4.  (a) 
State the mean value theorem.

 (b)  Show that for all x > 1, where n is any
integer .

 (c)  Show that
satisfies ,
for every pair of numbers .

5.  Suppose that the equation

 

 represents a curve. [Note: You may assume that
and ]

 (a)  Find the points where this curve intersects the coordinate axes.

 (b)  Explain why the shape of the curve in the first quadrant
determines its shape in the other three quadrants.

 (c)  Find the tangent slope for any
point on this curve.

 (d)  At what points on the curve is the tangent line
horizontal? At what points is the tangent vertical?

 (e)  Show that is on the curve and
compute the tangent slope there.

 (f)  From the information in parts (a) to (e), draw a rough sketch of
the curve.

 (g)  Equation (1) expresses conservation of energy during motion of a
pendulum, with x as angular position variable and y as angular
velocity. What is the amplitude of this motion (i.e., the maximum
value of x)?
