For what values of a and b is the line -4x + y = b tangent to the curve y = ax^3 when x = -2

Question

johnnydicamillo · Answer

Would I just need to find the derivative of $$y = ax ^{3}$$?

luffingsails · Answer

You need to start with the derivative and then finding the slope along the curve. That slope would then be what you would need to solve for in the formula of the line.

johnnydicamillo · Answer

$$y' = a (3)x^{2}$$ because a is some constant

luffingsails · Answer

Right, so after you find the derivative, you can substitute -2 in for x. That should give you the slope along the curve, right?

johnnydicamillo · Answer

right, $$y' = a12$$

luffingsails · Answer

Okay, so now you know the slope of tangency, you just need to use the formula for the line (it will have the same slope... you will just need to solve for a and b to find the points.

johnnydicamillo · Answer

well if I rewrite the original -4x + 7 = b as y = 4x + b I can set a12 = 4 and solve for a

johnnydicamillo · Answer

that would get me 1/3

johnnydicamillo · Answer

how would I find b?

johnnydicamillo · Answer

oh duh, we know y = $$ax^3$$ and we just solved for a so we can do
$$-4(2) + (\frac{ 1 }{ 3 } -2^{3}) = b$$