CPhill

Doing OK, man.......glad you're doing well, too......keep playing that music....!!!!!!

Unfortunately, GL.....I don't remember how to do this from my Calculus days...!!!....see if this helps some......

http://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx

BTW....how are you  ????

(f - g) (x)   =

f(x)  - g(x)  =

[5x + 7 ] -  [2 - 3x]  =

5x + 7 - 2 + 3x  =

8x + 5

    [ g ]     -   [  f  ]     =

  [4x + 14]  - [ x^2 - 4 ] =  

-x^2 + 4x  +18

So

(g - f) ( -1)    =  - (-1)^2 + 4(-1) + 18    =  -1 - 4 + 18   =  13

z³ =   1 / 343        take the cube root of both sides

z = 1/ 7

The horizontal distance =  sqrt ( d²  - 8800² )

(8x⁸)(6x⁹)  =

(8 * 6) ( x⁸ * x⁹)   =

48 x¹⁷

The side  will =  +cube root [27 / 512]    =  [3 / 8 ]  ft

sin(3x)  = sin(4x -x )  = sin4xcosx - sinxcos4x

sin(5x)  = sin(4x + x)  = sin4xcosx + sinxcos4x     .......so....

sin4xcosx + sinxcos4x  =  sin4xcosx - sinxcos4x

2sinxcos4x  = 0      divide both sides by 2

sinxcos4x = 0     

So either  

sinx  = 0     which happens at 0 + nPi    where n is an integer

or

cos(4x)  = 0

cos(x)  = 0   at   pi/2 and  3pi/2

So.....dividing each angle by 4 we have that

cos(4x)  =  0   at    pi/8 + n(pi/4)   where n is an integer

And   at    3pi/8 + n(pi/4).....however  the previous solution covers this one as well....so we have.......pi/8 + n(pi/4)

Here's the graph  of the intersection points [ in degrees].......https://www.desmos.com/calculator/jv5zqyexum

The distance to be closed in each case is the same.........=  k   km

And it will obviously take more time for the faster rider to catch the slower when they go in the same direction than when they close this distance traveling in the opposite direction

So......let S be the rate of the faster cyclist and s  be the rate of the slower

When they are going in the same direction, the effective rate rate of closure = S - s........when they are closing the distance from the opposite direction, the effective rate of closure  = S + s

And we have

k  km       =     k km

And Rate * Time   = Distance......... so.......

[S  - s] r    =   [ S + s] t

Sr - sr =  St + st

Sr - St  =  st + sr

S[r - t]  = s[r + t]

S/s   = [r + t ] / [ r - t ]