Calculus 1  Final Exam    -   Math 220   Fall, 2006

 

 

 

 

1.

 

 

Compute the following integrals.

 

        2                                                                                 √π/2

a.     ∫ [1/(4 + x2)] dx               b.  ∫  a3x dx                c.        ∫  x cos (x2)  dx            

       0                                                                                  0

 

Hint for a:  x = 2u      Answer for a:  I  =  π/4

Answer for b:  I  =  ae3x/3 lna    

Hint for c:  u  =  x2      Answer for c:   I  =  √2 /4

 

 

 

 

2.

A chemical engineer needs to build a closed (that is, it has both a top and bottom in

addition to its sides) cylindrical tank with a volume of 2000 L.  She’s on a tight budget

and wants to minimize the cost C, which is directly proportional to the surface area

A of the tank (that is,  C = kA for some fixed k).  What are the dimensions of the

tank (radius and height) that the engineer should choose?

Hint:  V  =  π r2 h  ,     Surface area  =  A  =  2π r h   +  2π r2

Answer:   r  =  10 / (π)1/3     h  =  20 / (π)1/3     

 

 

 

 

3.

 

 
  1. The following is an equation for a circle.  State its center and radius.

 

                             x2  +  y2  =  -8x

 

Hint:  Use  (x – a)2  +  (y – b)2  =  r2,      Answer:  center at (-4,0)    radius  =  4

 

  1. Find an equation for the tangent line to this circle at the point (-2,2√3)

(HINT)  use implicit differentiation)

Answer:   y  =  -(1/√3) x  +  (4/3)√3

 

 

 

 

4.

 

Compute the following limits, or indicate that it does not exist.

 

a.  lim  cot(x) ln(1 + x)          b.  lim [(x + 2)3  -  8]/x     c.  lim [tan(3θ)]/θsec(2θ)

    x → 0                                    x → 0                               θ → 0

 

Answer for a:  1,    Answer for b:  12,    Answer for c:  3

 

 

 

 

 

5.

Differentiate the following functions

                                                                                                                      x

a.  h(t)  =  at tan(t) ,  a > 1      b.  g(x)  =  cos(x)/[x2 – 3x + 1]   c.  f(x) =  ∫ ln(t2 + t-2) dt

                                                                                                                     1

Hint for a:  Use product rule.  To find derivative of  at   use logarithmic differentiation.

Answer for a:  dh/dt  =  (at lna)tan t  +  at sec2 t

Answer for b:  dg/dx  =  - sin x / (x2 -3x + 1)   -  (2x – 3)cos x / (x2 -3x + 1)2 

Answer for c:  df/dx  =  ln(x2  +  x-2)

 

 

 

6.

Find an equation for the tangent line to the graph   y = x arcsin(x)   at  x = ½

 

Hint:  Use    y  -  y1  =  m(x  -  x1)  where   m  =  dy/dx  =  slope of line

Answer:  y  -  π/12  =  (π/6  +  1/√3)(x  -  ½)

 

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