     Kérdés

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

     Válasz

     When the functions are differentiable, the chain rule gives

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

101. 101. kártya

     Kérdés

     What local property lets a function have an inverse derivative?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     For $a>0$,

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

103. 103. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Position, velocity, and acceleration relationships?

     Válasz

     For position $s(t)$,

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

105. 105. kártya

     Kérdés

     Central idea of a related-rates problem?

     Válasz

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

106. 106. kártya

     Kérdés

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

     Válasz

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

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

107. 107. kártya

     Kérdés

     L’Hospital’s Rule: basic conditions?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

109. 109. kártya

     Kérdés

     Speed in terms of velocity?

     Válasz

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

     Velocity includes direction; speed is nonnegative magnitude.

110. 110. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Differential approximation connecting $dx$ and $dy$?

     Válasz

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

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

112. 112. kártya

     Kérdés

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

     Válasz

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

113. 113. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     What does positive acceleration say about velocity?

     Válasz

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

115. 115. kártya

     Kérdés

     Related rates: when should numerical values be substituted?

     Válasz

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

116. 116. kártya

     Kérdés

     How does concavity predict linearization error?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     When is a particle moving in the positive direction?

     Válasz

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

119. 119. kártya

     Kérdés

     How can velocity show a change of direction?

     Válasz

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

120. 120. kártya

     Kérdés

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

     Válasz

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

121. 121. kártya

     Kérdés

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

     Válasz

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

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

122. 122. kártya

     Kérdés

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

     Válasz

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

123. 123. kártya

     Kérdés

     What must a contextual derivative sentence include?

     Válasz

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

124. 124. kártya

     Kérdés

     Velocity negative and acceleration positive: what happens?

     Válasz

     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. kártya

     Kérdés

     How should a negative related rate be interpreted?

     Válasz

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

126. 126. kártya

     Kérdés

     When is local linearity a sound approximation tool?

     Válasz

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

127. 127. kártya

     Kérdés

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

     Válasz

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

128. 128. kártya

     Kérdés

     When is speed increasing?

     Válasz

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

129. 129. kártya

     Kérdés

     Volume changes with time: notation for its rate?

     Válasz

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

130. 130. kártya

     Kérdés

     Why are similar triangles useful in related rates?

     Válasz

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

131. 131. kártya

     Kérdés

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

     Válasz

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

132. 132. kártya

     Kérdés

     What conclusion does L’Hospital’s Rule permit?

     Válasz

     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. kártya

     Kérdés

     When is speed decreasing?

     Válasz

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

134. 134. kártya

     Kérdés

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

     Válasz

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

135. 135. kártya

     Kérdés

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

     Válasz

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

136. 136. kártya

     Kérdés

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

     Válasz

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

137. 137. kártya

     Kérdés

     Extreme Value Theorem: hypothesis and conclusion?

     Válasz

     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. kártya

     Kérdés

     What is a critical number of $f$?

     Válasz

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

139. 139. kártya

     Kérdés

     First derivative test for a local maximum?

     Válasz

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

140. 140. kártya

     Kérdés

     Second-derivative sign for concave up?

     Válasz

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

141. 141. kártya

     Kérdés

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

     Válasz

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

142. 142. kártya

     Kérdés

     First step in an optimization model?

     Válasz

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

143. 143. kártya

     Kérdés

     Mean Value Theorem: hypotheses and conclusion?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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 kártya

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

     Tanulj ingyen ebből a pakliból

     A Nibomo megnyílik, és elkezdheted a tanulást.

145. 145. kártya

     Kérdés

     First derivative test for a local minimum?

     Válasz

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

146. 146. kártya

     Kérdés

     What must happen at an inflection point?

     Válasz

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

147. 147. kártya

     Kérdés

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

     Válasz

     $f''$ 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. kártya

     Kérdés

     How do you confirm an optimization answer is absolute?

     Válasz

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

149. 149. kártya

     Kérdés

     Rolle’s Theorem: hypotheses and conclusion?

     Válasz

     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. kártya

     Kérdés

     Difference between absolute and relative extrema?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

152. 152. kártya

     Kérdés

     Second derivative test for a local minimum?

     Válasz

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

153. 153. kártya

     Kérdés

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

     Válasz

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

154. 154. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Which theorem links an average slope to an instantaneous slope?

     Válasz

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

156. 156. kártya

     Kérdés

     How can an implicit derivative locate a horizontal tangent?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

158. 158. kártya

     Kérdés

     Second derivative test for a local maximum?

     Válasz

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

159. 159. kártya

     Kérdés

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

     Válasz

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

160. 160. kártya

     Kérdés

     Why must an optimization domain be stated?

     Válasz

     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. kártya

     Kérdés

     Which theorem guarantees absolute extrema, not where they occur?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

165. 165. kártya

     Kérdés

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

     Válasz

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

166. 166. kártya

     Kérdés

     How can an implicit derivative locate a vertical tangent?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

168. 168. kártya

     Kérdés

     Why are endpoints included in the candidates test?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

170. 170. kártya

     Kérdés

     Second-derivative sign for concave down?

     Válasz

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

171. 171. kártya

     Kérdés

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

     Válasz

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

172. 172. kártya

     Kérdés

     What should the final line of an optimization solution state?

     Válasz

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

173. 173. kártya

     Kérdés

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

     Válasz

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

174. 174. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

176. 176. kártya

     Kérdés

     Left Riemann sum on equal subintervals?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. 180. kártya

     Kérdés

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

     Válasz

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

181. 181. kártya

     Kérdés

     Right Riemann sum on equal subintervals?

     Válasz

     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. kártya

     Kérdés

     How does reversing integral bounds change the value?

     Válasz

     It changes the sign:

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

183. 183. kártya

     Kérdés

     Net Change Theorem?

     Válasz

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

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

184. 184. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Power rule for antiderivatives?

     Válasz

     For $n\ne-1$,

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

186. 186. kártya

     Kérdés

     Midpoint Riemann sum on equal subintervals?

     Válasz

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

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

187. 187. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. 189. kártya

     Kérdés

     Antiderivative of $1/x$?

     Válasz

     On any interval not crossing zero,

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

190. 190. kártya

     Kérdés

     Trapezoidal approximation on equal subintervals?

     Válasz

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

191. 191. kártya

     Kérdés

     How do geometric regions help evaluate a definite integral?

     Válasz

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

192. 192. kártya

     Kérdés

     Basic antiderivatives of sine and cosine?

     Válasz

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

193. 193. kártya

     Kérdés

     Definite integral as a limit of Riemann sums?

     Válasz

     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. kártya

     Kérdés

     Constant-multiple rule for integrals?

     Válasz

     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. kártya

     Kérdés

     What pattern suggests $u$-substitution?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

198. 198. kártya

     Kérdés

     Sum-and-difference rule for definite integrals?

     Válasz

     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. kártya

     Kérdés

     Basic antiderivative of $e^x$?

     Válasz

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

200. 200. kártya

     Kérdés

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

     Válasz

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

201. 201. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     How does concavity predict trapezoidal and midpoint error?

     Válasz

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

203. 203. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     What denominator pattern suggests an arctangent antiderivative?

     Válasz

     After completing the square and scaling, a form like

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

205. 205. kártya

     Kérdés

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

     Válasz

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

206. 206. kártya

     Kérdés

     How does an initial condition determine an antiderivative?

     Válasz

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

207. 207. kártya

     Kérdés

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

     Válasz

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

208. 208. kártya

     Kérdés

     Why does an indefinite integral include $+C$?

     Válasz

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

209. 209. kártya

     Kérdés

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

     Válasz

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

210. 210. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

212. 212. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

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

214. 214. kártya

     Kérdés

     Riemann sum for unequal subinterval widths?

     Válasz

     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. kártya

     Kérdés

     Does continuity guarantee integrability on a closed interval?

     Válasz

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

216. 216. kártya

     Kérdés

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

     Válasz

     Where $g(x)\ne0$,

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

217. 217. kártya

     Kérdés

     What algebraic rewrites often reveal a basic antiderivative?

     Válasz

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

218. 218. kártya

     Kérdés

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

     Válasz

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

219. 219. kártya

     Kérdés

     What is a differential equation?

     Válasz

     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. kártya

     Kérdés

     How does a verbal rate statement become a differential equation?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

222. 222. kártya

     Kérdés

     General solution versus particular solution?

     Válasz

     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. kártya

     Kérdés

     What does one segment in a slope field show?

     Válasz

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

224. 224. kártya

     Kérdés

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

     Válasz

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

225. 225. kártya

     Kérdés

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

     Válasz

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

226. 226. kártya

     Kérdés

     What makes a first-order differential equation separable?

     Válasz

     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. kártya

     Kérdés

     What is an initial value problem?

     Válasz

     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. kártya

     Kérdés

     What is an isocline in a slope field?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     General solution of $dy/dt=ky$?

     Válasz

     $$y=Ce^{kt}$$

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

231. 231. kártya

     Kérdés

     Core method for solving a separable differential equation?

     Válasz

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

232. 232. kártya

     Kérdés

     How should a solution curve follow a slope field?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

234. 234. kártya

     Kérdés

     Why is one integration constant enough after integrating both sides?

     Válasz

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

235. 235. kártya

     Kérdés

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

     Válasz

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

236. 236. kártya

     Kérdés

     Can one differential equation have infinitely many solutions?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

238. 238. kártya

     Kérdés

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

     Válasz

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

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

239. 239. kártya

     Kérdés

     What can be lost when dividing to separate variables?

     Válasz

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

240. 240. kártya

     Kérdés

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

     Válasz

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

241. 241. kártya

     Kérdés

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

     Válasz

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

242. 242. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     For $k>0$,

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

     It is independent of the initial amount.

246. 246. kártya

     Kérdés

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

     Válasz

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

247. 247. kártya

     Kérdés

     How is an initial condition used after separation?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

250. 250. kártya

     Kérdés

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

     Válasz

     For $k<0$,

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

251. 251. kártya

     Kérdés

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

     Válasz

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

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

252. 252. kártya

     Kérdés

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

     Válasz

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

     Velocity below zero contributes negative displacement.

253. 253. kártya

     Kérdés

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

     Válasz

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

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

     Think top minus bottom.

254. 254. kártya

     Kérdés

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

     Válasz

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

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

255. 255. kártya

     Kérdés

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

257. 257. kártya

     Kérdés

     Cross-sectional area when each slice is a square?

     Válasz

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

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

258. 258. kártya

     Kérdés

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

     Válasz

     Integrate the net rate:

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

259. 259. kártya

     Kérdés

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

     Válasz

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

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

     Think right minus left.

260. 260. kártya

     Kérdés

     Disc-method volume formula?

     Válasz

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

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

261. 261. kártya

     Kérdés

     What units does average value have?

     Válasz

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

262. 262. kártya

     Kérdés

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

     Válasz

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

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

263. 263. kártya

     Kérdés

     Cross-sectional area when each slice is a rectangle?

     Válasz

     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. kártya

     Kérdés

     How do you determine bounds for area between curves?

     Válasz

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

265. 265. kártya

     Kérdés

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

     Válasz

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

266. 266. kártya

     Kérdés

     When is a particle moving to the right or left?

     Válasz

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

267. 267. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Why must an area integral be split where curves intersect?

     Válasz

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

269. 269. kártya

     Kérdés

     How can a velocity table approximate displacement?

     Válasz

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

270. 270. kártya

     Kérdés

     Washer-method volume formula?

     Válasz

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

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

271. 271. kártya

     Kérdés

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

     Válasz

     If $s(a)$ is known,

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

272. 272. kártya

     Kérdés

     How do you choose between vertical and horizontal area slices?

     Válasz

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

273. 273. kártya

     Kérdés

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

     Válasz

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

274. 274. kártya

     Kérdés

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

     Válasz

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

275. 275. kártya

     Kérdés

     Single expression for area between two curves?

     Válasz

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. 276. kártya

     Kérdés

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

     Válasz

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

277. 277. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     When should a volume integral use $dy$?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

280. 280. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Why must total distance split at velocity sign changes?

     Válasz

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

282. 282. kártya

     Kérdés

     When does an accumulated quantity reach a local maximum?

     Válasz

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

283. 283. kártya

     Kérdés

     What distinguishes area from a definite integral?

     Válasz

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

284. 284. kártya

     Kérdés

     How do position, velocity, and acceleration graphs correspond?

     Válasz

     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. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

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

     Válasz

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

287. 287. kártya

     Kérdés

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

     Válasz

     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. kártya

     Kérdés

     Why should a contextual integral answer include a sentence?

     Válasz

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