Question: consider a plane curve c given by the parametric equations...

Consider a plane curve C given by the parametric equations x(t) = 6t² and y(t) = 3t³ − 4t. Note: In the solution of this problem use symmetry wherever possible.

1. (a) Find the point A(x0,y0) where C crosses itself. Hint: Solve the system of equations 6t²₁ = 6t²2 and 3t³₁ − 4t₁ = 3t³₂ − 4t₂.

2. (b) Find equations of both tangents at A.

3. (c) Find the points on C where tangent is horizontal or vertical.

4. (d) Determine where C is concave upward or downward.

5. (e) Indicate by arrows the direction of C.

6. (f) Find the area enclosed by the loop. Hint: Look at the problem 77 in the section 10.3

   of the textbook. Note: If you integrate the loop counterclockwise, the integral will be

   negative. Take absolute value of your answer.

7. (g) Find the arc length of the loop.