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Unformatted text preview: → 0 ≠ 0 . It only TENDS TO zero as P → Q . Now we take
δy
 =
δx L imit { 3x2 + 3x δ x + δ x2 } = 3x2 δx → 0 99 we
saying
It is impor tant to note tha t in no w a y w e ar e sa ying the FIRST
DERIVA
alw
straight
DERIVATIVE is al w a ys a str aight line function of the ffor m m x + c .
or
,
Example: Let y(x) = x 3 . Then y (x) = 3x 2 i s not a straight line function.
Draw the curve of the function y(x) = x 3. Let P be a point on the curve of the
function x 3. Let x 1 be the x coordinate of the point P. The value of the FIRST
DERIVATIVE at x = x 1 of the function x 3 is:
,
y ( x 1) = 3 x 1 2 .
This numerical value 3x 12 = tan θ1, where θ1 is the angle that the tangent
tangent
at the point P on the curve of the function x 3 makes with the xaxis.
Draw the tangent z 1(x) to the cur ve at the point y(x 1 = 1) = 1 3 = 1.
tangent
tang
Measure the angle θ 1 t hat the tangent z 1(x) makes with the xaxis.
tangent
It should be just about 72 . And tan θ1 = t an 72 = 3 .078.
At the point on the cur ve x 3 = 1 t he x coordinate is x1 = 1 .
Evaluate the FIRST DERIVATIVE of x 3 at x 1 = 1 .
,
SLOPE
curv at
1
tan 72 = y (x1) = 3 = SLOPE of TANGENT to the cur ve at x 1 = 1..
Try this again at the point on the curve y(x2 = 2) = 23 = 8. tangent
Draw the tangent z 2(x) to the curve at this point.
tang
Measure the angle θ 2 t hat the tangent z 2 (x) makes with the xaxis.
tangent
It should be a little more than 85 . And tan θ2 = t an 85 = 1 2.
At the point on the cur ve x 3 = 8 t he x coordinate is x2 = 2 .
Evaluate the FIRST DERIVATIVE of x 3 at x2 = 2 .
,
SLOPE
curv at
2
tan 85 = y (x2) = 12 = SLOPE of TANGENT to the cur ve at x2 = 2..
100 zi ( x i ) = y (x i )
At x = xi slope of tangent zi (x) = y ’(x i )
slope
zi (x) = y ’(xi ) . x + c i is the tangent to y (x) at x = xi
tangent
The constant c i = zi ( x i ) − y ’(xi ) . xi
In point slope form the linear equation of the tangent is :
pointtangent
p oint
t ang
zi (x) − zi ( x i ) = y ’(x i ) . (x − x i )
At x = 1 the equation of tangent to y = x 3 is : z1(x) = 3x − 2
t angent
At x = 2 the equation of tangent to y = x3 is : z2(x) = 12x...
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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

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