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8/13/2019 H3-12MM- Calculus - Gradient of a Curve at a Point and
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Mathematical Methods Unit 2
CalculusH Gradient of a curve at a point and the gradient
functionExample 1
Consider the function() .Let P be the point (1, 0) and let Q be
a point h units to the right of P.
Find the gradient of the chord PQ and hence find the gradient at
point P.
Example 2
For a curve with equation .
a. Find the gradient of chord AB where A is the point (1, -1)
and B is the point b. Find the gradient of AB when c. Find the
gradient of the curve at A.
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8/13/2019 H3-12MM- Calculus - Gradient of a Curve at a Point and
the Gradient Function
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The Gradient of a curve at any point.Recall:The gradient of a
function represents the rate at which a function is changing. To
find thegradient of a curve at any point, we need to find the
gradient of the tangent of the curveat thatpoint.
This gradient can be approximated by calculating the gradient of
the chord that connects twopoints that lie close to one another
i.e. finding the average rate of changebetween the twopoints.
Consider the points A to E on the graph of .3)( 2xxf
Using the CAS calculator, the gradient of the tangent at A is
________.
Investigating the gradients of chords connecting A to points
close to A we get:
Gradient of AE:12
12
xx
yym
=
Gradient of AD:12
12
xx
yym
=
Similarly, the gradients of AC and AB become better estimatesof
the gradient of the tangentat A.
Chord Gradient of chord
AE
AD
AC
AB
Gradient of tangent at A
As the distance between the two points approaches zero, the line
connecting the two pointsresembles the gradient of the tangent at
.Ax The gradient of the chordbecomes an accurateestimate of the
gradient of the tangent at this point.
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8/13/2019 H3-12MM- Calculus - Gradient of a Curve at a Point and
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In general:
Let A be any point on the curve, where the coordinates of point
A are represented by )).(,( xfx
Let B be another point on the curve so that the difference
between the x - coordinates of A
and B is equal to .h The coordinates of point B would thus be
)).(,( hxfhx
Then, the gradient of the chord AB(the average rate of change)
is:
12
12
xx
yym
As point B moves closer to point A, the distance between their x
coordinates, or ,h
approaches zero. The gradient of the chord ABbecomes an accurate
estimate of the gradientof the tangent.
i.e as
h
xfhxfh
)()(,0 gradient of the tangent at point A
The gradient of the tangent to the curve (the instantaneous rate
of change) is:
0,)()(
lim0
hh
xfhxfh
This limit is referred to as the gradient function, derivative
function, )(),(' fDxf x ,dx
dyor the
derivative of ).(xf The process of obtaining )(' xf is called
differentiationand evaluating the
above limit is known as differentiation from first
principles.
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8/13/2019 H3-12MM- Calculus - Gradient of a Curve at a Point and
the Gradient Function
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Example 3
Find the derivative of() using first principles
Example 4
a. Find the derivative of() using first principles
b. Hence find the gradient when ie find ()
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Example 5
a. Find the derivative of() using first principles
b. Hence find the gradient when
Ex 19B 1
