Calculus 1  Hour Exam 1   -   Math 221     Fall, 2010

 

 

 

1.

 

 

 

Find   dy/dx              a.  y = exp [ sec(xe  +  3x ) ]          b.  y = log2 [ 5π + tan-1 (x2) ]

 

Answers:  a. dy/dx  =  exp(sec(xe  + 3x) sec (xe  +  3x) tan (xe  + 3x ) (e xe-1  +  (ln 3) 3x )

 

                 b. dy/dx  =  2x / [ (ln 2) (5π  +  tan-1 (x2)  ( 1 + x4 ) 

 

 

 

2.

 

Find the equation of the tangent line to the curve     x2  +  y2  =  (2x2  +  2y2  - x )2

 

  Answer:    y  - ½  =  x

 

 

 

 

 

3.

 

 

Calculate the following limits.   Hint:  Use algebraic manipulation.

 

a.        lim  (x2  +  x  - 6) / (x2  -  7x   + 10)

     x→2

 

b.       Lim  ( cos 5θ  - 1 ) / θ2

           θ →0

 

Answer:  a.  – 5/3           b.  – 25/2

 

 

 

4.

 

 

Let  f(x)  =  2x / (x + 1)          Use the definition of the derivative to calculate  f’(x).  

 

  Answer:       f’(x)  =  2 / (x + 1)2

 

 

 

 

 

 

 

 

5.

 

 

 

Write down a specific function, f(x), satisfying the following conditions:

a.                   f(x) is defined for all real numbers  x

b.                  f is not continuous at  x = -1 or at x = 2, but is continuous at all other

                        real numbers  x

c.                   lim  f(x)  =  + ∞  and   lim f(x)  and lim f(x)  both exist

                     x→2-                           x→2+

 

You need not justify that your function has the required properties.  There are many

possible answers.

 

                           1/(x + 1)2          for   x  ≠  -1, 2

      f(x)   =            6                     for  x  = -1

                             7                     for  x =  2