166 Calculus,         PRACTICE TEST #1

(* Actual Test #1 is Friday, January 31, 2003  7:30 to 9:00 AM *)

Sections 5.7, 6.1-6.4

Snail mail your papers to:
Irvin Roy Hentzel
Department of Mathematics
432 Carver Hall
Iowa State University
Ames, Iowa 50011-2064


1.  Find the most general antiderivative of the function.

                5      3
(a)  f(x) = 18 x  + 7 x - 13 x









                    12        3
                17 x    + 13 x     + 7
(b)  f(x) =   ------------------------- 
                           5
                          x










                    2            5
(c)  f(t) =  t Sin[t ] - 2 Sqrt[t ]












                7      2
             7 x  + 2x  + 1
(d)  f(x) = -------------
                  3/2 
                 x  










2.  A ball is thrown upwards from the edge of a 200 foot high cliff

with an upwards speed of 30 ft/sec.  How fast is it going when

it hits the ground at the base of the cliff?


























3.  A car is traveling at 60 m/h when the brakes are fully 

applied, producing a constant deceleration of 10 ft/s^2.

What is the distance covered before the car comes to a stop?


































4. Prove using induction that 


                 n              n(n+1)
                SUM  i     =   ------ .
                i=1               2  
































































5. Evaluate

      5       2
(a)  SUM  (3i)
     i=2















      29
(b)  SUM  2.1 
     i=7
















      6
(c)  SUM (2i+1)/2
     i=1
















      5     3
(d)  SUM  (x + x - 1)
     i=3









6.  Display the limit of this sum as n->infinity by drawing

a picture of the area that it converges to.  Do not give

a numerical answer.  Draw a picture and indicate the area.

                      _            _  2
                     |    / 2 i\  2 |
                     | 4-| ---- |   | 
          n          |_   \  n /   _|
         SUM    -----------------------
         i=1               n   





























7. Estimate the area of this lake.  Explain what you did.  Draw

rectangles showing the area that you actually computed  
                    
                                      . 
                                 .  ' | ' . .
                             . |      |      |'.
                         . '   |      |      |   ' .
                   .  ' |      |      |      |      |'.
              . ' |     |      |      |      |      |   ' .       
            .'    |     |      |      |      |      |      |'.      
          .'     20    25     35     40     35     25     15  '.    
          :.......|.....|......|......|......|......|......|.....:| 
          |<-10->|<-10->|<-10->|<-10->|<-10->|<-10->|<-10->|<-10->|















8. (a)  Give a proof to show that the derivative of the area

function is the cross section.

   (b)  Show how the theory applies to the area under y = 1/x

from x=1 to x=6.  What is A(x) in this case?  What is c(x) in

this case?





























































                                                     2     1/3
9.  Set up the limit to compute the area under y = (x  + 1)  

from x = 1 to 4 using a Riemann Sum.  Do not try to evaluate

the sum.

































































10.  Integrate the following

       Pi/4
(a)    INT    Sin[t] dt 
        0  
















        3        2 x
(b)    INT    8 e   dx
        0 















       
       Pi/3           
(c)    INT   Sec[t] Tan[t]   dt    
        0       
















       x=3
(d)    INT  x Sqrt[x+1] dx
       x=0