4.6 Graphing With Calculus and Calculators

 

Example Problems

Page 276

 

#2     F (x) = 8x⁵  + 45x⁴  + 80x³ +90x² +200x

 

First you need to put the equation into your calculator and then find the best fit window.   The table is used best in order to find zeros.  If you are unable to see clearly enough try graphing the first derivative instead.  ZoomFit under the zoom heading works best for finding a perfect window.

ZERO for F (x)  = 0

 

(First derivative >)     F ‘ (x) = 40x⁴ + 180x³ + 240x² + 180x + 200

ZERO = -2            Critical Number = -2

 

                                                                            +             +

 

                                                      

 

 

To check concavity you must use the second derivative

= 160x³ + 540x² + 480x + 180

ZEROS = -2.27

 

     -                  +

 

 

 

 

 

 

Concave Down   (-, -2 .27)

Concave Up     (-2 .27, )

Therefore a change in concavity would indicate a point of inflection.

Point of Inflection = -2 .27

 

 

 

 

 

 

#5    For a problem involving asymptotes you follow basically the same directions as you would with the first example problem.

 

ZEROS =  none      ASYMPTOTES =  -1.59 , 2.55 , 0.228

 

Now graph the derivative using the quotient rule to find the first derivative.

Quotient Rule =

 

U = x                   = 1

V = x³ – x² – 4x + 1     = 3x² – 2x² – 4x

 

 

ZEROS = 1    > to find zeros and observe the graph fix the window by pressing graph on                      your calculator and then zoom6

	+ = above x-axis - = below x-axis

 

     +    +    +    -    -

 

 

 

 

 

The graph is increasing from negative infinity to one, and decreasing from one to positive infinity.  This indicates a max at one.

 

To check concavity you must use the second derivative:

Now use the graph to determine concavity.  Anything above the x-axis is considered concave up, and anything below the x-axis is considered concave down.

 

CU (-, -1.59) (-0.51,0.228) (2.55,)

CD (-1.59,-0.51) (0.228,2.55_

 

The point of inflection is at (-0.51,-0.19), which is where the graph changes from increasing to decreasing.