Differntiaiton Flashcards
What happens when grsdient function above or below 0
Above is increasing function (gradient getting positbe ) below decreasing
What happens if second derivative above or below 0
ABIVE 0 means grsdient is increasing
This means the already positbe becomes more or already negative becomes less negative
If below 0 then grsdient is decreasing
So positbe becoming less kositobe
And negative becoming more negative
So what is conditond for concave up and down?
Up = is when second Derivative above 0
Down is when second is below 0
How to determine if it is up or down by graoh
Draw chord, of its abive tje graoh then up if below then down
How to find max or min grsdient
Second derivative , equate to 0 gives you values for when first derivative max or min
To determine nature third derivative test or either side test
How to determine nature of stationary points
Plug into second if negative maximum if positive then minimum if 0 MUST DO TEST
If ther grsdient is same on either side it’s a point of inflection .
But why do we have to do test
The second derivative of all inflections are 0 but not all 0 second derivatives are inflection
Thus need to do backwards and forwards tedt of max x value into grsdient function to decide
If it is (x-1)^3 what clue does thisngive you about nature of stationary
Likely to be point of inflection
What’s another thing about second derivative and points of inflection
Point of inflection goes from concave up to down or down to up, thus the sign of second derivative MUST change in a POINT IF INFLECTION
Okay what about NON STATIONARY POINTS OF INFLECTION
for this dy/dx not 0, but d26bdx2 MUST BE 0, as condition for a point of inflection is second derivative must equal to 0
- and the concavity changes throughout
So equate second to zero, see if concavity changes it it doesntit’s not a point of inflection at all
If it does see if negative to positbe etc
So how to find all points of inflection
Find second derrivsitv solutions and test if their sign changes in second derivier
To see nature out in first
To see stationary inflection solve for first and if they Match solution for second try it
Chain rule
That you can cancel out in derivatives for composite functions
And can do as many as you sant
KEY PROPERTY what does dy/ex equal in terms of dx/dy
How can find gradient etc if only have y In expression for dy/ dx
Why use
Dy/dx = 1/dx/dy
And dx/dy = 1/dy/dx
So if yiu have y, then jus sub in value for y and it will work!
Use because itshard to rearrange for other one and requires more ruled
Same thing before but don’t be dumb , dy/dx = 2 thrn what is dx/dy
1/2 just recirporcya e
For product rule whar do you don
Differentiate one and keep other and add to differentiate other and keep the one , might have to use chain rule
