Calculus 1  Hour Exam 2  (continued) -   Math 220     Spring, 2010

 

 

 

8

 

 

When a thin circular plate of metal is heated in an oven, its radius increases at the

rate of  0.01  centimeters per minute.  At what rate is the plate’s area increasing

when the radius is  80  centimeters?

 

Answer:    dA/dt  =  16 π  cm2 / min

 

 

 

9.

What are the coordinates  (x,y)  for the highest point on the graph of the function

 

g(x)  =  10 x e -2x

 

Answer:  ( ˝ , 5/e )

 

 

 

 

 

 

 

 

 

10.

 

A function   f(x)   is given below along with its first and second derivatives in factored

and unfactored forms.

 

f(x)  =  x4  -  4x3  +  16x  -  16  =  (x+2) (x – 2)3

 

df/dx  =  4 x3  -  12 x2  +  16   =   4( x + 1 )(x – 2)2

 

d2f/dx2  =  12 x2  - 24 x   =   12 x (x – 2)

 

The graph of  f(x)  is decreasing upon which one of the following intervals?

 

a.       (-2, 2)     b.  (-1, 2)   c.  (0,2)     d.  (-∞,2)   e.  (-∞,-1)

 

      f.   (-∞,0)     g.  (-2,∞)    h.  (-1,∞)   i.  (0,∞)     j.  (-∞,∞)

 

Answer:  e. 

 

 

 

 

 

11.

 

Evaluate the following limits.   Show sufficient work to justify each answer.

 

a.                   lim  cos x / sin2x

                      x→0

 

b.                  lim  (e6x  - 6x  - 1 ) / x2

 

Answers:  a.  ∞      b.  18

 

 

12.

Which point on the graph of  y  =  √5x   is closest to the point (10,0)?

 

Answer:  ( 7.5, √37.5 )