Question

For what values of a and b is the line -2x + y = b tangent to the curve y=ax^3 when x=5? No decimal answers.

Answer 1

Hello,


`y=2x+b \\\\ y=ax^3\\\\ slope\ =\ 3ax^2\\ if \ x=5\ then\ slope=75a\ =2\\\\ a= (2)/(75)\ and \ y= (2)/(75)* x^3\\\\ if \ x=5\ then\ y=(2)/(75)* 5^3= (10)/(3) \\\\ (10)/(3)=2*5+b\\\\ b= (-20)/(3)`

Answer 2

the tangent of the curve is determined by getting the first derivative of teh equation of the curve and substituting with the given data. in this case, the derivative of y=ax^3 is y' = 3a x^2. when x is equal to 5, y' = 3a (25) = 75 a

x=5; y = 125 a
y-y1 = m(x-x1)
y-125 a = 75 a (x-5)
y = 75 ax -500

-2x + y = b
-2(75/2) x + y = -500
a = 75/2
b = -500