Solutions to Math 1411- Test 2

 

 

1.         a.         Using the definition of derivative, find  if .

Answer:         

           

           

b.         Find the equation of the tangent line at

Answer:         

The equation of the tangent line is  or .        

 

 

2.         Given the function , find .

Answer:         

 

             

 

 

3.         Find the derivatives of

            a.        

Answer:

b.        

Answer:

 

c.        

Answer:

 

 

 

4.         Given the following function,

 

            a.         Indicate intervals where the function is increasing, decreasing, concave up, concave down.

Answer:          It appears the function is increasing on the intervals  and . 

                        It appears the function is decreasing on the intervals  and .

                        It appears the function is concave up on the intervals  and .

                        It appears the function is concave down on the interval     

            b.         Indicate the location of inflections points (if any)

Answer:          It appears that the inflection points are located at . 

c.         Sketch the graph of the derivative function.

Answer:          Note that  at  and .   on the intervals  and  and  on the intervals  and . The location of the maximum and minimum points on  are the locations of the inflection points of .  The are located at .      

d.         What is the equation of the function.

Answer:  It appears to be a fourth-degree polynomial with x-intercepts at .  The equation is .

                 Use the fact that the y-intercept is 8 to find the value of a.

 where  and

 

Note: for part c, the deriviative is .

The zeros for the graph of the derivative are .

 

The local maxima and minima points are when the derivative of equals zero.  The derivative equals zero when  when .  The inflection points are