Do the following problems and label correctly.  I haven’t been able to create an extra credit assignment.  But this assignment says its worth 20 points.  If you complete it correctly, you will receive 10 points extra credit.  So really the assignment is worth 20/10 points.  If you do not complete the assignment, you will NOT receive any extra credit points.  You will only receive extra credit if the assignment is complete.

Given the function ????(????)=????3ˆ’27????+12f(x)=x3ˆ’27z+12a. ????²(????)f²(x)
b. Find the critical values of the function ( hint set the f'(x) = 0)
2. Fill in the blanks ( Copy and past, put your answers in a different color)

Notation: Second Derivative
For ______________the second derivative of f, provided that it exists, is _______________________
Definition:      Concavity

The graph of a function f is concave upward on the interval (a, b) if _________is increasing on (a  b) and is concave downward on the interval (a, b) if _________is decreasing on (a, b).

Summary:      Concavity  
For the interval (a,b), if ________________, then ___________is increasing and the graph of f  is concave upward.  If __________then _______ is decreasing and the graph of f is concave downward.

Theorem:       Inflection Point

If ______________is continuous on (a, b) and has an inflection point at _____________then either  ___________or _____________ does not exist.

Procedure:     Graphing Strategy (first version)

Step 1  Analyze ___________Find the domain and the intercepts
Step 2  Analyze ____________Find the partition numbers for, and the critical values of, ___________ and determine local extrema.
Step 3  Analyze ___________Find the partition numbers for ____________and determine concavity.
            Step 4  Sketch the graph of the function f.
3. Fill in the blanks ( Copy and past, put your answers in a different color)  
Definition: Absolute Maxima and Minima

            If _________________for all x in the domain of f, then ________is called the absolute maximum.
            If _________________for all x in the domain of f, then ________is called the absolute minimum.

Theorems:

            1.         A function f that is continuous on a closed interval [a, b] has both an absolute __________ and an ____________ minimum on that interval.

            2.         Absolute ______________ (if they exist) must always occur at critical values or at endpoints.

            3.         Second Derivative Test
            Let f be continuous on an interval I with only one critical value c in I.

            If __________ and __________then ______________ is the absolute minimum of f on I.
            If ___________ and __________ then _____________ is the absolute maximum of f on I.

Procedure: Finding absolute extrema on closed intervals

            1.         Check to make certain that f is _)______________________ over [a, b].
            2.         Find the ______________ values in the interval (a, b).
            3.         Evaluate f at the ________________ a and b and at the critical values found in step 2.
            4.         The absolute _______________ is the largest value found in step 3.
            5.         The absolute _______________ is the smallest value found in step 3.
4.  What are the formulas and properties on indefinite integrals for C and k both a constant. PLease reference Chapter 5.1. This is an extension of Theorem 1 found in chapter 5
a.
b.
c.
d.
e.
5. Fill in the Blanks or write the formulas:

Formulas: General Indefinite Integral Formulas for Integration by substitution.  Can be found in Chapter 5.2 in your textbook

a.     
b.     
c.     
d.    
e.    
f.     
 Fill in the blanks of the Definition:      Differentials

            If ___________defines a differentiable function, then:
1.     The_____________________of the independent variable x is an arbitrary real number.
2.     The ______________________ of the dependent variable y is defined as the product of _______and 
dx:  ___________________

Procedure:     Integration by Substitution

a.     Select a _____________ that appears to simplify the integrand.  In particular, try to select u so the ________ is a factor in the integrand.
b.     Express the integrand entirely in terms of _____and ______, completely ________________ the original variable and its differential.
c.     __________________ the new integral if possible.
d.     Express the ________________ found in step 3 in terms of the original variable.
6. What is the Theorem for the Area between two curves:
7. What is the formula for Integration by parts and what are the 5 steps?