{"id":19833,"date":"2022-06-10T01:04:34","date_gmt":"2022-06-10T01:04:34","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=4160"},"modified":"2022-06-10T01:04:34","modified_gmt":"2022-06-10T01:04:34","slug":"differential-calculus","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/06\/10\/differential-calculus\/","title":{"rendered":"Differential Calculus"},"content":{"rendered":"\n<p>Differential calculus focuses on solving the problem of finding the rate of change of a function with respect to the other variables. To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Differential helps in the study of the limit of a quotient, dealing with variables such as x and y, functions f(x), and the corresponding changes in the variables x and y. The notations dy and dx are known as differentials. The process used to find the derivatives is called differentiation. The derivative of a function, y with respect to variable x, is represented by dy\/dx or f\u2019(x).<\/p>\n\n\n\n<p><strong>Limits<\/strong><\/p>\n\n\n\n<p>Limit helps in calculating the degree of closeness to any value or the approaching term. A limit is normally expressed using the\u00a0limit formula\u00a0as,<\/p>\n\n\n\n<p>lim<sub>x\u2192c<\/sub>f(x) = A<\/p>\n\n\n\n<p>This expression is read as \u201cthe limit of f of x as x approaches c equals A\u201d.<\/p>\n\n\n\n<p><strong>Derivatives<\/strong><\/p>\n\n\n\n<p>Derivatives represent the instantaneous rate of change of a quantity with respect to the other. The derivative of a function is represented as:<\/p>\n\n\n\n<p>lim<sub>x\u2192h<\/sub>[f(x + h) \u2212 f(x)]\/h = A<\/p>\n\n\n\n<p><strong>Continuity<\/strong><\/p>\n\n\n\n<p>A function f(x) is said to be continuous at a particular point x = a, if the following three conditions are satisfied \u2013<\/p>\n\n\n\n<ul><li>f(a) is defined<\/li><li>lim<sub>x\u2192a<\/sub>f(x) exists<\/li><li>lim<sub>x\u2192a<sup>\u2212&nbsp;<\/sup><\/sub>f(x) = lim<sub>x\u2192a<sup>+&nbsp;<\/sup><\/sub>f(x) = f(a)<\/li><\/ul>\n\n\n\n<p><strong>Continuity and Differentiability<\/strong><\/p>\n\n\n\n<p>A function is always continuous if it is differentiable at any point, whereas the vice-versa for this condition is not always true.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Integral Calculus<\/h2>\n\n\n\n<p>Integral calculus is the study of integrals and the properties associated to them. It is helpful in:<\/p>\n\n\n\n<ul><li>calculating f from f\u2019 (i.e. from its derivative). If a function, say f is differentiable in any given interval, then f\u2019 is defined in that interval.<\/li><li>calculating the area under a curve for any function.<\/li><\/ul>\n\n\n\n<p><strong>Integration<\/strong><\/p>\n\n\n\n<p>Integration is the reciprocal of differentiation. As differentiation can be understood as dividing a part into many small parts, integration can be said as a collection of small parts in order to form a whole. It is generally used for calculating areas.<\/p>\n\n\n\n<p><strong>Definite Integral<\/strong><\/p>\n\n\n\n<p>A definite integral has a specific boundary or limit for the calculation of the function. The upper and lower limits of the independent variable of a function are specified. A definite integral is given mathematically as,<\/p>\n\n\n\n<p>\u222b<sub>a<\/sub><sup>b<\/sup>&nbsp;f(x).dx = F(x)<\/p>\n\n\n\n<p><strong>Indefinite Integral<\/strong><\/p>\n\n\n\n<p>An indefinite integral does not have a specific boundary, i.e. no upper and lower limit is defined. Thus the integration value is always accompanied by a constant value (C). It is denoted as:<\/p>\n\n\n\n<p>\u222b f(x).dx = F(x) + C<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Differential calculus focuses on solving the problem of finding the rate of change of a function with respect to the other variables. To find the optimal solution, derivatives are used to calculate the maxima and minima values of a function. Differential helps in the study of the limit of a quotient, dealing with variables such [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[760],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/19833"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=19833"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/19833\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=19833"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=19833"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=19833"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}