Calculus 1  Hour Exam 2   -   Math 221     Fall, 2010

 

 

 

1.

 

 

 

Find the absolute maximum and minimum and also where these extreme values occur

for                   g(x)  =  x √(6 – x2 )

 

Answers:             Abs max  = 3  at  (√3,3)       Abs min  =  - √5  at  (-1, √5 )

 

Also, without using the Mean Value Theorem, show explicitly that the conclusion of this

Theorem holds for    f(x)  =  x2  - 4x  +  1   on  [2,4] .

 

 

 

 

 

2.

 

Evaluate the following limits.   Hint:  Apply l’Hospital’s Rule

 

a.        lim  ( cos(x)  -  3 sin(x) )1/x

     xŕ0+

 

b.       lim  ( ex  -  e-x ) / sin (3x)                               Answers:  a.  e-3            b.   2/3

           xŕ0

                                                                              

 

 

3.

 

 

The volume of a cube is decreasing at a rate of 6 cubic inches per minute.  At what rate is

the surface area changing when the length of the side of the cube is 2 inches?

 

Answer:   -12 in2 / minute

 

 

 

 

 

 

 

 

 

4.

 

 

Consider the following function  f(x)  with  f’(x)  and  f’’(x)  given.

 

f(x)  =  x / (x2 – 9) ,     f ’(x)  =  - (x2 – 9) / (x2 – 9)2   ,   f ‘’(x)  =  2x (x2 + 27) / (x2 – 9)3     

 

Supply the requested information about the graph of y = f(x).  You need not justify

your answers.

 

types of symmetry if any                       Answer:  symmetric with respect to origin

 

vertical asymptote(s)                              Answer:    x  ± 3

 

horizontal asymptote(s)                         Answer    y = 0

 

interval(s) of decrease                           Answers   ( - ∞, -3 ),   ( -3, 3 ),   ( 3, ∞ )

 

interval(s) of upward concavity            Answers   ( 3, + ∞ ),   ( -3, 0 )

 

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