I need some help on some of these problems...we very briefly went over the theory in class, with no real explanation or any examples!
1) Consider a parametric curve given by x(t)=t^2+28t+13 and y(t)=t^2+28t+4. How many units of distance are covered by the point P(t) = (x(t),y(t)) between t=0, and t=6 ?
I came up with 168, using the integral from 0 to 6 of sqrt(x'(t)^2+y'(t)^2)dt, but it's wrong
2) Find the length of the arc formed by x^2=5y^3 from point A to B, where A=(0,0) and B=(25,5).
I've never seen a problem like this, nor can I seem to find an example of one. Can anyone give me a push in the right direction?
3)Find the length of the parametrized curve formed by x(t)=0t^3+6t^2+6t and y(t)=-4t^3-6t^2+0t, where t goes from 0 to 1
I used the same formula as in #1, and ended up with a messy integral I couldnt really evaluate
4) Suppose a curve is traced by the parametric equations x= 5sin(t) and y=24-10cos(t)^2-20sin(t). At what point (x,y) is the tangent line horizontal?
I know that the tangent line is horizontal when the derivative is zero, but I don't know what to derive and such.
Thank you for any help!