Unified Calculus I

Lab 4

Theorems Extravaganza
Four Existence Theorems in Differential Calculus

Applications on Continuity

I.  The Intermediate Value Theorem (IVT)
1.    State the Intermediate Value Theorem  (IVT) and give a graphical illustration of all
       the givens in the statement.

2.    Prove that the function:  x^3 - 8x + 3  has a root in the interval [0,2]

3.    Let f(x) = 1/x.  Although f(-1) = -1 and f(1) = 1, there is no value c such that
       f(c) = 0    Does this situation contradict the  IVT?  Explain.  

II.  The   Extreme Value Theorem (EVT).
4.    State the Extreme Value Theorem (EVT) and give a graphical illustration of all the
       givens of the theorem.

5.    Let g(x) = x^3 - 4x^2 + x - 2 be a continuous function defined on the closed
       interval     [-3,3].  Show that g(x) satisfies the EVT.  What is the maximum of g(x)?
       What is the minimum of g(x)?

6.    Given f(x) = 1/x.  Although f(x) is continuous on [1,infinity), f(x) has no minimum
        value on this  interval.  Does this function f(x) contradict EVT?  Explain.  

Applications on Continuity and Differentiability

III.  Rolle's Theorem (RT)
7.    State Rolle's Theorem and draw a graph to illustrate all the givens in the theorem.

8.    Let h(x) = x^3 - 2x^2 - x + 2  be a function defined on the closed interval [-1,2]
   .  Verify that h(x) satisfies the conditions of Rolle's Theorem (RT) then find a suitable
       number c that satisfies the conclusion of (RT).

9.    Repeat exercise 8 for the function t(x) = 3 (cos x)^2  defined on [(pi)/2,3(pi)/2].

IV.  The Mean Value Theorem. (MVT).
10.    State the Mean Value Theorem as formally and rigorously as you can. 

11.    Using the function S(x) = [(x-5)^2 ](x - 1) + 7  defined on the closed interval [1,7],
         give a graphical representation of the ( MVT) and find a number c such that c
         satisfies the conclusion of the  MVT.  For better results use the Derive printout of
         the graph of the function to answer the question.  (Turn in the Derive printout of the
         function with all required information completed on it.)

          (Answer for this question should be on the Derive graph printout)

12.    For f(x) = x^3 - 2x + 4 defined on the closed interval [0, 2], find the number c that
         satisfies the Mean Value Theorem.

13.    Repeat exercise 12 for the function U(x) = 3sin 2x  defined on [0, (pi)/2]

14.    The function f(x) = 1/(x-3) is differentiable on (0,3).  Does f(x) satisfy the Mean
         Value Theorem? Give a full explanation.

15.    The function g(x) = |x-2| is continuous on [0,3].  Does g(x) satisfy the Mean Value
         Theorem?  Give a full explanation.

16.    If f(x) = x^2 - x + 4 defined on the closed interval [0,3] represents the distance
         traveled by a moving vehicle, find the value of x where the instantaneous velocity
         equals the average velocity over the given interval.    

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