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Z - AP Calculus > Unit Five: Logs > Flashcards

Unit Five: Logs Flashcards

1
Q

What is the integration of 1/x?

A

Ln |x| + C

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2
Q

What is the derivative of Ln x?

A

1/x. The derivative of the natural log of something is one over something times the derivative of the something.

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3
Q

Break apart Ln ab

A

Ln a + Ln b

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4
Q

Break apart Ln a/b

A

Ln a - Ln b

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5
Q

Break apart Ln a^P

A

P times Ln a

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6
Q

How do you find the derivative of an equation without the quotient rule?

Ex: Y=a/b

A

Natural log both sides, then derive both sides. Remember to include the derivative of y. (dy/dx) If you end up with a y then substitute what y equals back in.

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7
Q

What is the integration of tan(x)?

A

-Ln Icos(x)I + C

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8
Q

What is the integration of cot(x)?

A

Ln Isin(x)I + C

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9
Q

What is the integration of sec(x)?

A

Ln Isec(x) + tan(x)I + C

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10
Q

What is the integration of csc(x)?

A

-Ln Icsc(x) + cot(x)I + C

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11
Q

What is the integration of sin(x)?

A

-cos(x)

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12
Q

What is the integration of cos(x)?

A

sin(x)

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13
Q

What is the derivative of tan(x)?

A

sec^2(x)

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14
Q

What is the derivative of cot(x)?

A

-csc^2(x)

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15
Q

What is the derivative of sec(x)?

A

sec(x)*tan(x)

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16
Q

What is the derivative of csc(x)?

A

-csc(x)*cot(x)

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17
Q

What is the derivative of sin(x)?

A

cos(x)

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18
Q

What is the derivative of cos(x)?

A

-sin(x)

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19
Q

What is the formula for a trapezoidal approximation of an area under a curve?

A

(b-a)/n times 1/2 times (f(x0) + 2f(x1) +2f(x2) + … + 2f(xn-1) + f(xn))

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20
Q

What is the formula for Simpson’s rule of an approximation of an area under a curve?

A

(b-a)/n times 1/3 times (f(x0) + 4f(x1) +2f(x2) +4f(x3) + 2f(x4) + … + 4f(xn-1) + f(xn)) *N must be an even number

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21
Q

How do you find the inverse of a function?

A

Switch the x and y

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22
Q

What is the domain and range of f equal to relative to the inverse?

A

They switch

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23
Q

If f and g are inverses what are f(g(x)) and g(f(x)) equal to?

A

X

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24
Q

How do the graphs of f and its inverse relate?

A

The inverse is a reflection over y=x

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25
Q

What is the requirement for a function to have an inverse?

Do monotonic functions have an inverse?

A

One-to-one, yes

26
Q

If g(x) is equal to f^-1(x) then what is g’(x) equal to?

A

One over f’(g(x))

27
Q

If g(x) is equal to f^-1(x) then what is (f^-1(x))’ equal to?

A

One over f’(f^-1(x))

28
Q

What is the derivative of e^x?

A

E^x. The derivative of e^something is e^something times the derivative of the something

29
Q

What is the integration of e^x?

A

E^x + C

30
Q

Graph of Ln(x)

Ln(x+y)
Ln(x) + y
-Ln(x)
-Ln(-x)

A

High on the right (+x) down to -infinity (Ln 1 = 0)

Y up

-Y right

Going up to infinity (0+)

Flipped over the y-axis from previous

31
Q

What are the two rules to remember when solving Ln and e equations?

A

Ln e^x is x and e^ln(x) is x

32
Q

What is another form of logb(x)=C?

A

b^c=x

33
Q

What does b^logb(x) equal?

A

x for all x>0

34
Q

What does logb(b^x) equal?

A

x for all x

35
Q

Break apart logb(xy)

A

logb(x) + logb(y)

36
Q

Break apart logb(x/y)

A

logb(x) - logb(y)

37
Q

Break apart logb(x^p)

A

P*logb(x)

38
Q

How do you do a change of base from log to natural? (words)

A

logb(x)=lnx/lnb or logx/logb

39
Q

What is another way to put b^x? (in e)

A

e^((lnb)(x))

40
Q

What is the derivative of loga(x)? (a is a constant and x is a variable)

A

1/(lna * x)

41
Q

What is the derivative of a^x? (a is a constant and x is a variable)

A

lna * a^x

42
Q

What is the integration of a^x? (a is a constant and x is a variable)

A

1/lna * a^x + C

43
Q

What is the formula for continuously compounding interest?

A

Initial*e^rt
R is interest rate
T is time in years

44
Q

What is the formula for interest compounded less than continuously?

A

Initial(1+rate/times pr year)^ times per yearyears

45
Q

What is the domain and range for sin^-1?

A

[-1,1] and [-pie/2, pie/2]

46
Q

What is the domain and range for csc^-1?

A

(-infinity,-1] U [1, infinity) and [-pie/2,0) U (0, pie/2]

47
Q

What is the domain and range for cos^-1?

A

[-1,1] and [0, pie]

48
Q

What is the domain and range for sec^-1?

A

(-infinity,-1] U [1, infinity) and [0, pie/2) U (pie/2, pie]

49
Q

What is the domain and range for tan^-1?

A

(-infinity, infinity) and (-pie/2, pie/2)

50
Q

What is the domain and range for cot^-1?

A

(-infinity, infinity) and (0,pie)

51
Q

What quadrant do you use for the inverse trig of a positive or negative number?

A

Positive is 1,- is 2 or 4 depending on domain

52
Q

What is the derivative of arcsinu?

A

Derivative of u/the square root of 1-u^2

53
Q

What is the derivative of arccosu?

A

Negative Derivative of u/the square root of 1-u^2

54
Q

What is the derivative of arctanu?

A

Derivative of u/1+u^2

55
Q

What is the derivative of arccotu?

A

Negative Derivative of u/1+u^2

56
Q

What is the derivative of arccscu?

A

Negative Derivate of u/|u| times the square root of u^2 -1

57
Q

What is the derivative of arcsecu?

A

Derivate of u/|u| times the square root of u^2 -1

58
Q

What is the rule relating sin and cos?

A

Sin^2 + Cos^2 = 1

59
Q

What is the phrase to remember how the trigonomic functions relate to a triangle?

A

Soh-Cah-Toa

60
Q

What is the integral of 1/ the square root of a^2-u^2?

A

Arcsin(u/a) + C

61
Q

What is the integral of 1/a^2+u^2?

A

1/a * Arctan(u/a) + C

62
Q

What is the integral of 1/u*the square root of u^2-a^2?

A

1/a*Arcsec(|u|/a) + C

Z - AP Calculus (12 decks)

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