Tangents and Normals
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Find the tangents and the normals at any point of the following curves:
1) y=1/x^2+2 (1,1/2)
2) y=6/(x^2+1)^2 (1,3/2)
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Solution Summary
Finding the tangent and normal at a point to a curve defined by a Cartesian equation is one of the basic problems in elementary calculus. The derivative of y with respect to x gives the slope of the tangent and this can be used to find the equations of the tangent and the normal. This method is explained here using two examples. First, a brief account of the theory involved is given, then the problems are solved, showing each step accompanied by explanations. Links are also included for further reference, at each point in the solution.
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Theory:
Given the slope of a straight line, m, and any point it passes through, (x1, y1), the equation of the line is given by the point-slope formula:
(y - y1) = m(x- x1)
[
Refer: http://en.wikipedia.org/wiki/Linear_equation#Point.E2.80.93slope_form
http://www.purplemath.com/modules/strtlneq2.htm
]
The normal to a curve at a point is the line through the point that is perpendicular (or orthogonal) to the tangent to the curve at the same point. When two lines are perpendicular to each other, the product of their slopes is -1. That is, if the tangent has slope m, and the normal has slope n, then:
mn = -1
Or, given the slope of the tangent, m, the slope of the normal (at the same point) is:
n = -1/m
[ Refer: http://www.tpub.com/math2/28.htm ]
Using the above, we can solve the given problems. First, we find the slopes of the tangents, and use ...
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