Taylor blog #12

The majority of this week we were reviewing and studying for the chapter five test. However, friday we did learn something new.
We learned the second derivative test and the shortcut to the first derivative test.

Some key words to recognize when to use the second derivative test are:
"Use the second derivative test"
"which direction is the concavity"
"determine the intervals on which the graph is concave up or down"
"determine the points of inflection"


an easy way to try to remember the steps for the second derivative test is to remember that you will be doing the same steps as the first derivative test except with a second derivative

STEPS TO USING THE SECOND DERIVATIVE TEST:

Step #1:
take the first derivative set equal to zero and solve. ((keep in mind that this step will not always be necessary but can always be performed anyway. This step is not necessary when one looking for concavity or points of inflection.))

Step #2:

take the second derivative set = 0 and solve

Step #3:
Check differentiability

Step #4:
set up intervals

Step #5:
pick numbers on the interval and plug them into the SECOND derivative

Step #6:
if POSITIVE: the function is concave up on that interval
if NEGATIVE: the function is concave down on that interval

Step #7:
If there is a change it is a point of inflection ((This step only needs to be done when the problem is asking for the points of inflection specifically))

Example #1:

Determine the open intervals on which the graph is concave up or concave down:
F(x)= 6/ x^2+3

Step #1:
take the first derivative set equal to zero and solve. (( But this step is not necessary because the problem is looking for concavity.))

Step #2:

take the second derivative set = 0 and solve

(x^2+3)(0)-[(6)(2x)]/ (x^2+3)^2

F'(x)= -12x/ (x^2+3)^2

(x^2+3)^2(-12)-[(-12x)(2(x^2+3)(2x))]/ (x^2+3)^2

(x^2+3)(-12)+48x^2

-12x^2-36+48x^2

F"(x)= -36x^2-36=0

-36x^2=36

x^2= 1

x= +/- 1


Step #3:
Check differentiability

DIFFERENTIABLE?
yes.

Step #4:
set up intervals

(-infinity,-1)U(-1,1)U(1, infinity)


Step #5:
pick numbers on the interval and plug them into the SECOND derivative

F"(-2)= 36(-2)^2-36/ (-2^2+3)^2 = POSITIVE

F"(0)= 36(0)^2-36/ (0^2+3)^2 = NEGATIVE

F"(2)= 36(2)^2-36/ (2^2+3)^2= POSITIVE

Step #6:
if POSITIVE: the function is concave up on that interval

Therefore the function is concave up on the intervals (-infinity, -1)U(1,Infinity)

if NEGATIVE: the function is concave down on that interval

Therefore the function is concave down on the interval (-1,1)

Step #7:
If there is a change it is a point of inflection.

This step only needs to be done when the problem is asking for the points of inflection specifically therefore this step is not necessary for this problem because it is not asking for the points of inflection.




THE SHORTCUT TO THE FIRST DERIVATIVE TEST

Steps to the shortcut for the first derivative test:

Step #1:

take the derivative, set equal to zero, and solve

Step #2:

Check to see if differentiable

Step #3:

Plug in critical points into the second derivative

Step #4

If POSITIVE there is a min
If NEGATIVE there is a max
If 0 the test is false

(((NOTICE THAT FOR THE SHORTCUT WHETHER THE PLUG IN IS POSITIVE OR NEGATIVE HAS A REVERSED MEANING)))


Example:

Step #1:

take the derivative, set equal to zero, and solve


-15x^4+15x^2=0

15x^2(-x^2+1)

x= 0, 1,-1


Step #2:

Check to see if differentiable

differentiable?
yes.


Step #3:

Plug in critical points into the second derivative

F"(X)=60x^3 + 30x

-60(0)^3+30(0)=0
-60(-1)^3+30(-1)=POSITIVE
-60(1)^3+30(1)=NEGATIVE

Step #4

If POSITIVE there is a min

Therefore, at x= -1 there is a min


If NEGATIVE there is a max

Therefore, at x= 1 there is a max


If 0 the test is false

Therefore, at x=0 the test fails