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Contents
The Cartesian Plane
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The
Cartesian Plane
Calculus
can be applied to the cartesian plane. The derivative of an equation has a
geometrical representation. From here, the properties of a graph can be
determined. In this section, we will discuss equations of tangents and normals,
as well as stationary points. Integration also has a geometric representation
which we explain separately.
Geometrical
Representation of the Derivative
As
discussed in our section on differentiation by first principles, the
derivative represents the gradient of the graph. The general form of the
derivative, such has f'(x) = 3x2 + 4, represents the general
form of the gradient of that particular graph. When a numerical value is given
to the derivative, such as f'(1) = 7, it means the gradient at the
point with x-coordinate 1 is 7.
Increasing
& Decreasing Functions
1.
When f'(x) > 0, the implication is that y increases as x increases.
2.
Conversely, when f'(x) < 0, the implication is that y decreases as x
increases.
 Prove
that y = x3 - 6x2 + 15x - 8 is increasing for all values
of x.
Solution:
dy/dx = 3x2 - 12x + 15
= 3 (x2 - 4x + 5)
= 3 [(x - 2)2 + 1]
= 3 (x - 2)2 + 3
Since (x - 2)2 > 0 and 3 > 0, dy/dx > 0
Thus, the function is increasing for all x.
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