Multivariable Calculus

1. For a curve ~(s), the Frenet-Serret frame is f^t; ^p;^ bg. The plane with normal vector ^p iscalled the rectifying plane, the plane with normal vector ^b is called the osculating plane, andthe plane with normal vector ^t is called the normal plane. Find the equations of each of theseplanes in the cases of the circular helix,(t) = (cos t; sin t; t) at the point (p2=2;p2=2; =4):2. A curve is said to be a generalized helix if the ratio of curvature and torsion is constant.That is, (s)=(s) is independent of s. Show that any circular helix is also a generalized helix.Can you come up with any others?3. Use the formula for curvature in general parameter to nd a formula for the curvature of afunction y = f(x) using the parameterization, ~(t) = (t; f(t)):4. Given an example of a curves whose curvature is (s) = s2 and whose torsion is zero. Use acomputer to help you plot this graph.5. Find^t(s)d^pds(s)d^bds(s):6. Suppose that(s) is a curve whose curvature is (s) = s and whose torsion is (s) = s.Find d4~ds4 in terms of s; ^t(s); ^p(s); and ^b(s):7. Suppose that(s) is a curve whose curvature is (s) = es and whose torsion is (s) = eô€€€s.Find d4~ds4 in terms of s; ^t(s); ^p(s); and ^b(s):8. If a projectile is red with an angle of inclination and initial speed v0, then (a) show thatthe trajectory is x(t) = v0t cos and y(t) = v0t sin ô€€€ gt2=2, where g is the acceleration due togravity. (b) Find a value of which will maximum the total distance traveled by the projectile.9. Give an example of a curve with curvature (s) = 11+s2 :10. Is it possible to have a curve with zero curvature and nonzero torsion?