Lecture 4 3 June Corey Irving _ Let's go back to Section 1.3 just for a

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Lecture 4 3 June Corey Irving _ Let's go back to Section 1.3 just for a Powered By Docstoc
					                                            Lecture 4
                                             3 June
                                           Corey Irving


•    Let's go back to Section 1.3 just for a moment:

•    The correct stretching and shrinking rules are:

    y= f ( cx)                    SHRINK       y= f ( x) horizontally by factor of c
          x                       STRETCH        y= f ( x) horizontally by factor of c
    y= f ( )
          c


•    I had the words SHRINK and STRETCH reversed in Lecture 2.

•    Let's go back and do the examples in the notes that this error effected:
                ◦ Lecture 2, Section 1.3, example 1 part 6:

                         x 2
                      y=( )
                         2




•    This was the second to last example we did in yesterday's notes:
                ◦ Graph the following:

                                  x
                      f (x )=∣ln ( )∣
                                  2




                                                  1
Section 2.1: The Tangent and Velocity Problems

   •   Section 2.1 Suggested Exercises: 1-9

   •   The tangent line to a curve at a point, P, on the curve is the line that most closely approximates
       the curve near P. Near P, this tangent line will hit the curve at the point P, but at no other points.

   •   A secant line of a curve is formed by choosing two points on the curve and drawing the line
       determined by those two points.




   •   How do we construct a tangent line given a graph and a point on it?




The Velocity Problem:

Example 2.
The table shows distance d from Milner that I was at after time t on my walk to Blocker from Milner
this morning. (Average adult walking speed: 3mi/hr= 264 ft/sec.)

Time t (minutes)     1                     2                     3                      4
Distance d (feet)   300                   550                  700                      1000
   1. What is the slope of the secant line through the points where


         t=1,2


         t=2,3




                                                     2
        t=3,4


   2. What is the interpretation of these slopes?




   3. Describe how we can find the instantaneous velocity at the point where t=3.




Section 2.2: The Limit of a Function
          •   Section 2.2 Suggested Homework: 1-8, 9, 11, 12, 13-19 odd, 21-25

          •   Intuitively, the limit of a function, y= f ( x) , asks the following: as x gets closer and
              closer to some value a , what y value is y= f ( x) getting closer and closer too?

          •   The notation for the above intuitive notion is

                                              lim x → a f ( x)

          •   If y= f ( x) does not approach exactly one real number value as x approaches a, we
               say that the limit does not exist. (HW #1)

          •   At first glance, it may seem like as x gets closer to a,   y= f ( x) always gets closer to
               y= f ( a).

          •   This is very often what happens, but not always. For instance, it may be that a is not in
              the domain of f.. Consider the following examples:


Example 1.
   1. We have a denominator.




                                                     3
   2. Piecewise functions




   •    Definition. (HW #2) The left-hand limit of f (x ) as x approaches a is the value that
           f (x ) approaches as x approaches a from the left, or equivalently, the value that f ( x ) as
        x increases towards a. The notation for this is

                                              lim x → a f ( x)
                                                      −




   •    Similarly, The right-hand limit of f (x ) as x approaches a is the value that f ( x )
        approaches as x approaches a from the right, or equivalently, the value that f ( x ) as x
        decreases towards a. The notation for this is

                                              lim x → a f ( x)
                                                      +




             lim x → a f ( x)= L         ⇔      lim x → a f ( x) And
                                                          −            lim x → a f ( x)
                                                                               +




Example 2. (HW #3-8, 13-16)
For which values of a does a limit exist?




                 {                 }
                      1   x⩽−2
   1.     f ( x )= x 3   −2< x⩽1
                    2− x    x>1




                                                     4
   2. With f as above, evaluate


        lim x →−2 f ( x )
                 +




Section 2.3: Calculating Limits Using the Limit Laws

   •   Section 2.3 Suggested Homework: 1, 2, 3-7 odd, 8-24, 33-39, 49


   •   Assume that c is a constant and the limits

                               lim x → a f ( x) and lim x → a g ( x )           exist,
       then we have the following limit Laws:

The Limit Laws

                            lim x → a [ f ( x)+g ( x)]=lim x→ a f ( x )+lim x → a g (x )



                            lim x → a [ f ( x)−g (x )]=lim x→ a f (x )−lim x → a g (x )



                                        lim x → a f ( x)=c lim x → a f ( x)



                              lim x → a [ f ( x) g ( x)]=lim x →a f (x )lim x→ a g ( x)



                                                            5
                                                  f ( x) lim x → a f ( x)
                                      lim x → a         =
                                                  g (x ) lim x → a g ( x)




       Example 1.         If lim x → a f ( x)=π and lim x → a g ( x )=π 2 , evaluate the following



   •    lim x → a [ f ( x) g ( x)]




                    π f (x)
   •    lim x → a
                     g ( x)




Example 2.


                              n
   •    lim x → a [ f ( x)]




                      x 2−9 3
   •    lim x →−3 (        )
                      x+3




                                                          6
        Here are more properties of limits:

         lim x → a √ f ( x )=√ lim x → a f ( x)
   1.

   2. If f is a polynomial or rational function and a is in the domain of f , then

                                                  lim x → a f ( x)= f ( a)



Example 3.
                         2
                         x −2
   •     lim x → 4        2
                         x −9




Example 4. Evaluate the limit

                          4−x
   1.    lim x → 4   +
                         ∣4−x∣




                            x 2 −9
   2.    lim x →−3
                         x 2+2x−3




                                                             7
                  x 2+ x−12
3.    lim x → 3
                     x−3




4. Trick: Calculating Limits by Rationalizing:

      lim x → 3   √ x+6−x
                   x 3−3x 2




                  ∣x∣
5.    lim x → 0
                   x




•    You do not have to learn the Squeeze Theorem!




                                            8

				
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