Chapter 5 Flashcards
What is the definition of the natural log function
Lnx = integral from 1 to x of 1 over t dt, x>0
Describe the natural logarithmic function
Domain (0, infinity)
Range ( -infinity, infinity )
Continuous, increasing, one to one
Concave down
Log property
Ln(1)
0
Log property
Ln(ab)
Ln(a) + ln(b)
Log property
Ln(a^n)
n lna
Ln (a/b)
Ln(a) - ln(b)
Ln(e)
1
D
— [lnx]
Dx
1/x , x>0
D
— [lnu]
Dx
U prime
——– , u> 0
U
What do you so when finding derivatives
USE ln RULES FIRST!
Anti derivative of 1/x dx
Ln | x | + c
How do you know you have to find the derivative with natural log
Denominator is to the power of 1
Need derivative of the denominator on top
How do you know when it is reverse general power rule
Denominator is raised to power other than 1
Make it so there is no longer a denominator and then have the derivative of the quantity outside the parenthesis. Then multiply by the power, add 1 to power, put it all over new exponent
What happens if the degree of the numerator exceeds or is equal to the degree of the denominator
Have to do long division
Remainder is over original denominator
Antiderivative
Sinx dx
-cosx + c
Antiderivative
tanx dx
- ln |cosx| + c
Antiderivative
Secx dx
Ln |secx + tanx| + c
Antiderivative
Cosx dx
Sinx + c
Antiderivative
cotx dx
Ln | sinx | + c
Antiderivative
Cscx dx
- ln | cscx + cotx | + c
Average value
1 over b minus a from the integral of a to b of f of x dx
Finding area
Antiderivative of function
When are functions I verses of each other
When their composite function equals x
What is the requirement of inverse functions
They are one to one
Must pass horizontal line test
What are the steps to find an inverse
Switch x and y
Solve for y
What are inverse functions symmetrical to
Origin
Strictly monotonic
EITHER ALWAYS. Increasing or decreasing
How do you know when a function is strictly monotonic
By attaining the derivative and seeing if it is strictly pos or neg
When finding the inverse function always remember to…
State the domain and range
Finding tangent of an inverse we can’t find
(F^-1) ‘ a = 1/ f’ ( f^-1(a))
Answer is the slope of the tangent of the inverse
