		      FUNCTIONS, CURVES AND INVERSES
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To be even moderately successful in Calculus III you must know the basic
graphs of many functions.  This means not only recognizing functions
when you see their graphs, but also being able to sketch the graph when
given the function y = f(x).                                 

In addition to the graphs of 

        lines :      y = mx + b, 

        parabolas :  (y - h) = 4p(x - k)²  and  (x - k) = 4p(y - h)², and

        circles :    (x - x )² + (y - y )² = r²,
                           °           °      °

you should also know the graphs of functions such as y = ± 1/x,	y = exp x,
y = ln x, y = sin x, and y = cos x.  


You must also know the following results :


			Related Angle Theorem
	                 
Part 1 : The related angle of a given angle Θ is the positive acute
	 angle between the x axis and the terminal side of Θ.

Part 2:  Any trigonometric function of an angle is numerically equal
	 to the same function of its related angle.  The proper sign, 
	 + or -, is determined by the quadrant in which the given 
	 angle lies.

         QI   : All trig functions are positive.
                                                
         QII  : sine is positive (so is cosecant).                    
                                                
         QIII : tangent is positive (so is cotangent).                
                                                
         QIV  : cosine is positive (so is secant).


This sketch is a memory device to                        │
                                                     S   │   A
help you remember what trig functions                    │
                                                   ──────┼───────
are positive in which quadrants.                         │
                                                     T   │   C
                                                         │

Reading counterclockwise, beginning in the first quadrant, it spells ASTC.
The mnemonic I use is "All Students Take Calculus".


                        Inverse Functions

Theorem : If the domain of f is an interval, and if f is either an 
	  increasing function or a decreasing function on that 
	  interval, then f has an inverse.

                                
                      Inverse Trig Functions                                

        Θ = arccos x  if  O ≤ Θ ≤ π 
                
        Θ = arcsin x  if -π/2 ≤ Θ ≤ π/2                                 
        
        Θ = arctan x  if -π/2 < Θ < π/2


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