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					Chapter 2   Calculus: Hughes-Hallett




     The Derivative
Continuity of y = f(x)

A function is said to be continuous if there
 are no “breaks” in its graph.
A function is continuous at a point x = a if
 the value of f(x)  L, a number, as x  a
 for values of x either greater or less than
 a.
Continuous Functions-
The function f is continuous at x = c if f is
 defined at x = c and lim f ( x)  f (c).
                         xc

The function is continuous on an interval
 [a,b] if it is continuous at everypoint in
 the interval.
If f and g are continuous, and if the
 composite function f(g(x)) is defined on
 an interval, then f(g(x)) is continuous on
 that interval. (A theorem.)
  Definition of Limit-
Suppose a function f, is defined on an interval
 around c, except perhaps not at the point x = c.
The limit of f(x) as x approaches c:lim f ( x )  L
                                               x c
 is the number L (if it exists) such that f(x) is as
 close to L as we please whenever x is suffici-
 ently close to c (but x  c).
In        lim f ( x )  L    ,   0,   0,
            x c
 Symbols:             ,   ,   0,   0,
                   if 0 | x  c |  then | f ( x )  L | .
 Properties of Limits-

Assuming all the limits on the right hand side exist:
                                           
    1. lim bf ( x)   b lim f ( x) , b  a constant
       x c                  x c

     2.lim  f ( x)  g ( x)   lim f ( x)  lim g ( x)
       x c                          x c              x c

     3.lim  f ( x) g ( x)   (lim f ( x))(lim g ( x))
       x c                         x c           x c

            f ( x) lim f ( x)
     4.lim           x c       , provided lim g ( x)  0
       x c g ( x )   lim g ( x)             x c
                      x c

     5. lim b  b                           6. lim x  c
       x c                                     x c
Limits at Infinity-

If f(x) gets as close to a number L as we
 please when x gets sufficiently large, then
 we write: lim f ( x)  L
             x 


Similarly, if f(x) approaches L as x gets
 more and more negative, then we write:

              lim f ( x)  L
              x 
Average Rate of Change-

The average rate of change is the slope of the
secant line to two points on the
graph of the function.


          y 2  y1 f ( x 2 )  f ( x1 ) f ( x1  h )  f ( x1 )
m sec                                
          x 2  x1      x 2  x1                  h
The Derivative is --
 Physically- an instantaneous rate of change.

 Geometrically- the slope of the tangent line
  to the graph of the curve of the function at a
    point.

 Algebraically- the limit of the difference
  quotient as h 0 (if that exists!). In symbols:
dy df                      f(x1  h )  f ( x1 )         f(x1  x )  f ( x1 )
      f ' ( x1 )  lim                          lim
dx dx                 h 0         h               x 0         x
First Derivative Interpretation-

 If f’ > 0 on an interval, then f is
  increasing over the interval.

 If f’ < 0 on an interval, then f is decreas-
  ing over the interval.
Derivative Symbols:

If y = f(x) = x 3  3x  3 then each of the
  following symbols have the same meaning:
 dy df df ( x ) d( x 3  3x  3)
                               f ' ( x )  y'  D x f  D x y  3x 2  3
 dx dx   dx            dx

And at a particular point, say x = 2, these
 symbols are used:
   dy
               f ' (2)  y' (2)  D x f (2)  3(2) 2  3  9
   dx   x 2
Basic Formulas (1):

Derivative of a constant:
 If f(x) = k, the f’(x) = 0, k - a constant

Derivative of a linear function:
 If f(x) = mx + b, then f’(x) = m

Derivative of x to a power:
 If f ( x )  x n , then f ' ( x )  nx n 1
      Second Derivative
      Interpretation-
If f’’ > 0 on an interval, then f’ is increasing, so
 the graph of f is concave up there.
If f’’ < 0 on an interval, then f’ is decreasing, so
 the graph of f is concave down there.
If y = s(t) is the position of an object at time t,
 then:                              
Velocity: v(t) = dy/dt = s’(t) = s ( t )
                                                          
Acceleration: a(t) =d y / dt  s' ' ( t )  v' ( t )  s  v
                        2     2
Continuous Functions-
The function f is continuous at x = c if f is
 defined at x = c and lim f ( x)  f (c).
                         xc

The function is continuous on an interval
 [a,b] if it is continuous at everypoint in
 the interval.
If f and g are continuous, and if the
 composite function f(g(x)) is defined on
 an interval, then f(g(x)) is continuous on
 that interval. (A theorem.)
Continuity of y = f(x)

A function is said to be continuous if there
 are no “breaks” in its graph.
A function is continuous at a point x = a if
 the value of f(x)  L, a number, as x  a
 for values of x either greater or less than
 a.
Theorem on Continuity-

Suppose that f and g are continuous on an
 interval and that b is a constant. Then, on
 that same interval:
1. bf(x) is continuous.
2. f(x) + g(x) is continuous.
3. f(x)g(x) is continuous.
4. f(x)/g(x) is continuous, provided g( x)  0
 ` on the interval.
Differentiability-

A function f is said to be differentiable at
 x = a if f’(a) exists.

Theorem: If f(x) is differentiable at a point
 x = a, then f(x) is continuous at x = a.
     Linear Tangent Line
     Approximation-
Suppose f is differentiable at x = a. Then,
 for values of x near a, the tangent line
 approximation to f(x) is:
          f ( x)  f (a)  f ' (a)(x  a).
The expression f (a)  f ' (a)(x  a) is called
 the local linearization of f near x = a. We
  are thinking of a as fixed, so that f(a) and
  f’(a) are constant. The error E(x), is
  defined by E ( x)  f ( x)  f (a)  f ' (a)(x  a).
  and lim E ( x)  0.
        x a   xa

				
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posted:7/4/2012
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