Calculus 1  Hour Exam 1  (continued) -   Math 221     Fall, 2011

 

 

 

 

 

 

 

5.

 

 

 

a.       State the Intermediate Value Theorem.  Be sure to include hypotheses and

conclusion.

 

b.      Let  f(x) = x3 – 3x + 1.  Use this theorem to show that there is a number  c

between  1  and  2  such that  f(c)  =  0.  You should write sentences (in

addition to mathematical expressions) and be sure to verify, in writing,

the hypotheses of this theorem.

 

Answers:  If  f  is continuous on  [a,b] and N is between  f(a) and f(b), then there

                 Is a  c ε [a,b] with  f(c) = N.

 

f  is continuous since it is a polynomial;  f(1) = -1  and  f(2) = 3; 

0 is between f(1) and f(2).  So by this theorem there is a  c ε (1,2) with  f(c) = 0.

 

 

 

6.

 

Find  dy/dx for  y = 2 cos x.  Then find the equation of the tangent line to the graph

of  y = 2 cos x  at the point (0,2).

 

Answers:   dy/dx = - 2 sin x    and     y – 2  =  0

 

 

 

 

 

 

 

 

 

 

7.

 

 

For this problem answer true or false for each part.  You do not need to show work or

give any reason.  There is no partial credit on this problem.

 

a.        If   lim  [f(x) – f(1)] / (x-1)  =  3,  then  lim  f(x) = f(1).

                  x→1                                                x→1

 

b.      If  f(x) ≤ g(x) ≤ h(x) for all x and if  lim f(x) = 3 and if  lim h(x) = 4, then

                                                                    x→2                        x→2

 

          lim g(x) exists and   3 ≤  g(x) ≤ 4.

        x→2

 

c.       Polynomials are both continuous and differentiable for all x.

d.      If  f(x)  and  g(x) are differentiable, then  d/dx [f(x) g(x)]  =  f ‘(x) g ‘(x).

 

e.       If  f(x) and g(x) are differentiable and if  c  is a constant, then

        d/dx [c f(x) – g(x)]  =  c f ‘(x) – g ‘(x).

                

Answers:  a. T          b.  F          c.  T          d.  F          e.  T