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Differential
calculus - derivatives
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Applications of the derivative
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Tangent,
normal, subtangent and subnormal
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A
segment of a tangent to a curve lying between the tangency point (the
point at which a tangent is drawn to a
curve) and the intercept of the tangent with the x-axis
is called the length of the tangent.
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A
segment of the x-axis
lying between the x-coordinate
of the tangency point and the intercept of the tangent with
the axis is called the sub tangent. |
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Therefore,
the sub tangent is the projection of the segment of the tangent onto the x-axis.
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segment of a line normal to a tangent lying between the tangency point
and the intercept of the normal with the x-axis
is called the length of the normal. |
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A
segment of the x-axis
lying between the x-coordinate
of the tangency point and the intercept of the normal with
the axis is called the subnormal.
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Therefore,
the subnormal is the projection of the segment of the normal onto the x-axis.
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In
the figure below denoted are,
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length of the tangent tl
= PT,
the subtangent st
= TX, |
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length of the normal nl
= PN
and
the subnormal sn
= NX. |
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the right triangle PTX, |
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| In
the triangle PNX, |
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| Therefore, |
the subtangent |
and |
the
subnormal |
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The
length of the tangent |
and |
the
length of the normal |
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Properties
of the parabola
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every point of the parabola y2
= 2px the subnormal have the
same value p. |
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By
differentiating both sides of the equation of the parabola with
respect to x
we get
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= 2p
or yy'
= p = sn. |
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We use this property to construct the normal and
the
tangent at a point of the parabola.
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Thus,
the normal line passes through the given point P(x,
y) of the
parabola
and the point x
+ p,
lying on the x-axis.
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since |
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the
vertex of the parabola y2
= 2p bisects the sub
tangent. |
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Property
of power functions
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sub tangent of the power function y
= axn,
where n
is a positive integer, |
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Thus
for example, the tangent to the parabola y
= ax2
at P(x,
y)
bisects the abscissa of P
that is, st
= x/2,
as
shows the figure below. |
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Property
of the exponential function
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sub tangent of the exponential f
y
= erx,
r
Î
R and r ¹
0, since |
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Thus
for example, at every point of the exponential function
y
= ex the
sub tangent st
= 1,
as shows the figure
above. |
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Contents
J
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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