1. Consider the curve defined by Find Write an equation of each horizontal tangent line to the curve The line through the origin with slope –1 is tangent to the curve at point P. Find the x and y coordinates of point P

2. Let Be the fourth degree Taylor polynomial for the function f about 4. Assume f has derivatives of all orders for all real numbers. Find f(4) and f ’’’(4)

3. Let R be the region enclosed by the graphs of Find the area of R

4. h is a function defined of all nonzero x Find all values of x for which the graph of h has a horizontal tangent and determine whether h has a local maximum, a local minimum of neither at each of these values. Justify. On what intervals, if any, is the graph of g concave up? Justify Write an equation for the line tangent to the graph of h at x=4 Does the tangent line to the graph of h at x=4 lie above or below the graph of h for x>4? Justify.

5. Find the height and the radius of the largest cone that can be inscribed in a sphere with radius R centimeters. If jello is flowing into the cone in part 1 at a rate of J cubic centimeters per hour, how fast is the jello rising when the cone is one third full of jello?

6. The function f has a Taylor Series about x=2 that converges to f(x) for all x in the interval of convergence. The nth derivative of f at x=2 is given by a) Write the first four terms and the general term of the Taylor series for f about x=2 b) Find the radius of convergence for the Taylor series for f about x=2. c) Let g be a function satisfying g(2) =3 and g’(x) =f(x) for all x. Write the first four terms and the general term of the Taylor series for g about x=2 Does the Taylor series for g converge at x=-2? Justify

7. A particle moves in the xy-plane so that the position of the particle at any time t is given by Find the velocity vector for the particle in terms of t, and find the speed at t=0 Find in terms of t, and find Find each value of t at which the line tangent to the path of the particle is horizontal, or explain why none exists Find each value t at which the line tangent to the path of the particle is vertical, or explain why none exists

8. An object moving along a curve in the xy-plane has position At time with At time t=0, the object is at position (-13,5). At time t=2, the object is at point P with x coordinate 3 a) Find the acceleration vector at time t=2 and the speed at time t=2 b) Find the y-coordinate of P c) Write an equation for the line tangent to the curve at P d) For what value of t, in any, is the object at rest? Explain

9. The solution to subject to Has at least one critical point. Find both coordinates of any critical points on the graph of the solution. Justify

10. Is on the graph of the solution. Solve the differential equation Determine the x-intercept(s) of the graph of the solution

12. Determine the volume when the region in the first quadrant, on the interval is rotated about the x-axis About the y-axis Find the arc length of the region above Find the surface area for the regions you found the volume of