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2-1 Limit Definition of the Derivative

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					2-1: Limit Definition of the Derivative                     Name ________________________ P. ___

Hw in Book: P. 104# 5,7,15,19,21,25,81,83, 84, 91 and attached!

The geometric meaning of the derivative is ……..




Let's look for this slope at P:




The secant line through P and Q has slope




We can approximate the tangent line through P

by moving Q towards P, decreasing x .

In the limit as x  0 , we get the tangent line

through P with slope




We define the derivative of f(x) as




** We derive all the basic differentiation formulas using this definition!


                                                                                                   1
Ex. 1   Find the derivative of f(x) = x2

using the limit definition of derivative.




Ex. 2    Find the derivative of f(x) = 1/x




                                             2
Ex.3 Find the derivative of f ( x)  x




Ex. 4   Find the tangent line of
        f ( x)  x3 at the point (1, 1).




Graph results to check solution!

                                           3
   Thm 2.1




Question:      Does continuity imply differentiability? Let’s see!

Ex. 5 Look at the graph of f ( x)  x  2 Notice that it is continuous on everywhere.
However, does a tangent line exist at the vertex, (2, 0)? In other words, is f(x) differentiable at x=2? To
answer this question, we need to use an alternative form of the limit definition of the derivative (p. 101 in the
text).
                                              f ( x )  f (c )
 The derivative of f(x) at c is f '(c)  lim                   , provided that this limit exists. If it does, then the right
                                         x c      xc
& left-hand limits (called the derivative from the left and from the right) must be equal. See graph to
understand.




Before we start, recall the definition of absolute value:




                                                                                                                               4
                                       1
Ex. 6 Look at the graph of f ( x)  x 3 Notice that it is continuous everywhere and specifically at x = 0.
However, is it differentiable at x = 0? The derivative of f(x) at 0 is




Derivatives Using the Limit Definition
The following problems require the use of the limit definition of a derivative, which is given by


                                                                            .
They range in difficulty from easy to somewhat challenging. If you are going to try these problems before
looking at the solutions, you can avoid common mistakes by making proper use of functional notation and
careful use of basic algebra. Keep in mind that the goal (in most cases) of these types of problems is to be able


to divide out the    term so that the indeterminant form       of the expression can be circumvented and the limit
can be calculated.

                                                                                         1    3
PROBLEM 1 : Use the limit definition to compute the derivative, f '( x) , for f ( x)      x
                                                                                         2    5

PROBLEM 2 : Use the limit definition to compute the derivative, f '( x) , for f ( x)  5 x 2  3x  7 .

PROBLEM 3 : Use the limit definition to compute the derivative, f '( x) , for f ( x)  4  x  3 .

                                                                                         x 1
PROBLEM 4 : Use the limit definition to compute the derivative, f '( x) , for f ( x)         .
                                                                                         2 x

PROBLEM 5 : Use the limit definition to compute the derivative, f '( x) ,for f ( x)  cos(3x)

                                                                                          x 1
PROBLEM 6 : Use the limit definition to compute the derivative, f '( x) , for f ( x) 
                                                                                         x  3x
                                                                                           2



Attach all Work!




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