This preview shows page 1. Sign up to view the full content.
Unformatted text preview: nd go directly to Par t 4 without loss of
continuity. 97 DERIVA
angent
1 9 . FIRST DERIVATIVE = Slope of Tang ent
y Q P
θ
(0 , 0 ) x1 y x)
z( δy P δx
x1 + δ x x θ1
0,0)
(0,0) )=
z 1(x t
gen
tan
+ c1
). x
y’(x 1 x1 x We obser ve that the line z(x) thru P and Q makes an angle θ w ith the xaxis.
δy
We also obser ve that : tan θ = x
δ
What is the Limit o f the line z(x) as Q → P ?
L imit tangent
The tangent t o the cur ve of the function y(x) at the point P or at x = x 1.
tang
Let us call it : z1(x) = m1x + c1 , where m1 is the SLOPE .
Note that at P : z1(x 1) = y(x1) This tangent to the curve of the function y(x) at the point P intersects the xaxis
tangent
tang
at an angle. Let us call this angle θ1 .
So, when we take the Limit as Q → P the line z(x) becomes the tangent to the
Limit
tangent
curve of y(x) at the point P. So : z1(x) = tan θ 1 . x + c1 .
δy
This is equivalent to saying: δL imit0  at the point P = tan θ 1
x→
δx
= S LOPE of TANGENT z1(x) t o the cur ve of y(x) at x = x1.
98 From the Analysis point of view :
Analysis
δy
  is the AVERAGE RATE OF CHANGE of the function y(x) over the
δx
small inter val δ x.
,
y (x) = L imit δ y is the INSTANTANEOUS RATE OF CHANGE.
δx → 0 δx
,
y (x 1) = FIRST DERIVATIVE of the function y(x) at x = x 1
= G RADIENT or SLOPE of the TANGENT z1(x)
to the curve of y(x) at x = x 1 . y ,(x 1) = tan θ1 Hence : On closer observation of the diagrams, as δ x → 0 it would appear that δ y also
TENDS TO zero. Hence:
Limit δ y = L imit δ y should = 0/ .
0
δx → 0 δ x
P → Q δx
What we can depict in a diagram has its limitations. So let us look at what is happening
Analysis
from an Analysis point of view.
Anal
Let y(x) = x3 What is δ y ? (x + δ x)3 − x3
= { x3 + 3x2 δ x + 3x δ x2 + δ x3 } − x3
So : = 3x2 δ x + 3x δ x2 + δ x3
δy
3x2 δ x + 3x δ x2 + δ x3
=
δx
δx
2 + 3x δ x + δ x2
= 3x We may simplify because δ x
the LIMIT as P → Q .
Limit δ y = L imit
P → Q δx
δx...
View
Full
Document
This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

Click to edit the document details