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f(x)=3x^5+6x^2−5

g(x)=2x^4+7x^2−x+16

(f - g) x  =  (3x^5+6x^2−5)  - ( 2x^4+7x^2−x+16 )  =

3x^5 - 2x^4 - x^2 + x - 21

avg rate of change over interval [7, 9]

=   slope between  ( 7, f(7) )  and  ( 9, f(9) )

=   slope between  ( 7, 1852 )  and  ( 9, 3878 )

=   \(\frac{3878-1852}{9-7}\)

=   \(\frac{2026}2\)

=   1013

avg rate of change over interval [4, 6]

=   slope between  ( 4, f(4) )  and  ( 6, f(6) )

=   slope between  ( 4, 358 )  and  ( 6, 1178 )

=   \(\frac{1178-358}{6-4}\)

=   \(\frac{820}2\)

=    410

How much greater is  1013  than  410   ?

1013 - 410  =  603

1013  is  603  greater than  410 .

Let's use a little analyses, NSS

Note that when x  = 0   .......y  is positiive  (-12/ -25 )  = 12/25   which is  < 1

So....this rules out the second graph and the third graph

Note that we can  find out where the graph is  = 0  by solving this

x^2 + 5x  - 12  = 0 

x^2 + 5x + 25/4  =  12 + 25/4

(x + 5/2)^2  =  73/4       take both roots

x + 5/2  =  ±√73 / 2

x  =  √73 / 2  -  5/2  ≈  1.77

x  = -√73 / 2  - 5/2   ≈ -6.77 

Notice that the 4th graph appears to have "zeroes" at about these values

Let's confirm that the 4th graph is correct :

https://www.desmos.com/calculator/rfpfjuqtrb

Yes he does need a break. Look at all the math questions he has answered!

Sorry, no LaTex !!.

1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...))))))))))))))))))))))).........etc.

Thanks

your breath?

1.  Not neccessarily

For instance

y  = x^2 - 25             (1)

x  - 5   = 0    ⇒    x  = 5    (2)

Is best solved  by   substituting  (2)   into (1)  ......so....False

2.  False.....look at the graph of these :

https://www.desmos.com/calculator/3jcfobc82b

x^2 + y^2  =  25

x - 5   = 0

Both of the equations have real roots

Sorry here they are

Look at the graphs , NSS

First  one :

https://www.desmos.com/calculator/llbnkmet5q

Second one   ....the asymptotes are at   y  = 5    (horizontal)   and   x = 1  ( vertical)

https://www.desmos.com/calculator/yfj1szyfym

No graphs shown , NSS .....  

No prob... MT   !!!

This is definitely even....it is symmetric about the y axis

Also...if  even   f (x)    =  f (-x)

f (-x)   =  -2(-x)^4  + 3 (-x)^2   =    -2x^4  +  3x^2    =  f (x)

Thanks so much, CPhill! 

A fire! 

Best to draw  a  "mock" pic

x

         10

z     √51             y

We need to find  xz  =    √ [ 10^2  - 51  ]  =  √  [ 100 - 51 ] =  √ 49  = 7

So....tan  X    =  zy / xz  =  √51  / 7

Circumference = pi * d    or    pi * 2 r

pi * 2 * 12.6 = pi 25.2

We are looking for the fraction of the circumference that 120 out of 360 dgrees covers

120 /360  *  pi 25.2 =   25.2 pi / 3

Answered here :

https://web2.0calc.com/questions/help_82215