{"id":5591,"date":"2021-09-22T15:16:48","date_gmt":"2021-09-22T09:46:48","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5591"},"modified":"2021-11-13T23:39:03","modified_gmt":"2021-11-13T18:09:03","slug":"slopes-of-tangent-and-normal-to-the-curve","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/slopes-of-tangent-and-normal-to-the-curve\/","title":{"rendered":"Slopes of Tangent and Normal to the Curve"},"content":{"rendered":"

Here you will learn slopes of tangent and normal to the curve with examples.<\/p>\n

Let’s begin –<\/p>\n

Slopes of Tangent and Normal to the Curve<\/h2>\n

(a) Slopes of Tangent<\/h3>\n

Let y = f(x) be a continuous curve, and let \\(P(x_1, y_1)\\) be a point on it. Then,\u00a0<\/p>\n

\n

\\(({dy\\over dx})_P\\) is the tangent to the curve y = f(x) at point P.<\/p>\n

i.e. \\(({dy\\over dx})_P\\) = tan \\(\\psi\\) = Slope of the tangent at P,<\/p>\n<\/blockquote>\n

where \\(\\psi\\) is the angle which the tangent at \\(P(x_1, y_1)\\) makes with the positive direction of x-axis.<\/p>\n

If the tangent at P is parallel to x-axis<\/strong>, then<\/p>\n

\\(\\psi\\) = 0 \\(\\implies\\) tan \\(\\psi\\) = 0 \\(\\implies\\) Slope = 0 \\(\\implies\\) \\(({dy\\over dx})_P\\) = 0<\/p>\n

If the tangent at P is perpendicular to x-axis, or parallel to y-axis<\/strong>, then<\/p>\n

\\(\\psi\\) = \\(\\pi\\over 2\\) \\(\\implies\\) cot \\(\\psi\\) = 0 \\(\\implies\\) \\(1\\over tan \\psi\\) = 0 \\(\\implies\\) \\(({dx\\over dy})_P\\) = 0<\/p>\n

(b) Slopes of Normal<\/h3>\n

The normal to the curve at \\(P(x_1, y_1)\\) is a line perpendicular to the tangent at P and passing through P.<\/p>\n

\n

\\(\\therefore\\)\u00a0 Slope of the normal at P = \\(-1\\over Slope of the tangent at P\\) = \\(-({dx\\over dy})_P\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : find the slopes of the tangent and the normal to the curve \\(x^2 + 3y + y^2\\) = 5 at (1, 1).<\/p>\n

Solution<\/strong><\/span> : The equation of the curve is \\(x^2 + 3y + y^2\\) = 5<\/p>\n

Differentiating with respect to x, we get<\/p>\n

2x + 3\\(dy\\over dx\\) + 2y\\(dy\\over dx\\) = 0<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(-2x\\over 2y + 3\\)<\/p>\n

\\(\\implies\\) \\(({dy\\over dx})_{(1, 1)}\\) = -(\\(2\\over 2 + 3\\)) = -\\(2\\over 5\\)<\/p>\n

\\(\\therefore\\)\u00a0 Slope of the tangent at (1, 1) = -\\(2\\over 5\\)<\/p>\n

and, Slope of normal at (1, 1) = \\(-1\\over slope of tangent at (1, 1)\\) = \\(5\\over 2\\)<\/p>\n

\n

Related Questions<\/h3>\n

Show that the tangents to the curve y = \\(2x^3 \u2013 3\\) at the points where x =2 and x = -2 are parallel.<\/a><\/p>\n

Find the slope of normal to the curve x = 1 \u2013 \\(a sin\\theta\\), y = \\(b cos^2\\theta\\) at \\(\\theta\\) = \\(\\pi\\over 2\\).<\/a><\/p>\n

Find the slope of the normal to the curve x = \\(a cos^3\\theta\\), y = \\(a sin^3\\theta\\) at \\(\\theta\\) = \\(\\pi\\over 4\\).<\/a><\/p>\n\n\n

\n
Next – Equation of Tangent and Normal to the Curve<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Lagrange\u2019s Mean Value Theorem<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn slopes of tangent and normal to the curve with examples. Let’s begin – Slopes of Tangent and Normal to the Curve (a) Slopes of Tangent Let y = f(x) be a continuous curve, and let \\(P(x_1, y_1)\\) be a point on it. Then,\u00a0 \\(({dy\\over dx})_P\\) is the tangent to the curve …<\/p>\n

Slopes of Tangent and Normal to the Curve<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[37],"tags":[76,103,101,102,99,100],"yoast_head":"\nSlopes of Tangent and Normal to the Curve - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn slopes of tangent and normal to the curve with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/slopes-of-tangent-and-normal-to-the-curve\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Slopes of Tangent and Normal to the Curve - 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