{"id":2463,"date":"2025-06-06T06:54:26","date_gmt":"2025-06-06T06:54:26","guid":{"rendered":"https:\/\/diznr.com\/?p=2463"},"modified":"2025-06-06T06:54:26","modified_gmt":"2025-06-06T06:54:26","slug":"advance-engineering-maths-function-of-variables-real","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/advance-engineering-maths-function-of-variables-real\/","title":{"rendered":"Advance Engineering Maths: Function of real Variables"},"content":{"rendered":"<p>Advance Engineering Maths: Function of real Variables<\/p>\n<p>[fvplayer id=&#8221;42&#8243;]<\/p>\n<h3 data-start=\"0\" data-end=\"72\"><strong data-start=\"4\" data-end=\"70\">Functions of Real Variables \u2013 Advanced Engineering Mathematics<\/strong><\/h3>\n<p data-start=\"74\" data-end=\"275\">In <strong data-start=\"77\" data-end=\"113\">Advanced Engineering Mathematics<\/strong>, the study of <strong data-start=\"128\" data-end=\"159\">functions of real variables<\/strong> is fundamental. It deals with real-valued functions, their properties, limits, continuity, and differentiability.<\/p>\n<h3 data-start=\"282\" data-end=\"337\"><strong data-start=\"285\" data-end=\"335\">1. Definition of a Function of a Real Variable<\/strong><\/h3>\n<p data-start=\"338\" data-end=\"457\">A <strong data-start=\"340\" data-end=\"357\">real function<\/strong> is a rule that assigns a unique real number <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> to each real number <span class=\"katex\"><span class=\"katex-mathml\">xx<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span> in its domain.<\/p>\n<p data-start=\"459\" data-end=\"490\"><strong data-start=\"462\" data-end=\"488\">Mathematical Notation:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">f:D\u2192R,x\u21a6f(x)f: D \\to \\mathbb{R}, \\quad x \\mapsto f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mrel\">:<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">D<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathbb\">R<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u21a6<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"539\" data-end=\"616\">where <strong data-start=\"545\" data-end=\"556\"><span class=\"katex\"><span class=\"katex-mathml\">DD<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">D<\/span><\/span><\/span><\/span><\/strong> is the domain of <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>, and <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is the range.<\/p>\n<p data-start=\"618\" data-end=\"703\"><strong data-start=\"621\" data-end=\"633\">Example:<\/strong><br data-start=\"633\" data-end=\"636\" \/>If <span class=\"katex\"><span class=\"katex-mathml\">f(x)=x2f(x) = x^2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, then for <span class=\"katex\"><span class=\"katex-mathml\">x=2x = 2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span>, we get <span class=\"katex\"><span class=\"katex-mathml\">f(2)=4f(2) = 4<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><\/span><\/span><\/span>.<\/p>\n<h3 data-start=\"710\" data-end=\"740\"><strong data-start=\"713\" data-end=\"738\">2. Types of Functions<\/strong><\/h3>\n<p data-start=\"741\" data-end=\"795\">Functions of a real variable can be classified into:<\/p>\n<p data-start=\"797\" data-end=\"1089\"><strong data-start=\"799\" data-end=\"822\">Algebraic Functions<\/strong> \u2013 Polynomial, rational, root functions<br data-start=\"861\" data-end=\"864\" \/><strong data-start=\"866\" data-end=\"893\">Trigonometric Functions<\/strong> \u2013 <span class=\"katex\"><span class=\"katex-mathml\">sin\u2061x,cos\u2061x,tan\u2061x\\sin x, \\cos x, \\tan x<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">sin<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mop\">cos<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mop\">tan<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span>, etc.<br data-start=\"930\" data-end=\"933\" \/><strong data-start=\"935\" data-end=\"976\">Exponential and Logarithmic Functions<\/strong> \u2013 <span class=\"katex\"><span class=\"katex-mathml\">ex,ln\u2061xe^x, \\ln x<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mop\">ln<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><br data-start=\"995\" data-end=\"998\" \/><strong data-start=\"1000\" data-end=\"1023\">Piecewise Functions<\/strong> \u2013 Defined in different intervals<br data-start=\"1056\" data-end=\"1059\" \/><strong data-start=\"1061\" data-end=\"1085\">Even &amp; Odd Functions<\/strong> \u2013<\/p>\n<ul data-start=\"1093\" data-end=\"1189\">\n<li data-start=\"1093\" data-end=\"1139\">Even: <span class=\"katex\"><span class=\"katex-mathml\">f(\u2212x)=f(x)f(-x) = f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> (e.g., <span class=\"katex\"><span class=\"katex-mathml\">x2x^2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/li>\n<li data-start=\"1143\" data-end=\"1189\">Odd: <span class=\"katex\"><span class=\"katex-mathml\">f(\u2212x)=\u2212f(x)f(-x) = -f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> (e.g., <span class=\"katex\"><span class=\"katex-mathml\">x3x^3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/li>\n<\/ul>\n<h3 data-start=\"1196\" data-end=\"1241\"><strong data-start=\"1199\" data-end=\"1239\">3. Important Properties of Functions<\/strong><\/h3>\n<h3 data-start=\"1242\" data-end=\"1272\"><strong data-start=\"1246\" data-end=\"1270\">(i) Domain and Range<\/strong><\/h3>\n<ul data-start=\"1273\" data-end=\"1396\">\n<li data-start=\"1273\" data-end=\"1339\"><strong data-start=\"1275\" data-end=\"1285\">Domain<\/strong>: Set of all values for which <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is defined.<\/li>\n<li data-start=\"1340\" data-end=\"1396\"><strong data-start=\"1342\" data-end=\"1351\">Range<\/strong>: Set of all possible values of <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<p data-start=\"1398\" data-end=\"1444\"><strong data-start=\"1401\" data-end=\"1413\">Example:<\/strong><br data-start=\"1413\" data-end=\"1416\" \/>For <span class=\"katex\"><span class=\"katex-mathml\">f(x)=xf(x) = \\sqrt{x}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/p>\n<ul data-start=\"1445\" data-end=\"1560\">\n<li data-start=\"1445\" data-end=\"1529\"><strong data-start=\"1447\" data-end=\"1457\">Domain<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">x\u22650x \\geq 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2265<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span> (since square root of a negative number is not real).<\/li>\n<li data-start=\"1530\" data-end=\"1560\"><strong data-start=\"1532\" data-end=\"1541\">Range<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">y\u22650y \\geq 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">y<\/span><span class=\"mrel\">\u2265<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span>.<\/li>\n<\/ul>\n<h3 data-start=\"1567\" data-end=\"1601\"><strong data-start=\"1571\" data-end=\"1599\">(ii) Limit of a Function<\/strong><\/h3>\n<p data-start=\"1602\" data-end=\"1663\">The <strong data-start=\"1606\" data-end=\"1615\">limit<\/strong> of <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> as <span class=\"katex\"><span class=\"katex-mathml\">x\u2192ax \\to a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> is written as:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192af(x)=L\\lim_{x \\to a} f(x) = L<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><span class=\"mop\">lim<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">L<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1696\" data-end=\"1713\"><strong data-start=\"1699\" data-end=\"1711\">Example:<\/strong><\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u21922(3x+5)=3(2)+5=11\\lim_{x \\to 2} (3x + 5) = 3(2) + 5 = 11<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span>2<\/span><\/span><span class=\"mop\">lim<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">5<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">11<\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 data-start=\"1768\" data-end=\"1794\"><strong data-start=\"1772\" data-end=\"1792\">(iii) Continuity<\/strong><\/h3>\n<p data-start=\"1795\" data-end=\"1969\">A function is <strong data-start=\"1809\" data-end=\"1838\">continuous at <span class=\"katex\"><span class=\"katex-mathml\">x=ax = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><\/strong> if:<br data-start=\"1842\" data-end=\"1845\" \/><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192a\u2212f(x)=lim\u2061x\u2192a+f(x)\\lim_{x \\to a^-} f(x) = \\lim_{x \\to a^+} f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"vlist-t\"><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mbin mtight\">\u2212<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"vlist-t\"><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mbin mtight\">+<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><br data-start=\"1900\" data-end=\"1903\" \/><span class=\"katex\"><span class=\"katex-mathml\">f(a)f(a)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is defined<br data-start=\"1928\" data-end=\"1931\" \/><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192af(x)=f(a)\\lim_{x \\to a} f(x) = f(a)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"1971\" data-end=\"2122\"><strong data-start=\"1974\" data-end=\"1986\">Example:<\/strong><br data-start=\"1986\" data-end=\"1989\" \/><span class=\"katex\"><span class=\"katex-mathml\">f(x)=x2f(x) = x^2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is continuous everywhere, but <span class=\"katex\"><span class=\"katex-mathml\">f(x)=1xf(x) = \\frac{1}{x}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is not continuous at <span class=\"katex\"><span class=\"katex-mathml\">x=0x = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span> (because it is undefined).<\/p>\n<h3 data-start=\"2129\" data-end=\"2161\"><strong data-start=\"2133\" data-end=\"2159\">(iv) Differentiability<\/strong><\/h3>\n<p data-start=\"2162\" data-end=\"2248\">A function <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is <strong data-start=\"2187\" data-end=\"2205\">differentiable<\/strong> at <span class=\"katex\"><span class=\"katex-mathml\">x=ax = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> if the derivative exists:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(x)=lim\u2061h\u21920f(x+h)\u2212f(x)hf'(x) = \\lim_{h \\to 0} \\frac{f(x+h) &#8211; f(x)}{h}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">h<\/span><span class=\"mrel mtight\">\u2192<\/span>0<\/span><\/span><span class=\"mop\">lim<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"mord mathnormal\">h<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><span class=\"mord mathnormal\">h<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2305\" data-end=\"2346\"><strong data-start=\"2308\" data-end=\"2320\">Example:<\/strong><br data-start=\"2320\" data-end=\"2323\" \/>For <span class=\"katex\"><span class=\"katex-mathml\">f(x)=x2f(x) = x^2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(x)=2xf'(x) = 2x<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2366\" data-end=\"2408\"><span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is differentiable everywhere.<\/p>\n<h3 data-start=\"2415\" data-end=\"2453\"><strong data-start=\"2419\" data-end=\"2451\">(v) Mean Value Theorem (MVT)<\/strong><\/h3>\n<p data-start=\"2454\" data-end=\"2578\">If <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is continuous on <span class=\"katex\"><span class=\"katex-mathml\">[a,b][a, b]<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span> and differentiable on <span class=\"katex\"><span class=\"katex-mathml\">(a,b)(a, b)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mpunct\">,<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>, then there exists a point <span class=\"katex\"><span class=\"katex-mathml\">cc<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span> such that:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(c)=f(b)\u2212f(a)b\u2212af'(c) = \\frac{f(b) &#8211; f(a)}{b &#8211; a}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2212<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p data-start=\"2622\" data-end=\"2676\"><strong data-start=\"2625\" data-end=\"2637\">Example:<\/strong><br data-start=\"2637\" data-end=\"2640\" \/>For <span class=\"katex\"><span class=\"katex-mathml\">f(x)=x2f(x) = x^2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> on <span class=\"katex\"><span class=\"katex-mathml\">[1,3][1,3]<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">[<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">]<\/span><\/span><\/span><\/span>,<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(c)=9\u221213\u22121=4\u21d22c=4\u21d2c=2f'(c) = \\frac{9 &#8211; 1}{3 &#8211; 1} = 4 \\Rightarrow 2c = 4 \\Rightarrow c = 2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">3<span class=\"mbin\">\u2212<\/span>19<span class=\"mbin\">\u2212<\/span>1<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">4<\/span><span class=\"mrel\">\u21d2<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 data-start=\"2760\" data-end=\"2811\"><strong data-start=\"2763\" data-end=\"2809\">4. Applications in Engineering Mathematics<\/strong><\/h3>\n<p data-start=\"2812\" data-end=\"3072\"><strong data-start=\"2814\" data-end=\"2835\">Signal Processing<\/strong> \u2013 Continuous functions in Fourier Analysis<br data-start=\"2878\" data-end=\"2881\" \/><strong data-start=\"2883\" data-end=\"2902\">Control Systems<\/strong> \u2013 Stability of real functions in system response<br data-start=\"2951\" data-end=\"2954\" \/><strong data-start=\"2956\" data-end=\"2974\">Thermodynamics<\/strong> \u2013 Behavior of state functions like entropy<br data-start=\"3017\" data-end=\"3020\" \/><strong data-start=\"3022\" data-end=\"3041\">Fluid Mechanics<\/strong> \u2013 Velocity profile functions<\/p>\n<h3 data-start=\"3079\" data-end=\"3120\"><strong data-start=\"3082\" data-end=\"3118\">5. Summary Table of Key Formulas<\/strong><\/h3>\n<div class=\"overflow-x-auto contain-inline-size\">\n<table data-start=\"3121\" data-end=\"3392\">\n<thead data-start=\"3121\" data-end=\"3142\">\n<tr data-start=\"3121\" data-end=\"3142\">\n<th data-start=\"3121\" data-end=\"3131\">Concept<\/th>\n<th data-start=\"3131\" data-end=\"3142\">Formula<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"3165\" data-end=\"3392\">\n<tr data-start=\"3165\" data-end=\"3210\">\n<td><strong data-start=\"3167\" data-end=\"3176\">Limit<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192af(x)=L\\lim_{x \\to a} f(x) = L<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">L<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"3211\" data-end=\"3264\">\n<td><strong data-start=\"3213\" data-end=\"3227\">Continuity<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192af(x)=f(a)\\lim_{x \\to a} f(x) = f(a)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"3265\" data-end=\"3338\">\n<td><strong data-start=\"3267\" data-end=\"3281\">Derivative<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(x)=lim\u2061h\u21920f(x+h)\u2212f(x)hf'(x) = \\lim_{h \\to 0} \\frac{f(x+h) &#8211; f(x)}{h}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">h<\/span><span class=\"mrel mtight\">\u2192<\/span>0<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">h<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">f<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathnormal mtight\">h<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">f<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"3339\" data-end=\"3392\">\n<td><strong data-start=\"3341\" data-end=\"3348\">MVT<\/strong><\/td>\n<td><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(c)=f(b)\u2212f(a)b\u2212af'(c) = \\frac{f(b) &#8211; f(a)}{b &#8211; a}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">b<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">f<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">b<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">f<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3 data-start=\"3399\" data-end=\"3418\"><strong data-start=\"3402\" data-end=\"3416\">Conclusion<\/strong><\/h3>\n<p data-start=\"3419\" data-end=\"3639\">Functions of real variables form the <strong data-start=\"3456\" data-end=\"3470\">foundation<\/strong> of Advanced Engineering Mathematics, and their concepts are widely applied in engineering fields. Understanding their properties helps in solving real-world problems.<\/p>\n<p data-start=\"3641\" data-end=\"3720\" data-is-last-node=\"\" data-is-only-node=\"\">Would you like <strong data-start=\"3656\" data-end=\"3677\">practice problems<\/strong> or <strong data-start=\"3681\" data-end=\"3700\">detailed proofs<\/strong> for any theorem?<\/p>\n<h3 data-start=\"3641\" data-end=\"3720\"><a href=\"https:\/\/wp.kntu.ac.ir\/dfard\/ebook\/em\/Advanced%20Engineering%20Mathematics%2010th%20Edition.pdf\" target=\"_blank\" rel=\"noopener\">Advance Engineering Maths: Function of real Variables<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/elasticbeanstalk-us-east-2-344375731421.s3.us-east-2.amazonaws.com\/StudyChat\/Dennis-G.-Zill-Advanced-Engineering-Mathematics-2016-Jones-Bartlett.pdf\" target=\"_blank\" rel=\"noopener\">Dennis-G.-Zill-Advanced-Engineering-Mathematics- &#8230;<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/static2.wikia.nocookie.net\/math\/ro\/images\/1\/16\/Advanced-Engineering-Mathematics.pdf\" target=\"_blank\" rel=\"noopener\">Advanced Engineering Mathematics<\/a><\/h3>\n<p data-start=\"0\" data-end=\"205\">Sure! Here&#8217;s a detailed and easy-to-understand explanation of <strong data-start=\"62\" data-end=\"95\">&#8220;Functions of Real Variables&#8221;<\/strong> from <strong data-start=\"101\" data-end=\"137\">Advanced Engineering Mathematics<\/strong>, suitable for GATE, university exams, and engineering applications.<\/p>\n<hr data-start=\"207\" data-end=\"210\" \/>\n<h2 data-start=\"212\" data-end=\"261\">\ud83d\udcd8 <strong data-start=\"218\" data-end=\"261\">Functions of Real Variables \u2013 Explained<\/strong><\/h2>\n<h3 data-start=\"263\" data-end=\"308\">\ud83d\udd39 What is a Function of a Real Variable?<\/h3>\n<p data-start=\"310\" data-end=\"432\">A <strong data-start=\"312\" data-end=\"343\">function of a real variable<\/strong> is a rule that assigns a <strong data-start=\"369\" data-end=\"391\">real number output<\/strong> to each real number input in its domain.<\/p>\n<h3 data-start=\"434\" data-end=\"451\">\u2705 Definition:<\/h3>\n<blockquote data-start=\"452\" data-end=\"591\">\n<p data-start=\"454\" data-end=\"591\">A function <span class=\"katex\"><span class=\"katex-mathml\">f:R\u2192Rf: \\mathbb{R} \\rightarrow \\mathbb{R}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mrel\">:<\/span><\/span><span class=\"base\"><span class=\"mord mathbb\">R<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathbb\">R<\/span><\/span><\/span><\/span> is a <strong data-start=\"513\" data-end=\"544\">function of a real variable<\/strong> if both the input and output are real numbers.<\/p>\n<\/blockquote>\n<hr data-start=\"593\" data-end=\"596\" \/>\n<h2 data-start=\"598\" data-end=\"617\">\ud83d\udcd0 <strong data-start=\"604\" data-end=\"617\">Examples:<\/strong><\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"619\" data-end=\"940\">\n<thead data-start=\"619\" data-end=\"657\">\n<tr data-start=\"619\" data-end=\"657\">\n<th data-start=\"619\" data-end=\"630\" data-col-size=\"sm\">Function<\/th>\n<th data-start=\"630\" data-end=\"643\" data-col-size=\"sm\">Expression<\/th>\n<th data-start=\"643\" data-end=\"657\" data-col-size=\"sm\">Graph Type<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"697\" data-end=\"940\">\n<tr data-start=\"697\" data-end=\"745\">\n<td data-start=\"697\" data-end=\"706\" data-col-size=\"sm\">Linear<\/td>\n<td data-col-size=\"sm\" data-start=\"706\" data-end=\"728\"><span class=\"katex\"><span class=\"katex-mathml\">f(x)=2x+3f(x) = 2x + 3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mbin\">+<\/span><\/span><span class=\"base\"><span class=\"mord\">3<\/span><\/span><\/span><\/span><\/td>\n<td data-col-size=\"sm\" data-start=\"728\" data-end=\"745\">Straight line<\/td>\n<\/tr>\n<tr data-start=\"746\" data-end=\"789\">\n<td data-start=\"746\" data-end=\"758\" data-col-size=\"sm\">Quadratic<\/td>\n<td data-start=\"758\" data-end=\"777\" data-col-size=\"sm\"><span class=\"katex\"><span class=\"katex-mathml\">f(x)=x2f(x) = x^2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td data-start=\"777\" data-end=\"789\" data-col-size=\"sm\">Parabola<\/td>\n<\/tr>\n<tr data-start=\"790\" data-end=\"836\">\n<td data-start=\"790\" data-end=\"806\" data-col-size=\"sm\">Trigonometric<\/td>\n<td data-start=\"806\" data-end=\"828\" data-col-size=\"sm\"><span class=\"katex\"><span class=\"katex-mathml\">f(x)=sin\u2061xf(x) = \\sin x<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">sin<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/td>\n<td data-start=\"828\" data-end=\"836\" data-col-size=\"sm\">Wave<\/td>\n<\/tr>\n<tr data-start=\"837\" data-end=\"888\">\n<td data-start=\"837\" data-end=\"851\" data-col-size=\"sm\">Exponential<\/td>\n<td data-start=\"851\" data-end=\"870\" data-col-size=\"sm\"><span class=\"katex\"><span class=\"katex-mathml\">f(x)=exf(x) = e^x<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">e<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<td data-start=\"870\" data-end=\"888\" data-col-size=\"sm\">Rapid increase<\/td>\n<\/tr>\n<tr data-start=\"889\" data-end=\"940\">\n<td data-start=\"889\" data-end=\"903\" data-col-size=\"sm\">Logarithmic<\/td>\n<td data-col-size=\"sm\" data-start=\"903\" data-end=\"925\"><span class=\"katex\"><span class=\"katex-mathml\">f(x)=log\u2061xf(x) = \\log x<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">log<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/td>\n<td data-col-size=\"sm\" data-start=\"925\" data-end=\"940\">Slow growth<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"942\" data-end=\"945\" \/>\n<h2 data-start=\"947\" data-end=\"975\">\ud83e\udde0 <strong data-start=\"953\" data-end=\"975\">Important Concepts<\/strong><\/h2>\n<h3 data-start=\"977\" data-end=\"1002\">1. <strong data-start=\"984\" data-end=\"1002\">Domain &amp; Range<\/strong><\/h3>\n<ul data-start=\"1003\" data-end=\"1127\">\n<li data-start=\"1003\" data-end=\"1068\">\n<p data-start=\"1005\" data-end=\"1068\"><strong data-start=\"1005\" data-end=\"1015\">Domain<\/strong>: Set of all <span class=\"katex\"><span class=\"katex-mathml\">xx<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span> for which <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is defined.<\/p>\n<\/li>\n<li data-start=\"1069\" data-end=\"1127\">\n<p data-start=\"1071\" data-end=\"1127\"><strong data-start=\"1071\" data-end=\"1080\">Range<\/strong>: Set of all possible output values <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1129\" data-end=\"1150\">2. <strong data-start=\"1136\" data-end=\"1150\">Continuity<\/strong><\/h3>\n<p data-start=\"1151\" data-end=\"1215\">A function <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> is <strong data-start=\"1176\" data-end=\"1190\">continuous<\/strong> at point <span class=\"katex\"><span class=\"katex-mathml\">x=ax = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span> if:<\/p>\n<p><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192af(x)=f(a)\\lim_{x \\to a} f(x) = f(a)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><span class=\"mop\">lim<\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<h3 data-start=\"1250\" data-end=\"1278\">3. <strong data-start=\"1257\" data-end=\"1278\">Differentiability<\/strong><\/h3>\n<p data-start=\"1279\" data-end=\"1371\">If the derivative <span class=\"katex\"><span class=\"katex-mathml\">f\u2032(x)f'(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> exists, then the function is <strong data-start=\"1338\" data-end=\"1356\">differentiable<\/strong> at that point.<\/p>\n<h3 data-start=\"1373\" data-end=\"1396\">4. <strong data-start=\"1380\" data-end=\"1396\">Monotonicity<\/strong><\/h3>\n<ul data-start=\"1397\" data-end=\"1506\">\n<li data-start=\"1397\" data-end=\"1451\">\n<p data-start=\"1399\" data-end=\"1451\">Increasing: <span class=\"katex\"><span class=\"katex-mathml\">f(x1)&lt;f(x2)f(x_1) &lt; f(x_2)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> if <span class=\"katex\"><span class=\"katex-mathml\">x1&lt;x2x_1 &lt; x_2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li data-start=\"1452\" data-end=\"1506\">\n<p data-start=\"1454\" data-end=\"1506\">Decreasing: <span class=\"katex\"><span class=\"katex-mathml\">f(x1)&gt;f(x2)f(x_1) &gt; f(x_2)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">&gt;<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> if <span class=\"katex\"><span class=\"katex-mathml\">x1&lt;x2x_1 &lt; x_2<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">&lt;<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<h3 data-start=\"1508\" data-end=\"1541\">5. <strong data-start=\"1515\" data-end=\"1541\">Even and Odd Functions<\/strong><\/h3>\n<ul data-start=\"1542\" data-end=\"1655\">\n<li data-start=\"1542\" data-end=\"1599\">\n<p data-start=\"1544\" data-end=\"1599\"><strong data-start=\"1544\" data-end=\"1552\">Even<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">f(\u2212x)=f(x)f(-x) = f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> \u21d2 symmetric about Y-axis<\/p>\n<\/li>\n<li data-start=\"1600\" data-end=\"1655\">\n<p data-start=\"1602\" data-end=\"1655\"><strong data-start=\"1602\" data-end=\"1609\">Odd<\/strong>: <span class=\"katex\"><span class=\"katex-mathml\">f(\u2212x)=\u2212f(x)f(-x) = -f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> \u21d2 symmetric about origin<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"1657\" data-end=\"1660\" \/>\n<h2 data-start=\"1662\" data-end=\"1710\">\u270f\ufe0f <strong data-start=\"1668\" data-end=\"1710\">Typical Problems in Engineering Maths:<\/strong><\/h2>\n<ol data-start=\"1711\" data-end=\"1940\">\n<li data-start=\"1711\" data-end=\"1763\">\n<p data-start=\"1714\" data-end=\"1763\">Determine the continuity of a piecewise function.<\/p>\n<\/li>\n<li data-start=\"1764\" data-end=\"1799\">\n<p data-start=\"1767\" data-end=\"1799\">Check if a function is even\/odd.<\/p>\n<\/li>\n<li data-start=\"1800\" data-end=\"1849\">\n<p data-start=\"1803\" data-end=\"1849\">Evaluate limit of <span class=\"katex\"><span class=\"katex-mathml\">f(x)f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span> as <span class=\"katex\"><span class=\"katex-mathml\">x\u2192ax \\to a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">x<\/span><span class=\"mrel\">\u2192<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span>.<\/p>\n<\/li>\n<li data-start=\"1850\" data-end=\"1892\">\n<p data-start=\"1853\" data-end=\"1892\">Differentiate and find critical points.<\/p>\n<\/li>\n<li data-start=\"1893\" data-end=\"1940\">\n<p data-start=\"1896\" data-end=\"1940\">Graphing and interpreting function behavior.<\/p>\n<\/li>\n<\/ol>\n<hr data-start=\"1942\" data-end=\"1945\" \/>\n<h2 data-start=\"1947\" data-end=\"1985\">\ud83d\udd27 <strong data-start=\"1953\" data-end=\"1985\">Applications in Engineering:<\/strong><\/h2>\n<ul data-start=\"1986\" data-end=\"2155\">\n<li data-start=\"1986\" data-end=\"2022\">\n<p data-start=\"1988\" data-end=\"2022\">Signal processing (wave functions)<\/p>\n<\/li>\n<li data-start=\"2023\" data-end=\"2058\">\n<p data-start=\"2025\" data-end=\"2058\">Control systems (response curves)<\/p>\n<\/li>\n<li data-start=\"2059\" data-end=\"2108\">\n<p data-start=\"2061\" data-end=\"2108\">Mechanics (velocity, acceleration as functions)<\/p>\n<\/li>\n<li data-start=\"2109\" data-end=\"2155\">\n<p data-start=\"2111\" data-end=\"2155\">Electrical circuits (voltage\/time functions)<\/p>\n<\/li>\n<\/ul>\n<hr data-start=\"2157\" data-end=\"2160\" \/>\n<h2 data-start=\"2162\" data-end=\"2182\">\ud83d\udccc Summary Chart:<\/h2>\n<div class=\"_tableContainer_16hzy_1\">\n<div class=\"_tableWrapper_16hzy_14 group flex w-fit flex-col-reverse\">\n<table class=\"w-fit min-w-(--thread-content-width)\" data-start=\"2184\" data-end=\"2474\">\n<thead data-start=\"2184\" data-end=\"2212\">\n<tr data-start=\"2184\" data-end=\"2212\">\n<th data-start=\"2184\" data-end=\"2194\" data-col-size=\"sm\">Concept<\/th>\n<th data-start=\"2194\" data-end=\"2212\" data-col-size=\"md\">Formula \/ Rule<\/th>\n<\/tr>\n<\/thead>\n<tbody data-start=\"2241\" data-end=\"2474\">\n<tr data-start=\"2241\" data-end=\"2278\">\n<td data-start=\"2241\" data-end=\"2249\" data-col-size=\"sm\">Limit<\/td>\n<td data-start=\"2249\" data-end=\"2278\" data-col-size=\"md\"><span class=\"katex\"><span class=\"katex-mathml\">lim\u2061x\u2192af(x)\\lim_{x \\to a} f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mrel mtight\">\u2192<\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2279\" data-end=\"2348\">\n<td data-start=\"2279\" data-end=\"2292\" data-col-size=\"sm\">Derivative<\/td>\n<td data-start=\"2292\" data-end=\"2348\" data-col-size=\"md\"><span class=\"katex\"><span class=\"katex-mathml\">f\u2032(x)=lim\u2061h\u21920f(x+h)\u2212f(x)hf'(x) = \\lim_{h \\to 0} \\frac{f(x+h) &#8211; f(x)}{h}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mop\">lim<span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">h<\/span><span class=\"mrel mtight\">\u2192<\/span>0<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">h<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">f<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mbin mtight\">+<\/span><span class=\"mord mathnormal mtight\">h<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">f<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">x<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2349\" data-end=\"2378\">\n<td data-start=\"2349\" data-end=\"2356\" data-col-size=\"sm\">Even<\/td>\n<td data-start=\"2356\" data-end=\"2378\" data-col-size=\"md\"><span class=\"katex\"><span class=\"katex-mathml\">f(\u2212x)=f(x)f(-x) = f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2379\" data-end=\"2408\">\n<td data-start=\"2379\" data-end=\"2385\" data-col-size=\"sm\">Odd<\/td>\n<td data-start=\"2385\" data-end=\"2408\" data-col-size=\"md\"><span class=\"katex\"><span class=\"katex-mathml\">f(\u2212x)=\u2212f(x)f(-x) = -f(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr data-start=\"2409\" data-end=\"2474\">\n<td data-start=\"2409\" data-end=\"2422\" data-col-size=\"sm\">Chain Rule<\/td>\n<td data-start=\"2422\" data-end=\"2474\" data-col-size=\"md\"><span class=\"katex\"><span class=\"katex-mathml\">ddxf(g(x))=f\u2032(g(x))\u22c5g\u2032(x)\\frac{d}{dx}f(g(x)) = f'(g(x)) \\cdot g'(x)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><span class=\"mord mathnormal mtight\">x<\/span><\/span><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">d<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">))<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">))<\/span><span class=\"mbin\">\u22c5<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">g<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"sticky end-(--thread-content-margin) h-0 self-end select-none\">\n<div class=\"absolute end-0 flex items-end\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<hr data-start=\"2476\" data-end=\"2479\" \/>\n<h2 data-start=\"2481\" data-end=\"2497\">\ud83d\udce5 Want More?<\/h2>\n<p data-start=\"2498\" data-end=\"2512\">I can provide:<\/p>\n<ul data-start=\"2513\" data-end=\"2667\">\n<li data-start=\"2513\" data-end=\"2559\">\n<p data-start=\"2515\" data-end=\"2559\">\ud83d\udcd8 A PDF of Advanced Engineering Maths notes<\/p>\n<\/li>\n<li data-start=\"2560\" data-end=\"2615\">\n<p data-start=\"2562\" data-end=\"2615\">\ud83d\udcca Graphical illustrations of real-variable functions<\/p>\n<\/li>\n<li data-start=\"2616\" data-end=\"2667\">\n<p data-start=\"2618\" data-end=\"2667\">\ud83d\udcdd Practice questions with step-by-step solutions<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2669\" data-end=\"2780\" data-is-last-node=\"\" data-is-only-node=\"\">Would you like this in <strong data-start=\"2692\" data-end=\"2701\">Hindi<\/strong>, or prefer topic-wise breakdowns (like limits, continuity, differentiability)?<\/p>\n<h3 data-start=\"2669\" data-end=\"2780\"><a href=\"https:\/\/elektrolibraria.wordpress.com\/wp-content\/uploads\/2015\/12\/197-advanced-engineering-mathematics.pdf\" target=\"_blank\" rel=\"noopener\">Advance Engineering Maths: Function of real Variables<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/students.aiu.edu\/submissions\/profiles\/resources\/onlinebook\/j2v3y7_advanced_engineering_mathematics-_3.pdf\" target=\"_blank\" rel=\"noopener\">Advanced Engineering Mathematics<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/people.math.harvard.edu\/~shlomo\/docs\/Real_Variables.pdf\" target=\"_blank\" rel=\"noopener\">Theory of functions of a real variable.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/drspmaths.wordpress.com\/wp-content\/uploads\/2020\/01\/advanced-engineering-mathematics-peter-v.-o-neil.pdf\" target=\"_blank\" rel=\"noopener\">advanced-engineering-mathematics-peter-v.-o-neil.pdf<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Advance Engineering Maths: Function of real Variables.<\/p>\n","protected":false},"author":64,"featured_media":2464,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1495],"tags":[1497,1498],"class_list":["post-2463","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-advance-engineering-mathematics","tag-advance-engineering-maths","tag-function-of-real-variables"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2463","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/64"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2463"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2463\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media\/2464"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2463"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2463"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}