1) If s(t) = 2t^3 - 21t^2 + 60t is the position function of a particle moving in a straight line, would you be able to find its total distance traveled in, say 3 seconds, by finding s(0), s(1), s(2), s(3), and calculating the absolute value between each of them and then summing those values, as opposed to differentiating the function first, setting the derivative to 0, and solving for t?

Would you get the same answer?

I've tried it and it seems to give the same answer. I was just wondering if it was true for any position function. I would say yes.

2) The function of a line is y^2 + x^3 = 9. I calculated the slope of its tangent to be -3x^2/2y. It asked us to find a point(s) so that its tangent line is y + 6x = 13. So it's slope must be -6 at that point.

I got (2,1) as a point. Are there more than one, or is that the only one?