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Mathematics · Ch 7 — Applications of Differential Calculus

Equations of Tangent and Normal

7.2.4

Equations of Tangent and Normal

Note

"According to Leibniz, tangent is the line through a pair of very close points on the curve."

Definition 7.1 (Tangent). The tangent line to a plane curve at a given point is the straight line that just touches the curve at that point.

Definition 7.2 (Normal). The normal at a point on the curve is the straight line perpendicular to the tangent at that point.

For the curve y=f(x)y=f(x), the equation of the tangent at the point (a,b)(a,b) is

y−b=(x−a)(dydx)(a,b)or equivalentlyy−b=f′(a) (x−a).y-b=(x-a)\left(\frac{dy}{dx}\right)_{(a,b)}\qquad\text{or equivalently}\qquad y-b=f'(a)\,(x-a).

Since the normal is perpendicular to the tangent, its slope is the negative reciprocal −(1dy/dx)(a,b)-\left(\dfrac{1}{dy/dx}\right)_{(a,b)}, so the equation of the normal is

y−b=−(1dy/dx)(a,b)(x−a)or equivalently(y−b)(dydx)(a,b)=−(x−a).y-b=-\left(\frac{1}{dy/dx}\right)_{(a,b)}(x-a)\qquad\text{or equivalently}\qquad (y-b)\left(\frac{dy}{dx}\right)_{(a,b)}=-(x-a).

Note

Special cases.

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