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

Kartītes šajā kavā

1. 1. kartīte

   Jautājums

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

   Atbilde

   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. kartīte

   Jautājums

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

   Atbilde

   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. kartīte

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   When does direct substitution evaluate a limit?

   Atbilde

   When the function is continuous at the target input. Then

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

4. 4. kartīte

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   Three conditions for continuity at $x=a$?

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

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

5. 5. kartīte

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   Intermediate Value Theorem: hypotheses and conclusion?

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   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. kartīte

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

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   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. kartīte

   Jautājums

   How do you read a finite limit from a graph?

   Atbilde

   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. kartīte

   Jautājums

   Limit law for a sum or difference?

   Atbilde

   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. kartīte

   Jautājums

   What makes a discontinuity removable?

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   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. kartīte

    Jautājums

    Squeeze Theorem: usable form?

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    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. kartīte

    Jautājums

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

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

12. 12. kartīte

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

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    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. kartīte

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    Limit law for a product?

    Atbilde

    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. kartīte

    Jautājums

    Graph signature of a jump discontinuity?

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

15. 15. kartīte

    Jautājums

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

    Atbilde

    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. kartīte

    Jautājums

    Horizontal asymptote from a limit at infinity?

    Atbilde

    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. kartīte

    Jautājums

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

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    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. kartīte

    Jautājums

    Limit law for a quotient—and its condition?

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    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. kartīte

    Jautājums

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

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    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. kartīte

    Jautājums

    When is the Squeeze Theorem a natural choice?

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

21. 21. kartīte

    Jautājums

    Vertical asymptote from one-sided behavior?

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

22. 22. kartīte

    Jautājums

    Limit at infinity of equal-degree rational functions?

    Atbilde

    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. kartīte

    Jautājums

    When can a limit pass through a continuous outer function?

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    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. kartīte

    Jautājums

    What makes a discontinuity infinite?

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

25. 25. kartīte

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

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

26. 26. kartīte

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

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    With angles in radians,

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

    Equivalent scaled forms follow by substitution.

27. 27. kartīte

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    Continuity of a composition?

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

28. 28. kartīte

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    Limit at infinity when a rational numerator has lower degree?

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

29. 29. kartīte

    Jautājums

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

    Atbilde

    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. kartīte

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

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    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. kartīte

    Jautājums

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

    Atbilde

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

32. 32. kartīte

    Jautājums

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

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

33. 33. kartīte

    Jautājums

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

    Atbilde

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

34. 34. kartīte

    Jautājums

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

    Atbilde

    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. kartīte

    Jautājums

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

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

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

36. 36. kartīte

    Jautājums

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

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

    The derivative exists only if this finite limit exists.

37. 37. kartīte

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

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

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

38. 38. kartīte

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    What does differentiability imply about continuity?

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

39. 39. kartīte

    Jautājums

    Power rule for derivatives?

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

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

40. 40. kartīte

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    Units of $f'(x)$?

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

41. 41. kartīte

    Jautājums

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

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

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

42. 42. kartīte

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

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    $f'(x)>0$ 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. kartīte

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    Derivative of a constant?

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

    A constant function has zero rate of change.

44. 44. kartīte

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    Derivative of $\sin x$?

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

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

45. 45. kartīte

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    Product rule?

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

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

46. 46. kartīte

    Jautājums

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

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    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. kartīte

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

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

48. 48. kartīte

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

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

49. 49. kartīte

    Jautājums

    Derivative of a sum or difference?

    Atbilde

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

50. 50. kartīte

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    Derivative of $\cos x$?

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

    The standard formula assumes radians.

51. 51. kartīte

    Jautājums

    Quotient rule?

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    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. kartīte

    Jautājums

    Common notations for the first derivative?

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

53. 53. kartīte

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    Derivative of $e^x$?

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

54. 54. kartīte

    Jautājums

    What graph features can make $f$ nondifferentiable?

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

55. 55. kartīte

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    Derivative of $\tan x$?

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    Where $\tan x$ is defined,

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

    Angles are in radians.

56. 56. kartīte

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

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

57. 57. kartīte

    Jautājums

    Derivative of $\ln x$?

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    For $x>0$,

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

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

58. 58. kartīte

    Jautājums

    How does the power rule handle roots or negative powers?

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    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. kartīte

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    Derivative of $\csc x$?

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    Where $\csc x$ is defined,

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

    Angles are in radians.

60. 60. kartīte

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

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    $f$ is increasing on that interval.

61. 61. kartīte

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

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    For $a>0$,

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

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

62. 62. kartīte

    Jautājums

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

    Atbilde

    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. kartīte

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    Derivative of $\sec x$?

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    Where $\sec x$ is defined,

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

    Angles are in radians.

64. 64. kartīte

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

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

65. 65. kartīte

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    Derivative of $\log_a x$?

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

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

66. 66. kartīte

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

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    For $h=fg$,

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

    Use the four table entries at the same input.

67. 67. kartīte

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    Derivative of $\cot x$?

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    Where $\cot x$ is defined,

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

    Angles are in radians.

68. 68. kartīte

    Jautājums

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

    Atbilde

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

69. 69. kartīte

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    Constant-multiple rule?

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    For a constant $c$,

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

70. 70. kartīte

    Jautājums

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

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

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

71. 71. kartīte

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

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

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

72. 72. kartīte

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

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

73. 73. kartīte

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

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    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. kartīte

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    Derivative of an inverse function at $x$?

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    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. kartīte

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    Derivative of $\arcsin x$?

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    For $|x|<1$,

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

76. 76. kartīte

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    Notation for the third derivative of $f$?

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

77. 77. kartīte

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

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

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

78. 78. kartīte

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

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    Where $y\ne0$,

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

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

79. 79. kartīte

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

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    Provided $f'(a)\ne0$,

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

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

80. 80. kartīte

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    Derivative of $\arctan x$?

    Atbilde

    For every real $x$,

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

81. 81. kartīte

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

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

    The extra factor is the chain rule.

82. 82. kartīte

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

    Atbilde

    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. kartīte

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

    Atbilde

    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. kartīte

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    Derivative of $\arccos x$?

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    For $|x|<1$,

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

85. 85. kartīte

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

    Atbilde

    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. kartīte

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

    Atbilde

    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. kartīte

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

    Atbilde

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

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

88. 88. kartīte

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    How are tangent slopes of inverse graphs related?

    Atbilde

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

89. 89. kartīte

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

    Atbilde

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

90. 90. kartīte

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

    Atbilde

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

91. 91. kartīte

    Jautājums

    Horizontal tangent on an implicit curve: derivative condition?

    Atbilde

    $dy/dx=0$ 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. kartīte

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

    Atbilde

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

93. 93. kartīte

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

    Atbilde

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

94. 94. kartīte

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

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

    This combines the power rule with the chain rule.

95. 95. kartīte

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    Vertical tangent on an implicit curve: derivative clue?

    Atbilde

    $dy/dx$ 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. kartīte

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

    Atbilde

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

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

97. 97. kartīte

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

    Atbilde

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

98. 98. kartīte

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

    Atbilde

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

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

99. 99. kartīte

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

    Atbilde

    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. kartīte

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

     Atbilde

     When the functions are differentiable, the chain rule gives

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

101. 101. kartīte

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     What local property lets a function have an inverse derivative?

     Atbilde

     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. kartīte

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

     Atbilde

     For $a>0$,

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

103. 103. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Position, velocity, and acceleration relationships?

     Atbilde

     For position $s(t)$,

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

105. 105. kartīte

     Jautājums

     Central idea of a related-rates problem?

     Atbilde

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

106. 106. kartīte

     Jautājums

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

     Atbilde

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

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

107. 107. kartīte

     Jautājums

     L’Hospital’s Rule: basic conditions?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

109. 109. kartīte

     Jautājums

     Speed in terms of velocity?

     Atbilde

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

     Velocity includes direction; speed is nonnegative magnitude.

110. 110. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Differential approximation connecting $dx$ and $dy$?

     Atbilde

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

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

112. 112. kartīte

     Jautājums

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

     Atbilde

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

113. 113. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     What does positive acceleration say about velocity?

     Atbilde

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

115. 115. kartīte

     Jautājums

     Related rates: when should numerical values be substituted?

     Atbilde

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

116. 116. kartīte

     Jautājums

     How does concavity predict linearization error?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     When is a particle moving in the positive direction?

     Atbilde

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

119. 119. kartīte

     Jautājums

     How can velocity show a change of direction?

     Atbilde

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

120. 120. kartīte

     Jautājums

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

     Atbilde

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

121. 121. kartīte

     Jautājums

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

     Atbilde

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

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

122. 122. kartīte

     Jautājums

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

     Atbilde

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

123. 123. kartīte

     Jautājums

     What must a contextual derivative sentence include?

     Atbilde

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

124. 124. kartīte

     Jautājums

     Velocity negative and acceleration positive: what happens?

     Atbilde

     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. kartīte

     Jautājums

     How should a negative related rate be interpreted?

     Atbilde

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

126. 126. kartīte

     Jautājums

     When is local linearity a sound approximation tool?

     Atbilde

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

127. 127. kartīte

     Jautājums

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

     Atbilde

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

128. 128. kartīte

     Jautājums

     When is speed increasing?

     Atbilde

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

129. 129. kartīte

     Jautājums

     Volume changes with time: notation for its rate?

     Atbilde

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

130. 130. kartīte

     Jautājums

     Why are similar triangles useful in related rates?

     Atbilde

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

131. 131. kartīte

     Jautājums

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

     Atbilde

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

132. 132. kartīte

     Jautājums

     What conclusion does L’Hospital’s Rule permit?

     Atbilde

     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. kartīte

     Jautājums

     When is speed decreasing?

     Atbilde

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

134. 134. kartīte

     Jautājums

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

     Atbilde

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

135. 135. kartīte

     Jautājums

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

     Atbilde

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

136. 136. kartīte

     Jautājums

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

     Atbilde

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

137. 137. kartīte

     Jautājums

     Extreme Value Theorem: hypothesis and conclusion?

     Atbilde

     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. kartīte

     Jautājums

     What is a critical number of $f$?

     Atbilde

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

139. 139. kartīte

     Jautājums

     First derivative test for a local maximum?

     Atbilde

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

140. 140. kartīte

     Jautājums

     Second-derivative sign for concave up?

     Atbilde

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

141. 141. kartīte

     Jautājums

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

     Atbilde

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

142. 142. kartīte

     Jautājums

     First step in an optimization model?

     Atbilde

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

143. 143. kartīte

     Jautājums

     Mean Value Theorem: hypotheses and conclusion?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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 kartītes

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

     Mācies šo kavu bez maksas

     Atvērsies Nibomo, lai vari sākt mācīties.

145. 145. kartīte

     Jautājums

     First derivative test for a local minimum?

     Atbilde

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

146. 146. kartīte

     Jautājums

     What must happen at an inflection point?

     Atbilde

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

147. 147. kartīte

     Jautājums

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

     Atbilde

     $f''$ 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. kartīte

     Jautājums

     How do you confirm an optimization answer is absolute?

     Atbilde

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

149. 149. kartīte

     Jautājums

     Rolle’s Theorem: hypotheses and conclusion?

     Atbilde

     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. kartīte

     Jautājums

     Difference between absolute and relative extrema?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

152. 152. kartīte

     Jautājums

     Second derivative test for a local minimum?

     Atbilde

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

153. 153. kartīte

     Jautājums

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

     Atbilde

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

154. 154. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Which theorem links an average slope to an instantaneous slope?

     Atbilde

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

156. 156. kartīte

     Jautājums

     How can an implicit derivative locate a horizontal tangent?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

158. 158. kartīte

     Jautājums

     Second derivative test for a local maximum?

     Atbilde

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

159. 159. kartīte

     Jautājums

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

     Atbilde

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

160. 160. kartīte

     Jautājums

     Why must an optimization domain be stated?

     Atbilde

     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. kartīte

     Jautājums

     Which theorem guarantees absolute extrema, not where they occur?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

165. 165. kartīte

     Jautājums

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

     Atbilde

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

166. 166. kartīte

     Jautājums

     How can an implicit derivative locate a vertical tangent?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

168. 168. kartīte

     Jautājums

     Why are endpoints included in the candidates test?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

170. 170. kartīte

     Jautājums

     Second-derivative sign for concave down?

     Atbilde

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

171. 171. kartīte

     Jautājums

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

     Atbilde

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

172. 172. kartīte

     Jautājums

     What should the final line of an optimization solution state?

     Atbilde

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

173. 173. kartīte

     Jautājums

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

     Atbilde

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

174. 174. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

176. 176. kartīte

     Jautājums

     Left Riemann sum on equal subintervals?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Fundamental Theorem of Calculus: evaluate a definite integral?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     If $f$ is continuous, then

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

     This connects accumulation with instantaneous rate.

180. 180. kartīte

     Jautājums

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

     Atbilde

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

181. 181. kartīte

     Jautājums

     Right Riemann sum on equal subintervals?

     Atbilde

     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. kartīte

     Jautājums

     How does reversing integral bounds change the value?

     Atbilde

     It changes the sign:

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

183. 183. kartīte

     Jautājums

     Net Change Theorem?

     Atbilde

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

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

184. 184. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Power rule for antiderivatives?

     Atbilde

     For $n\ne-1$,

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

186. 186. kartīte

     Jautājums

     Midpoint Riemann sum on equal subintervals?

     Atbilde

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

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

187. 187. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     If $f$ is continuous, then

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

     The variable lower bound produces the negative sign.

189. 189. kartīte

     Jautājums

     Antiderivative of $1/x$?

     Atbilde

     On any interval not crossing zero,

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

190. 190. kartīte

     Jautājums

     Trapezoidal approximation on equal subintervals?

     Atbilde

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

191. 191. kartīte

     Jautājums

     How do geometric regions help evaluate a definite integral?

     Atbilde

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

192. 192. kartīte

     Jautājums

     Basic antiderivatives of sine and cosine?

     Atbilde

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

193. 193. kartīte

     Jautājums

     Definite integral as a limit of Riemann sums?

     Atbilde

     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. kartīte

     Jautājums

     Constant-multiple rule for integrals?

     Atbilde

     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. kartīte

     Jautājums

     What pattern suggests $u$-substitution?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

198. 198. kartīte

     Jautājums

     Sum-and-difference rule for definite integrals?

     Atbilde

     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. kartīte

     Jautājums

     Basic antiderivative of $e^x$?

     Atbilde

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

200. 200. kartīte

     Jautājums

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

     Atbilde

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

201. 201. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     How does concavity predict trapezoidal and midpoint error?

     Atbilde

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

203. 203. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     What denominator pattern suggests an arctangent antiderivative?

     Atbilde

     After completing the square and scaling, a form like

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

205. 205. kartīte

     Jautājums

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

     Atbilde

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

206. 206. kartīte

     Jautājums

     How does an initial condition determine an antiderivative?

     Atbilde

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

207. 207. kartīte

     Jautājums

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

     Atbilde

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

208. 208. kartīte

     Jautājums

     Why does an indefinite integral include $+C$?

     Atbilde

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

209. 209. kartīte

     Jautājums

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

     Atbilde

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

210. 210. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

212. 212. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

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

214. 214. kartīte

     Jautājums

     Riemann sum for unequal subinterval widths?

     Atbilde

     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. kartīte

     Jautājums

     Does continuity guarantee integrability on a closed interval?

     Atbilde

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

216. 216. kartīte

     Jautājums

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

     Atbilde

     Where $g(x)\ne0$,

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

217. 217. kartīte

     Jautājums

     What algebraic rewrites often reveal a basic antiderivative?

     Atbilde

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

218. 218. kartīte

     Jautājums

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

     Atbilde

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

219. 219. kartīte

     Jautājums

     What is a differential equation?

     Atbilde

     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. kartīte

     Jautājums

     How does a verbal rate statement become a differential equation?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

222. 222. kartīte

     Jautājums

     General solution versus particular solution?

     Atbilde

     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. kartīte

     Jautājums

     What does one segment in a slope field show?

     Atbilde

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

224. 224. kartīte

     Jautājums

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

     Atbilde

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

225. 225. kartīte

     Jautājums

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

     Atbilde

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

226. 226. kartīte

     Jautājums

     What makes a first-order differential equation separable?

     Atbilde

     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. kartīte

     Jautājums

     What is an initial value problem?

     Atbilde

     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. kartīte

     Jautājums

     What is an isocline in a slope field?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     General solution of $dy/dt=ky$?

     Atbilde

     $$y=Ce^{kt}$$

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

231. 231. kartīte

     Jautājums

     Core method for solving a separable differential equation?

     Atbilde

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

232. 232. kartīte

     Jautājums

     How should a solution curve follow a slope field?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

234. 234. kartīte

     Jautājums

     Why is one integration constant enough after integrating both sides?

     Atbilde

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

235. 235. kartīte

     Jautājums

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

     Atbilde

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

236. 236. kartīte

     Jautājums

     Can one differential equation have infinitely many solutions?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

238. 238. kartīte

     Jautājums

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

     Atbilde

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

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

239. 239. kartīte

     Jautājums

     What can be lost when dividing to separate variables?

     Atbilde

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

240. 240. kartīte

     Jautājums

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

     Atbilde

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

241. 241. kartīte

     Jautājums

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

     Atbilde

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

242. 242. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     For $k>0$,

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

     It is independent of the initial amount.

246. 246. kartīte

     Jautājums

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

     Atbilde

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

247. 247. kartīte

     Jautājums

     How is an initial condition used after separation?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

250. 250. kartīte

     Jautājums

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

     Atbilde

     For $k<0$,

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

251. 251. kartīte

     Jautājums

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

     Atbilde

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

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

252. 252. kartīte

     Jautājums

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

     Atbilde

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

     Velocity below zero contributes negative displacement.

253. 253. kartīte

     Jautājums

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

     Atbilde

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

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

     Think top minus bottom.

254. 254. kartīte

     Jautājums

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

     Atbilde

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

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

255. 255. kartīte

     Jautājums

     Mean Value Theorem for Integrals: hypotheses and conclusion?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

257. 257. kartīte

     Jautājums

     Cross-sectional area when each slice is a square?

     Atbilde

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

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

258. 258. kartīte

     Jautājums

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

     Atbilde

     Integrate the net rate:

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

259. 259. kartīte

     Jautājums

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

     Atbilde

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

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

     Think right minus left.

260. 260. kartīte

     Jautājums

     Disc-method volume formula?

     Atbilde

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

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

261. 261. kartīte

     Jautājums

     What units does average value have?

     Atbilde

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

262. 262. kartīte

     Jautājums

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

     Atbilde

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

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

263. 263. kartīte

     Jautājums

     Cross-sectional area when each slice is a rectangle?

     Atbilde

     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. kartīte

     Jautājums

     How do you determine bounds for area between curves?

     Atbilde

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

265. 265. kartīte

     Jautājums

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

     Atbilde

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

266. 266. kartīte

     Jautājums

     When is a particle moving to the right or left?

     Atbilde

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

267. 267. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Why must an area integral be split where curves intersect?

     Atbilde

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

269. 269. kartīte

     Jautājums

     How can a velocity table approximate displacement?

     Atbilde

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

270. 270. kartīte

     Jautājums

     Washer-method volume formula?

     Atbilde

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

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

271. 271. kartīte

     Jautājums

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

     Atbilde

     If $s(a)$ is known,

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

272. 272. kartīte

     Jautājums

     How do you choose between vertical and horizontal area slices?

     Atbilde

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

273. 273. kartīte

     Jautājums

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

     Atbilde

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

274. 274. kartīte

     Jautājums

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

     Atbilde

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

275. 275. kartīte

     Jautājums

     Single expression for area between two curves?

     Atbilde

     When the functions are integrable,

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

     For hand evaluation, split where their order changes.

276. 276. kartīte

     Jautājums

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

     Atbilde

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

277. 277. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     When should a volume integral use $dy$?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

280. 280. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Why must total distance split at velocity sign changes?

     Atbilde

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

282. 282. kartīte

     Jautājums

     When does an accumulated quantity reach a local maximum?

     Atbilde

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

283. 283. kartīte

     Jautājums

     What distinguishes area from a definite integral?

     Atbilde

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

284. 284. kartīte

     Jautājums

     How do position, velocity, and acceleration graphs correspond?

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

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

     Atbilde

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

287. 287. kartīte

     Jautājums

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

     Atbilde

     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. kartīte

     Jautājums

     Why should a contextual integral answer include a sentence?

     Atbilde

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