Unit Five: Logs Flashcards by Jessica Brooks

What is the integration of 1/x?

Ln |x| + C

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What is the derivative of Ln x?

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

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Break apart Ln ab

Ln a + Ln b

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Break apart Ln a/b

Ln a - Ln b

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Break apart Ln a^P

P times Ln a

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

Ex: Y=a/b

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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What is the integration of tan(x)?

-Ln Icos(x)I + C

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What is the integration of cot(x)?

Ln Isin(x)I + C

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What is the integration of sec(x)?

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

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What is the integration of csc(x)?

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

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What is the integration of sin(x)?

-cos(x)

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What is the integration of cos(x)?

sin(x)

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What is the derivative of tan(x)?

sec^2(x)

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What is the derivative of cot(x)?

-csc^2(x)

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What is the derivative of sec(x)?

sec(x)*tan(x)

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What is the derivative of csc(x)?

-csc(x)*cot(x)

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What is the derivative of sin(x)?

cos(x)

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What is the derivative of cos(x)?

-sin(x)

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

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

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

(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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How do you find the inverse of a function?

Switch the x and y

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

They switch

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

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How do the graphs of f and its inverse relate?

The inverse is a reflection over y=x

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

One-to-one, yes

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

One over f'(g(x))

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

One over f'(f^-1(x))

What is the derivative of e^x?

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

What is the integration of e^x?

E^x + C

Graph of Ln(x) Ln(x+y) Ln(x) + y -Ln(x) -Ln(-x)

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

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

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

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

b^c=x

What does b^logb(x) equal?

x for all x>0

What does logb(b^x) equal?

x for all x

Break apart logb(xy)

logb(x) + logb(y)

Break apart logb(x/y)

logb(x) - logb(y)

Break apart logb(x^p)

P*logb(x)

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

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

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

e^((lnb)(x))

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

1/(lna * x)

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

lna * a^x

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

1/lna * a^x + C

What is the formula for continuously compounding interest?

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

What is the formula for interest compounded less than continuously?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What is the derivative of arcsinu?

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

What is the derivative of arccosu?

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

What is the derivative of arctanu?

Derivative of u/1+u^2

What is the derivative of arccotu?

Negative Derivative of u/1+u^2

What is the derivative of arccscu?

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

What is the derivative of arcsecu?

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

What is the rule relating sin and cos?

Sin^2 + Cos^2 = 1

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

Soh-Cah-Toa

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

Arcsin(u/a) + C

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

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

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

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