CPhill

-33 degrees 45 minutes = -33.75 degrees....to convert to radians, we can use

-33.75 x pi / 180 = about -.589 rads

What number do we need to multiply by 10 to get 20???

That number will be your answer.......

i'm sure heureka (or Alan or Bertie) can explain this better, but "implicit"  means that we can't "directly" take the derivative of this function because there is no way to separate the x and y terms so that we can isolate y - as we might normally do.

So....we use the Chain/Product Rules  by treating y as an "unknown" function of x....... and we can differentiate with respect to "x," first, and then with respect to "y".....  

This allows us to "collect" all the terms that have y '  - (or.....dy/dx) - associated with them and then "solve" for y'.........

(I tend to use y ' for dy/dx.......whatever.......this procedure has always seemed to me to have a bit of " black magic" about it.......LOL!!!!  }

Remember, Melody, that when we have a "same over same" situation, the horizontal asymptote will just be the ratio of the coefficients on  the highest powers of the variable x.....this will be (9x^2)/(x^2), or just, 9.

Seen another way, just take the limit of the function as x approaches infinity.........

So ... lim as x →∞   [9x^2 + 18x - 72]  / [x^2 - 10x + 24].........divide all terms by x^2........18x, -72, -10x and 24  will approach 0 ..... and  we are left with  9x^2 / x^2    which approaches 9 as x →∞

The basic equation would be

y = [(x + 5)(x  - 1)] / [(x + 2) ( x + 6) ]

Note, however, that when x = 0, y = -5/12....but we need to have y = 2,  when x = 0.......this implies that, if we just multiipied the basic function by -24/5, this would produce what we want since (-5/12)(-24/5) = 2

So we have...

y = [-24(x + 5)(x  - 1)] / [5(x + 2) ( x + 6) ]  =  (-24x^2 - 96x + 120) / (5x^2 + 40x  + 60)

Write an equation for a rational function with: Vertical asymptotes at x = 4 and x = 6 x intercepts at x = -4 and x = 2 Horizontal asymptote at y = 9

y = [9(x+4) (x -2)] / [(x -4) (x -6)]   =  [9x^2 + 18x - 72]  / [x^2 - 10x + 24]

3x^2y^2 − 3y −17 = 5x +14

6xy^2 + 6x^2yy ' - 3y '  = 5

y' ( 6x^2y - 3)  =  5 - 6xy^2

y' =   ( 5 - 6xy^2) / (6x^2y - 3)        and the slope at (1, -3)  = [(5 - 6(1)(-3)^2] / [(6(1)^2(-3) - 3) ] = -49 /  -21   =   7/3

And the equation of the tangent line at this point is.....

y = (7/3)(x - 1) -3  

y =(7/3)x - (7/3) - 3

y = (7/3)x - 16/3

f(4) = 5 + √x  =   5 + √4  =  5 + 2 = 7