Unit Five: Logs Flashcards
What is the integration of 1/x?
Ln |x| + C
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.
Break apart Ln ab
Ln a + Ln b
Break apart Ln a/b
Ln a - Ln b
Break apart Ln a^P
P times Ln a
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.
What is the integration of tan(x)?
-Ln Icos(x)I + C
What is the integration of cot(x)?
Ln Isin(x)I + C
What is the integration of sec(x)?
Ln Isec(x) + tan(x)I + C
What is the integration of csc(x)?
-Ln Icsc(x) + cot(x)I + C
What is the integration of sin(x)?
-cos(x)
What is the integration of cos(x)?
sin(x)
What is the derivative of tan(x)?
sec^2(x)
What is the derivative of cot(x)?
-csc^2(x)
What is the derivative of sec(x)?
sec(x)*tan(x)
What is the derivative of csc(x)?
-csc(x)*cot(x)
What is the derivative of sin(x)?
cos(x)
What is the derivative of cos(x)?
-sin(x)
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))
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
How do you find the inverse of a function?
Switch the x and y
What is the domain and range of f equal to relative to the inverse?
They switch
If f and g are inverses what are f(g(x)) and g(f(x)) equal to?
X
How do the graphs of f and its inverse relate?
The inverse is a reflection over y=x
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 yearyears
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
