Chapter 5.3, Problem 56E

Parameterize the following portions of curves: (a) e The straight line between (-5, 3) and (11, 11). (b) A portion of the function f(x) = r* + 1 from (1, 2) to (-2,-7) Three-quarters of the unit circle traversed counter-clockwise, starting at (1,0).

Find the values of h, k, and a that make the circle (x - h)2 + (y - k)2 = a2 tangent to the parabola y = x2 + 1 at the point (1, 2) and that also make the second derivatives d2y/dx2 have the same value on both curves there. Circles like this one that are tangent to a curve and have the same second derivative as the curve at the point of tangency are called osculating circles.

(6) Let a be a positive real number. Consider the curve defined by r? + y² – xy = a². Find all points on the curve with horizontal tangent line.