Due:  Monday, September 22

Rolle's Theorem states:   Let g be a continuous function defined on the closed interval [a,b] and differentiable on the open interval (a,b). If g(a) = g(b) = 0, then there exists at least one number c in (a,b) for which

The Mean Value Theorem states:  Let g be a continuous function defined on the closed interval [a,b] and differentiable on the open interval (a,b). There exists at least one number c in (a,b) for which

Cauchy's Extended Mean Value Theorem states:  If g and h are continuous functions each defined on the closed interval [a,b] and each differentiable on the open interval (a,b), then there exists at least one number c in the interval (a,b) such that

These three results are extremely important in the foundation of calculus.

(A)   Let g(x) = x³ - 7x + 6.  Check that g satisfies the hypotheses of Rolle's Theorem for the interval [-3,2].  Graph g.  How many points x=c do there appear to be in the interval [-3,2] for which g'(c) = 0?  Find all such points. Graph the tangent line to g at each (and all) of these points.

(B)   Let f (x) = (5x⁶-x⁵+2x⁴-3x)sin(px).  Check that f satisfies the hypotheses of Rolle's Theorem for the interval [0,1].  Graph f.  How many points x=c do there appear to be in the interval [0,1] for which f '(c) = 0?  Find all such points. Graph the tangent line to f at each (and all) of these points.

(C)   Let g(x) = cos(3x) / (x² + 1) be defined on the interval [0, p].  Find all points x=c for which the MVT is satisfied.  Graph (on the same set of axes) the function y = g(x), the line passing through the end points (0,g(0)) and (p,g(p)), and all the tangent lines at the points x=c which you have found.

(D)   Suppose that Albertine drives from Paris to LeHavre, leaving Paris at time t=0, and arriving in LeHavre at time t=3.3 hours.  Her distance function (in km) is given by h(t) = 11.3(t + t^1.3+t^1.5)(2.51+cos t).   How far has Albertine traveled?  Find Albertine's average velocity over her journey from Paris to LeHavre.  Using the MVT, find a point, t^*, at which her instantaneous velocity equals her average velocity. Plot all relevant graphs (i.e., her distance function, the line connecting the end points of her distance function, and the tangent line to her distance function at the special point t^*).

(E)   Define a parametric curve by:  x(t) = sin(2pt), and y(t) = t cos(2pt) for t in the interval [0, 1.05].  Sketch this curve over its domain.  Draw the line that connects the end points of this curve.  Clearly the slope of this line is given by m = (y(1.05)-y(0)) / (x(1.05)-x(0)).  By Cauchy's extended MVT, there exists a number c in the interval (0,1.05) for which  y'(c)/x'(c) = m.   By the Chain Rule, y'(c)/x'(c) represents the slope of the tangent line to the parameterized curve at the point (x(c), y(c)).  Find this point c.  Graph (on the same set of axes) the original curve, the line joining the two end points, and the tangent line to the curve at the point c.
(Note:  You will need the plots library and the display command to graph the several curves on the same set of axes.  Note that the x-coordinate of the parameterized curve varies between -1 and 1.  Thus you should make sure that the tangent lines and the line joining the endpoints are also defined for a similar interval on the x-axis.)

(F)   Discuss briefly the history of Rolle's Theorem and the Mean Value Theorem.  Discuss the contributions of Michel Rolle and Augustin Louis Cauchy.