Name                                    

 

MATH 106 PRACTICE MIDTERM 1

 

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

     Printable Key

 

 PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why or state the proper theorem.

A)  Suppose that f(x) and g(x) are differentiable inverse functions, and f '(2) = 3.  Then g'(2) = 1/3.

 

 

B)   If f is a differentiable function such that both f and f ' are positive for all x, then g(x)  =  ln(f(x)) is increasing for all values of x.

 

 

PROBLEM 2  Calculate the derivatives of the following functions.

A)            d
                       (2x)1-x
               dx

 

B)        d
                     eln(sin x)
            dx

 

C)         d         23t
                                     
             dt        t

 

D)      d
                 [ arcsin(1-3x) - (ln x)(arctan x)]
         dx

 

PROBLEM 3 Find the following integrals

A)      

   

B)    

 

 

C)      
 

 

D)     
 

 

 

PROBLEM 4    Let  f(x) = 2x3 + 4x + 5

A.  Prove that f(x) has an inverse.
    


B.    Find       d
                             f -1(11)
                     dx
    

PROBLEM 5   

Show that

          d
                     tanh x   =   sech2 x
         dx

 

 

Extra Credit:  Write down one thing that your instructor can do to make the class better and one thing that you do not want changed with the class. 

 

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