{"id":5581,"date":"2021-09-21T18:34:20","date_gmt":"2021-09-21T13:04:20","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5581"},"modified":"2021-11-15T00:14:07","modified_gmt":"2021-11-14T18:44:07","slug":"differentials-errors-and-approximations","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentials-errors-and-approximations\/","title":{"rendered":"Differentials Errors and Approximations"},"content":{"rendered":"

Here you will learn what is differentials errors and approximations with examples.<\/p>\n

Let’s begin –<\/p>\n

Differentials Errors and Approximations<\/h2>\n

In order to calculate the approximate value of a function, differentials may be used where the differential of a function is equal to its derivative multiplied by the differential of the independent variable,<\/p>\n

\n

In general dy = f'(x)dx or df(x) = f'(x)dx<\/p>\n

Or \\(\\Delta y\\) = \\(dy\\over dx\\) \\(\\Delta x\\)\u00a0<\/p>\n

Because [ f'(x) = \\(dy\\over dx\\) ]<\/p>\n<\/blockquote>\n

Absolute Error\u00a0<\/strong><\/h4>\n
\n

The error \\(\\Delta x\\) in x is called the absolute error in x.<\/p>\n<\/blockquote>\n

Relative Error\u00a0<\/strong><\/h4>\n
\n

If \\(\\Delta x\\) is an error in x, then \\(\\Delta x\\over x\\) is called the relative error in x<\/p>\n<\/blockquote>\n

Percentage Error\u00a0<\/strong><\/h4>\n
\n

If \\(\\Delta x\\) is an error in x, then \\({\\Delta x\\over x} \\times 100\\) is called the percentage error in x.<\/p>\n<\/blockquote>\n

Remark<\/strong> : Let y = f(x) be a function of x, and let \\(\\Delta x\\) be a small change in x. Let the corresponding change in y be \\(\\Delta y\\). Then,<\/p>\n

y + \\(\\Delta y\\) = \\(f(x + \\Delta x)\\)<\/p>\n

But, \\(\\Delta y\\) = \\(dy\\over dx\\) \\(\\Delta x\\) = f'(x) \\(\\Delta x\\) , approximately<\/p>\n

\\(\\therefore\\) \\(f(x + \\Delta x)\\) = y + \\(\\Delta y\\)<\/p>\n

\\(\\implies\\) \\(f(x + \\Delta x)\\) = y + f'(x) \\(\\Delta x\\) , approximately<\/p>\n

\\(\\implies\\) \\(f(x + \\Delta x)\\) = y + \\(dy\\over dx\\) \\(\\Delta x\\) , approximately<\/p>\n

Let x be the independent variable and y be the dependent variable connected by the relation y = f(x). We use the following algorithm to find an approximate change \\(\\Delta y\\) in y due to small change \\(\\Delta x\\) in x.<\/p>\n

Algorithm :<\/strong><\/p>\n

\n

1). Choose the initial value of the independent variable as x and the changed value as x + \\(\\Delta x\\).<\/p>\n

2). find \\(\\Delta x\\) and assume that dx = \\(\\Delta x\\).<\/p>\n

3). find \\(dy\\over dx\\) from the given relation y = f(x).<\/p>\n

4). find the value of \\(dy\\over dx\\) at (x, y).<\/p>\n

5). find dy by using the relation dy = \\(dy\\over dx\\)dx.<\/p>\n

6). Put \\(\\Delta y\\) = dy to obtain an approximate change in y.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : If y = \\(x^4\\) – 10\u00a0 and x changes from 2 to 1.99, what is the approximate change in y ? Also, find the changed value of y.<\/p>\n

Solution<\/strong><\/span> : Let x\u00a0 = 2, x + \\(\\Delta x\\) = 1.99. Then, \\(\\Delta x\\) = 1.99 – 2 = -0.01<\/p>\n

Let dx = \\(\\Delta x\\)\u00a0 = -0.01<\/p>\n

We have,\u00a0<\/p>\n

y = \\(x^4\\) – 10<\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = \\(4x^3\\) \\(\\implies\\) \\(({dy\\over dx})_{x=2}\\) = \\(4(2)^3\\) = 32<\/p>\n

\\(\\therefore\\) dy = \\(dy\\over dx\\) dx<\/p>\n

\\(\\implies\\) dy = 32(-0.01) = -0.32<\/p>\n

\\(\\implies\\) \\(\\Delta y\\) = -0.32 approximately<\/p>\n

So, approximate change in y = -0.32<\/p>\n

When, x = 2, we have<\/p>\n

y = \\(2^4\\) – 10 = 6<\/p>\n

So, changed value of y = y + \\(\\Delta y\\) = 6 + (-0.32) = 5.68<\/p>\n

\n

Related Questions<\/h3>\n

If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximating error in calculating its volume.<\/a><\/p>\n

Find the approximate value of f(3.02), where f(x) = \\(3x^2 + 5x + 3\\).<\/a><\/p>\n\n\n

\n
Next – Mean Value Theorems Class 12<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Derivative as a Rate Measure Class 12<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is differentials errors and approximations with examples. Let’s begin – Differentials Errors and Approximations In order to calculate the approximate value of a function, differentials may be used where the differential of a function is equal to its derivative multiplied by the differential of the independent variable, In general dy …<\/p>\n

Differentials Errors and Approximations<\/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":[76,116,114,115],"yoast_head":"\nDifferentials Errors and Approximations - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is differentials errors and approximations 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\/differentials-errors-and-approximations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentials Errors and Approximations - 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