1. Find the Cartesian equation of the curve defined by parametric equations
x = 2cos t - cos 2t, y = 2sint - sin 2t.
What curve is it?
2. Consider the curve
x = t4 - 2t, y = t + t4.
(a) Graph the curve using a computational software package.
(b) Use your graph to estimate the coordinates of the lowest point and the leftmost point on the curve.
(c) Find the exact coordinates of your points from (b).
3. The astronomer Giovanni Cassini (1625-1712) studied the family of curves with polar equations
r4 - 2c2r2 cos 2θ + c4 - a4 = 0
where a and c are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of a and c. (Cassini thought that these curves might represent planetary orbits better than Kepler's ellipses).
(a) Investigate the variety of shapes that these curves may have by considering different values for a and c.
(b) Determine how a and c are related to each other when the curve splits into two parts.
4. (a) Show that the equation of the tangent line to the parabola
y2 = 4px at the point (x0, y0) can be written as
y0y = 2p(x + x0)
(b) What is the x-intercept of this tangent line? Use this fact to draw the tangent line.
5. Find the volume of the solid generated by rotating the top half (y≥0) of the ellipse
x2/a2 + y2/b2 = 1
about the x-axis