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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.

(i) If the tangent to a curve is horizontal at (x1,y1)(x_1,y_1) (the derivative is 00 there), the tangent is y=y1y=y_1 and the normal is x=x1x=x_1.

…

Figure 7.7A curve y = f(x) with the tangent line (which just touches the curve) and the normal line (perpendicular to the tangent) drawn at a point on the curve.
Fig. 7.7 — A curve y = f(x) with the tangent line (which just touches the curve) and the normal line (perpendicular to the tangent) drawn at a point on the curve.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A curve y = f(x) with the tangent line (which just touches the curve) and the normal line (perpendicular to the tangent) drawn at a point …

Figure 7.8The parabola y = x^2 + 3x - 2 with the tangent line 5x - y - 3 = 0 and the normal line x + 5y - 11 = 0 at the point (1, 2).
Fig. 7.8 — The parabola y = x^2 + 3x - 2 with the tangent line 5x - y - 3 = 0 and the normal line x + 5y - 11 = 0 at the point (1, 2).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. The parabola y = x^2 + 3x - 2 with the tangent line 5x - y - 3 = 0 and the normal line x + 5y - 11 = 0 at the poi …

Figure 7.9The Lissajous curve given parametrically by x = 2 cos 3t, y = 3 sin 2t, which is neither a circle nor an ellipse.
Fig. 7.9 — The Lissajous curve given parametrically by x = 2 cos 3t, y = 3 sin 2t, which is neither a circle nor an ellipse.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. The Lissajous curve given parametrically by x = 2 cos 3t, y = 3 sin 2t, which is neither a circle nor an …

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