Tangents and Normals
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TANGENTS AND NORMALS
WHAT ARE TANGENTS AND NORMALS
A tangent can be simply described as a line that touches a curve or a plane that touches a surface. A normal on the other hand is described as a line drawn at right angle to a surface. The tangent and the normal at the same point on any surface are always at right angles to each other.
Slope of the tangent at a point (x1,y1) on the curve y = f(x) is dy/dx (at x = x1) = f’(x1)
Therefore, the equation of the tangent at a point (x1, y1) on the curve y = f(x) is
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y-y1 = (dy/dx) (x-x1) |
Similarly, the equation of normal at a point (x1, y1) on the curve y = f(x) is
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y- y1 = (-dx/dy) (x-x1) |
NOTE: dy/dx and dx/dy are at x = x1}
THE ANGLE BETWEEN TWO CURVES
In order to find the angle of intersection between two curves, find the point of intersection and the lope of the tangents at that point (m1 and m2). Then calculate the angle by the following formula.
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q = tan-1{(m1-m2)/(1 + m1m2) } |
MONOTONOCITY
MONOTONICALLY INCREASING FUNCTION
A function f(x) is said to be monotonically increasing over a set D if
f(x1) >= f(x2) for all x1 > x2 in the set D.
The necessary and sufficient conditions for f(x) to be monotonically increasing over D is that f’ (x) >= 0 on D.
The function is said to be strictly monotonically increasing if
f(x1) > f(x2) for all x1 > x2
and in such a case, f’ (x) > 0 over D.
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MONOTONICALLY DECREASING FUNCTION
A function f is said to be monotonically decreasing over a set D if
f(x1) < = f(x2) when x1 > x2
The necessary and sufficient condition for f(x) to be monotonically decreasing over D is that
f’ (x) < = 0 on D.
The function is said to be strictly monotonically decreasing on D if
f(x1) < f(x2) for all x1 > x2
and in such a case, f’ (x) < 0 over D.