Math 1507

October 2, 1997
Tom Morley
Name: ____________________________

Open Book and Notes. You have one half hour. Please explain all answers.

Problem 1

Find all x such that | 8 - 4x | ≲ | x + 3|

Solve[Abs[8  -  4 x]  ==  Abs[x  +  3], x]

So to draw a picture

\(Show[
      Graphics[{
		\ Line[{{\(-3\), 0}, {4, 0}}], 
			
          Table[Line[{{k,  .2}, {k, \(- .2\)}}], 

            				\ \ \ \ {k, \(-3\), 4}], 

          		{PointSize[ .03], 
			RGBColor[1, 0, 0], 
			
            Point[{\(-1\), 0}], 
			Point[{11/3, 0}]}}], 

      PlotRange\  -> \ {{\(-4\), 4}, {\(-2\), 2}}]\)

Chech the signs in the various intervals :

Problem  2

Let  f(x) = (Sqrt[x]), and g(x)  = 1 - (x^2). Find (f⋄g)(x) and (g ⋄f) (x). Find the domain of each. (LABEL!)

f[x_]  =  Sqrt[x]

g[x_]  :=  1  -  x^2

f[g[x]]

g[f[x]]

This should actually be 1  -   | x |

Domains...f[g[x] = (f⋄g)(x)  is |x| ⩽ 1, 
Domain of  g[f[x]] = (g ⋄f) (x) is all real numbers

Problem 3

Find the limit as h tends to zero of:


\(TraditionalForm
      \`\(\((5\  + \ 3\ h)\)\^3\  - \ 125\  - \ 27\ h\^3\)\/h\)

Limit[(((((5  +  3 h))^3)  -  125  -  27 (h^3))/h), h  ->  0]

This is not hard to do if you just expand the 
      numerator :

ExpandAll[(((5  +  3 h))^3)  -  125  -  27 (h^3)]