We started off class today going over the quiz #3. After going over next weeks schedule and things to remember for the TEST on Monday we immediately went into a complicated lesson on derivatives of inverse trig functions.
On question 3 of the quiz it should be remembered that
is similar to 
just with different variables and they are both asking you to find the derivative. Using the quotient rule ( (u*v'-u'*v)/ u^2 ) we can find the derivative when 
which is 
or 
and after simplifying equals 
Question 5 was just a matter of understanding what velocity, speed and acceleration. Velocity is the rate of change of a function or the derivative which in terms of seconds and meters is m/sec. Speed at a certain time is the absolute value of the velocity of the certain value or abs(m/sec). Acceleration is the second derivative or the rate of change of velocity. Therefore when finding the velocity each time the acceleration is zero means finding the zero of (second derivative)
which is 
. Then plug it into the velocity equation 
to get -12 m/sec. The speed at t=2 is the absolute value of velocity which equals 12 m/sec.Question 7 When using the h approaches 0 form of the derivative remember to use a FUFOO to simplify and use the derivatives rules to check your answer.
The Bonus question uses the idea that we can take the constant multiple out of the parenthesis of an equation to find the derivative more easily.
taking out the constant multiple and taking the derivative of inside the parenthesis using the product rule 
to get 
which simplifies to 
*Things to memorize before the test on Monday, November 5th:
-Trig Values
-Trig Derivatives
-Product Rule
-Quotient Rule
-Chain Rule
-h approaches zero form
-h approaches c form
At the end of the unit we should be able to find any derivative!
Inverse Trig Functions
Mr. O'Brien says that the best way to think of inverse functions is through the function machine (what make comes into the function and what comes out). In a regular sinx function the input is a number and the output is a ration. ex- when x=1/2, sinx = √2/2. Yet in a inverse function like
has an input of a number and an output of an angle. ex- when x=1/2, = π/6.You can also compute this:
, 
using Math - Frac on your calculator you will find the answer is 
. Using radian mode will make your life a lot easier. Also not that negatives reflect to the other side of the function ex- 
Graphing Inverse Trig Functions:
Sinx- The inverse of sinx doesn't pass the vertical line test and therefore is not a function so in order to graph the function there needs to be restrictions. Domain:[-1,1] and Range:[-π/2.π/2]
Cosx- The inverse of cosx doesn't pass the vertical line test either therefore it's not a function unless there are restrictions. To make the calculations easier the range starts at zero. Domain:[-1,1] and Range:[o,π]
Tanx- The inverse of tanx has asymptotes at y=-π/2 and π/2 rather x=-π/2 and π/2. Yet for it to be a function it also needs restrictions. Domain:[all real #'s] and Range:[-π/2 and π/2]
To prove the derivative of the inverse trig functions we use inverse properties to cancel out the inverse and then we use implicit differentiation (meaning we take the derivative of each side) then using the identities below we simplify and use division to equal out both sides and find the derivative. Since the derivatives of the inverse trig functions are angles we used a triangle to show what angle y equals.
CHea!
Homework for next class is to work on the IW #10 and #11 packet.
Here is a list of all the derivatives of inverse trig values
*Update- in certain functions it is easier to find the derivative after simplifying or reworking the original function. ex-
in which the derivative is
that is a lot easier than applying the product rule.
