Mathematics (113) \u2013 DAE Mechanical 1st Year \u2013 Pak Notes Hub<\/title>\r\n <link href=\"https:\/\/fonts.googleapis.com\/css2?family=Inter:wght@400;500;600;700&family=Fira+Code:wght@400;500&display=swap\" rel=\"stylesheet\"\/><style>*,*::before,*::after{box-sizing:border-box;margin:0;padding:0;}:root{--green:#1a7a4a;--green-dark:#145f38;--green-light:#e8f5ee;--teal:#17a589;--accent:#f0b127;--dark:#1c2833;--mid:#566573;--light:#f4f6f7;--code-bg:#f0f4f8;--code-border:#2e86c1;--note-bg:#fef9e7;--note-border:#f0b127;--white:#ffffff;--radius:10px;--shadow:0 4px 24px rgba(26,122,74,.10);}html{scroll-behavior:smooth;}body{font-family:'Inter',sans-serif;background:#f2f6f3;color:var(--dark);line-height:1.7;font-size:15px;}nav{position:sticky;top:0;z-index:100;background:var(--green-dark);padding:0 2rem;display:flex;align-items:center;justify-content:space-between;height:56px;box-shadow:0 2px 12px rgba(0,0,0,.25);}.nav-brand{color:#fff;font-weight:700;font-size:1.05rem;}.nav-brand 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p{color:#d5f5e3;font-size:1rem;}.progress-bar{position:fixed;top:0;left:0;height:3px;background:var(--accent);z-index:200;transition:width .1s linear;width:0%;}#back-top{position:fixed;bottom:2rem;right:2rem;background:var(--green);color:#fff;width:42px;height:42px;border-radius:50%;border:none;cursor:pointer;font-size:1.1rem;box-shadow:0 4px 14px rgba(26,122,74,.35);display:flex;align-items:center;justify-content:center;opacity:0;transition:opacity .25s,transform .25s;pointer-events:none;}#back-top.visible{opacity:1;pointer-events:auto;}#back-top:hover{transform:translateY(-2px);}@media(max-width:720px){.page-wrap{grid-template-columns:1fr;}.sidebar{position:static;display:none;}.hero::before{display:none;}nav .nav-links{display:none;}}<\/style>\r\n<\/head>\r\n<body>\r\n<div class=\"progress-bar\" id=\"progress\"><\/div>\r\n<nav><div class=\"nav-brand\">Pak <span>Notes Hub<\/span><\/div><div class=\"nav-links\"><a href=\"#unit-1\">Algebra<\/a><a href=\"#unit-3\">Trigonometry<\/a><a href=\"#unit-5\">Calculus<\/a><\/div><\/nav>\r\n<section class=\"hero\"><div class=\"hero-tag\">\u2211 DAE Mechanical \u2014 1st Year \u2014 Subject Code 113<\/div><h1>Mathematics<br\/><span>Complete Notes \u2014 Easy English<\/span><\/h1><p>Algebra \u00b7 Trigonometry \u00b7 Calculus \u00b7 Vectors \u00b7 Complete Curriculum<\/p><div class=\"hero-pills\"><div class=\"pill\">Algebra & Functions<\/div><div class=\"pill\">Trigonometry<\/div><div class=\"pill\">Calculus<\/div><\/div><\/section>\r\n<div class=\"page-wrap\"><aside class=\"sidebar\"><div class=\"sidebar-title\">\ud83d\udcd0 Table of Contents<\/div><ul class=\"toc-list\"><li><a href=\"#unit-1\"><span class=\"toc-num\">1<\/span> Algebra<\/a><\/li><li><a href=\"#unit-2\"><span class=\"toc-num\">2<\/span> Functions<\/a><\/li><li><a href=\"#unit-3\"><span class=\"toc-num\">3<\/span> Trigonometry<\/a><\/li><li><a href=\"#unit-4\"><span class=\"toc-num\">4<\/span> Vectors<\/a><\/li><li><a href=\"#unit-5\"><span class=\"toc-num\">5<\/span> Limits & Continuity<\/a><\/li><li><a href=\"#unit-6\"><span class=\"toc-num\">6<\/span> Differentiation<\/a><\/li><li><a href=\"#unit-7\"><span class=\"toc-num\">7<\/span> Integration<\/a><\/li><li><a href=\"#unit-8\"><span class=\"toc-num\">8<\/span> Series & Applications<\/a><\/li><\/ul><\/aside><main><div class=\"unit\" id=\"unit-1\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 1<\/div><h2>Algebra Fundamentals<\/h2><p>Equations and Inequalities<\/p><\/div><div class=\"unit-body\"><h3>Linear Equations<\/h3><p>Form: ax + b = 0<br>Solution: x = -b\/a<\/p><h3>Quadratic Equations<\/h3><p>Form: ax\u00b2 + bx + c = 0<br>Using quadratic formula: x = [-b \u00b1 \u221a(b\u00b2 - 4ac)] \/ 2a<\/p>\r\n\r\n<svg width=\"100%\" height=\"180\" viewBox=\"0 0 600 180\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"30\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Quadratic Formula Visualization<\/text>\r\n <rect x=\"50\" y=\"50\" width=\"500\" height=\"110\" fill=\"#e8f5ee\" stroke=\"#1a7a4a\" stroke-width=\"2\" rx=\"8\"\/>\r\n <text x=\"300\" y=\"85\" font-size=\"20\" font-weight=\"700\" fill=\"#1a7a4a\" text-anchor=\"middle\">x = [-b \u00b1 \u221a(b\u00b2 - 4ac)] \/ 2a<\/text>\r\n <text x=\"130\" y=\"120\" font-size=\"14\" fill=\"#566573\" font-weight=\"600\">Discriminant: \u0394 = b\u00b2 - 4ac<\/text>\r\n <circle cx=\"100\" cy=\"145\" r=\"4\" fill=\"#17a589\"\/>\r\n <text x=\"110\" y=\"150\" font-size=\"13\" fill=\"#1c2833\">\u0394 > 0: Two real roots<\/text>\r\n <circle cx=\"280\" cy=\"145\" r=\"4\" fill=\"#17a589\"\/>\r\n <text x=\"290\" y=\"150\" font-size=\"13\" fill=\"#1c2833\">\u0394 = 0: One root<\/text>\r\n <circle cx=\"450\" cy=\"145\" r=\"4\" fill=\"#17a589\"\/>\r\n <text x=\"460\" y=\"150\" font-size=\"13\" fill=\"#1c2833\">\u0394 < 0: Complex roots<\/text>\r\n<\/svg>\r\n\r\n<h3>Types of Solutions<\/h3><table class=\"data-table\"><thead><tr><th>Discriminant (\u0394)<\/th><th>Roots<\/th><\/tr><\/thead><tbody><tr><td>\u0394 > 0<\/td><td>Two real distinct roots<\/td><\/tr><tr><td>\u0394 = 0<\/td><td>One repeated real root<\/td><\/tr><tr><td>\u0394 < 0<\/td><td>Complex conjugate roots<\/td><\/tr><\/tbody><\/table><h3>Inequalities<\/h3><ul><li>Linear inequalities: ax + b > c<\/li><li>Quadratic inequalities: ax\u00b2 + bx + c > 0<\/li><li>Solution represented on number line<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Solve linear and quadratic equations with applications<\/div><\/div><\/div><div class=\"unit\" id=\"unit-2\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 2<\/div><h2>Functions and Relations<\/h2><p>Function Concepts<\/p><\/div><div class=\"unit-body\"><h3>Function Definition<\/h3><p>Relation where each input has exactly one output. Notation: f(x) = y<\/p><h3>Types of Functions<\/h3><ul><li><strong>Linear:<\/strong> f(x) = mx + c<\/li><li><strong>Quadratic:<\/strong> f(x) = ax\u00b2 + bx + c<\/li><li><strong>Polynomial:<\/strong> f(x) = a\u2099x\u207f + ... + a\u2081x + a\u2080<\/li><li><strong>Exponential:<\/strong> f(x) = a\u02e3<\/li><li><strong>Logarithmic:<\/strong> f(x) = log\u2090(x)<\/li><\/ul><h3>Function Operations<\/h3><ul><li>Addition: (f + g)(x) = f(x) + g(x)<\/li><li>Multiplication: (f\u00b7g)(x) = f(x)\u00b7g(x)<\/li><li>Composition: (f\u2218g)(x) = f(g(x))<\/li><\/ul><h3>Inverse Functions<\/h3><p>If f(a) = b, then f\u207b\u00b9(b) = a<\/p><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Analyze functions and find their properties<\/div><\/div><\/div><div class=\"unit\" id=\"unit-3\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 3<\/div><h2>Trigonometry<\/h2><p>Trigonometric Ratios and Identities<\/p><\/div><div class=\"unit-body\"><h3>Trigonometric Ratios<\/h3>\r\n\r\n<svg width=\"100%\" height=\"280\" viewBox=\"0 0 600 280\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"25\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Right Triangle - Trigonometric Ratios<\/text>\r\n \r\n <line x1=\"150\" y1=\"220\" x2=\"450\" y2=\"220\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n <line x1=\"150\" y1=\"70\" x2=\"150\" y2=\"220\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n <line x1=\"150\" y1=\"70\" x2=\"450\" y2=\"220\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n \r\n <text x=\"300\" y=\"245\" font-size=\"15\" font-weight=\"600\" fill=\"#1a7a4a\" text-anchor=\"middle\">Adjacent<\/text>\r\n <text x=\"115\" y=\"150\" font-size=\"15\" font-weight=\"600\" fill=\"#1a7a4a\" text-anchor=\"middle\">Opposite<\/text>\r\n <text x=\"320\" y=\"135\" font-size=\"15\" font-weight=\"600\" fill=\"#1a7a4a\" text-anchor=\"middle\">Hypotenuse<\/text>\r\n \r\n <path d=\"M 180 220 Q 185 210 195 210\" fill=\"none\" stroke=\"#f0b127\" stroke-width=\"2\"\/>\r\n <text x=\"200\" y=\"215\" font-size=\"14\" font-weight=\"700\" fill=\"#f0b127\">\u03b8<\/text>\r\n \r\n <rect x=\"140\" y=\"210\" width=\"10\" height=\"10\" fill=\"none\" stroke=\"#17a589\" stroke-width=\"2\"\/>\r\n \r\n <rect x=\"20\" y=\"50\" width=\"110\" height=\"20\" fill=\"#e8f5ee\" rx=\"4\"\/>\r\n <text x=\"75\" y=\"64\" font-size=\"13\" font-weight=\"600\" fill=\"#145f38\" text-anchor=\"middle\">sin \u03b8 = opp\/hyp<\/text>\r\n <rect x=\"20\" y=\"80\" width=\"110\" height=\"20\" fill=\"#e8f5ee\" rx=\"4\"\/>\r\n <text x=\"75\" y=\"94\" font-size=\"13\" font-weight=\"600\" fill=\"#145f38\" text-anchor=\"middle\">cos \u03b8 = adj\/hyp<\/text>\r\n <rect x=\"20\" y=\"110\" width=\"110\" height=\"20\" fill=\"#e8f5ee\" rx=\"4\"\/>\r\n <text x=\"75\" y=\"124\" font-size=\"13\" font-weight=\"600\" fill=\"#145f38\" text-anchor=\"middle\">tan \u03b8 = opp\/adj<\/text>\r\n<\/svg>\r\n\r\n<table class=\"data-table\"><thead><tr><th>Ratio<\/th><th>Definition<\/th><\/tr><\/thead><tbody><tr><td>sin \u03b8<\/td><td>opposite \/ hypotenuse<\/td><\/tr><tr><td>cos \u03b8<\/td><td>adjacent \/ hypotenuse<\/td><\/tr><tr><td>tan \u03b8<\/td><td>opposite \/ adjacent<\/td><\/tr><\/tbody><\/table><h3>Special Angles<\/h3><ul><li>sin 30\u00b0 = 1\/2, cos 30\u00b0 = \u221a3\/2<\/li><li>sin 45\u00b0 = \u221a2\/2, cos 45\u00b0 = \u221a2\/2<\/li><li>sin 60\u00b0 = \u221a3\/2, cos 60\u00b0 = 1\/2<\/li><\/ul><h3>Trigonometric Identities<\/h3><ul><li>sin\u00b2\u03b8 + cos\u00b2\u03b8 = 1<\/li><li>1 + tan\u00b2\u03b8 = sec\u00b2\u03b8<\/li><li>sin(A \u00b1 B) = sinA cosB \u00b1 cosA sinB<\/li><li>cos(A \u00b1 B) = cosA cosB \u2213 sinA sinB<\/li><\/ul><h3>Law of Sines and Cosines<\/h3><p>Sine: a\/sinA = b\/sinB = c\/sinC<br>Cosine: c\u00b2 = a\u00b2 + b\u00b2 - 2ab cosC<\/p><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Solve trigonometric equations and triangle problems<\/div><\/div><\/div><div class=\"unit\" id=\"unit-4\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 4<\/div><h2>Vectors<\/h2><p>Vector Algebra<\/p><\/div><div class=\"unit-body\"><h3>Vector Representation<\/h3><p>v = (x, y, z) or v = x\u00ee + y\u0135 + zk\u0302<\/p>\r\n\r\n<svg width=\"100%\" height=\"280\" viewBox=\"0 0 600 280\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"25\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Vector Representation in 2D Space<\/text>\r\n <defs>\r\n <marker id=\"arrowhead2\" markerWidth=\"10\" markerHeight=\"10\" refX=\"9\" refY=\"3\" orient=\"auto\">\r\n <polygon points=\"0 0, 10 3, 0 6\" fill=\"#1a7a4a\"\/>\r\n <\/marker>\r\n <\/defs>\r\n \r\n <line x1=\"100\" y1=\"220\" x2=\"550\" y2=\"220\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead2)\"\/>\r\n <line x1=\"100\" y1=\"250\" x2=\"100\" y2=\"50\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead2)\"\/>\r\n <text x=\"560\" y=\"225\" font-size=\"14\" font-weight=\"600\" fill=\"#566573\">x<\/text>\r\n <text x=\"105\" y=\"45\" font-size=\"14\" font-weight=\"600\" fill=\"#566573\">y<\/text>\r\n \r\n <line x1=\"100\" y1=\"220\" x2=\"380\" y2=\"100\" stroke=\"#1a7a4a\" stroke-width=\"4\" marker-end=\"url(#arrowhead2)\"\/>\r\n <text x=\"250\" y=\"150\" font-size=\"16\" font-weight=\"700\" fill=\"#1a7a4a\">v<\/text>\r\n \r\n <line x1=\"100\" y1=\"220\" x2=\"380\" y2=\"220\" stroke=\"#17a589\" stroke-width=\"2\" stroke-dasharray=\"5,5\"\/>\r\n <line x1=\"380\" y1=\"220\" x2=\"380\" y2=\"100\" stroke=\"#17a589\" stroke-width=\"2\" stroke-dasharray=\"5,5\"\/>\r\n <text x=\"240\" y=\"245\" font-size=\"14\" font-weight=\"600\" fill=\"#17a589\">v\u2093 (x-component)<\/text>\r\n <text x=\"395\" y=\"160\" font-size=\"14\" font-weight=\"600\" fill=\"#17a589\">v\u1d67 (y-component)<\/text>\r\n \r\n <text x=\"30\" y=\"140\" font-size=\"13\" fill=\"#1c2833\" font-weight=\"600\">|v| = \u221a(x\u00b2 + y\u00b2)<\/text>\r\n \r\n <circle cx=\"100\" cy=\"220\" r=\"4\" fill=\"#f0b127\"\/>\r\n <text x=\"85\" y=\"245\" font-size=\"13\" font-weight=\"600\" fill=\"#1c2833\">O(0,0)<\/text>\r\n<\/svg><h3>Vector Operations<\/h3><ul><li><strong>Addition:<\/strong> a + b = (a\u2081+b\u2081, a\u2082+b\u2082, a\u2083+b\u2083)<\/li><li><strong>Scalar Multiplication:<\/strong> k\u00b7a = (ka\u2081, ka\u2082, ka\u2083)<\/li><li><strong>Dot Product:<\/strong> a\u00b7b = |a||b|cos \u03b8<\/li><li><strong>Cross Product:<\/strong> a \u00d7 b (perpendicular to both)<\/li><\/ul><h3>Vector Properties<\/h3><ul><li>Magnitude: |v| = \u221a(x\u00b2 + y\u00b2 + z\u00b2)<\/li><li>Unit Vector: \u00fb = v\/|v|<\/li><li>Parallel vectors: a \u00d7 b = 0<\/li><\/ul><h3>Applications<\/h3><ul><li>Displacement and velocity<\/li><li>Force resolution<\/li><li>Work calculation<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Perform vector operations and solve applied problems<\/div><\/div><\/div><div class=\"unit\" id=\"unit-5\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 5<\/div><h2>Limits and Continuity<\/h2><p>Foundations of Calculus<\/p><\/div><div class=\"unit-body\"><h3>Limit Concept<\/h3><p>lim(x\u2192a) f(x) = L means f(x) approaches L as x approaches a<\/p><h3>Limit Laws<\/h3><ul><li>Sum: lim(f + g) = lim f + lim g<\/li><li>Product: lim(f\u00b7g) = (lim f)\u00b7(lim g)<\/li><li>Quotient: lim(f\/g) = (lim f)\/(lim g), if lim g \u2260 0<\/li><\/ul><h3>Continuity Definition<\/h3><p>Function f is continuous at x = a if:<\/p><ul><li>f(a) is defined<\/li><li>lim(x\u2192a) f(x) exists<\/li><li>lim(x\u2192a) f(x) = f(a)<\/li><\/ul><h3>Discontinuities<\/h3><ul><li><strong>Removable:<\/strong> Can be fixed<\/li><li><strong>Jump:<\/strong> Left and right limits differ<\/li><li><strong>Infinite:<\/strong> Function approaches infinity<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Calculate limits and analyze continuity<\/div><\/div><\/div><div class=\"unit\" id=\"unit-6\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 6<\/div><h2>Differentiation<\/h2><p>Rates of Change<\/p><\/div><div class=\"unit-body\"><h3>Derivative Definition<\/h3><p>f'(x) = lim(h\u21920) [f(x+h) - f(x)] \/ h<\/p>\r\n\r\n<svg width=\"100%\" height=\"260\" viewBox=\"0 0 600 260\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"25\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Derivative - Rate of Change (Slope of Tangent)<\/text>\r\n <defs>\r\n <marker id=\"arrowhead3\" markerWidth=\"10\" markerHeight=\"10\" refX=\"9\" refY=\"3\" orient=\"auto\">\r\n <polygon points=\"0 0, 10 3, 0 6\" fill=\"#566573\"\/>\r\n <\/marker>\r\n <\/defs>\r\n \r\n <line x1=\"80\" y1=\"220\" x2=\"550\" y2=\"220\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead3)\"\/>\r\n <line x1=\"80\" y1=\"230\" x2=\"80\" y2=\"50\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead3)\"\/>\r\n <text x=\"560\" y=\"225\" font-size=\"13\" font-weight=\"600\" fill=\"#566573\">x<\/text>\r\n <text x=\"85\" y=\"45\" font-size=\"13\" font-weight=\"600\" fill=\"#566573\">y<\/text>\r\n \r\n <path d=\"M 100 200 Q 200 180 300 130 T 500 80\" fill=\"none\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n \r\n <line x1=\"150\" y1=\"210\" x2=\"450\" y2=\"90\" stroke=\"#f0b127\" stroke-width=\"2\" stroke-dasharray=\"5,5\"\/>\r\n \r\n <circle cx=\"300\" cy=\"130\" r=\"5\" fill=\"#17a589\"\/>\r\n <text x=\"310\" y=\"125\" font-size=\"14\" font-weight=\"600\" fill=\"#17a589\">(x, f(x))<\/text>\r\n \r\n <text x=\"380\" y=\"110\" font-size=\"14\" font-weight=\"600\" fill=\"#f0b127\">Tangent line<\/text>\r\n <text x=\"350\" y=\"160\" font-size=\"13\" fill=\"#1c2833\">Slope = f'(x)<\/text>\r\n <text x=\"200\" y=\"210\" font-size=\"15\" font-weight=\"700\" fill=\"#1a7a4a\">y = f(x)<\/text>\r\n<\/svg><h3>Differentiation Rules<\/h3><ul><li><strong>Power Rule:<\/strong> d\/dx(x\u207f) = nx\u207f\u207b\u00b9<\/li><li><strong>Product Rule:<\/strong> d\/dx(fg) = f'g + fg'<\/li><li><strong>Quotient Rule:<\/strong> d\/dx(f\/g) = (f'g - fg')\/g\u00b2<\/li><li><strong>Chain Rule:<\/strong> d\/dx(f(g(x))) = f'(g(x))\u00b7g'(x)<\/li><\/ul><h3>Derivatives of Special Functions<\/h3><ul><li>d\/dx(e\u02e3) = e\u02e3<\/li><li>d\/dx(ln x) = 1\/x<\/li><li>d\/dx(sin x) = cos x<\/li><li>d\/dx(cos x) = -sin x<\/li><\/ul><h3>Applications<\/h3><ul><li>Finding maximum and minimum values<\/li><li>Determining increasing\/decreasing intervals<\/li><li>Analyzing motion (velocity, acceleration)<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Differentiate functions and solve optimization problems<\/div><\/div><\/div><div class=\"unit\" id=\"unit-7\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 7<\/div><h2>Integration<\/h2><p>Antiderivatives and Area<\/p><\/div><div class=\"unit-body\"><h3>Indefinite Integral<\/h3><p>\u222bf(x)dx = F(x) + C, where F'(x) = f(x)<\/p><h3>Integration Rules<\/h3><ul><li>\u222bx\u207f dx = x\u207f\u207a\u00b9\/(n+1) + C<\/li><li>\u222be\u02e3 dx = e\u02e3 + C<\/li><li>\u222b(1\/x) dx = ln|x| + C<\/li><li>\u222bsin x dx = -cos x + C<\/li><li>\u222bcos x dx = sin x + C<\/li><\/ul><h3>Definite Integral<\/h3><p>\u222b\u2090\u1d47 f(x)dx = F(b) - F(a)<\/p><h3>Integration Techniques<\/h3><ul><li><strong>Substitution:<\/strong> u-substitution<\/li><li><strong>Parts:<\/strong> \u222bu dv = uv - \u222bv du<\/li><\/ul><h3>Applications<\/h3><ul><li>Finding area under curves<\/li><li>Volume calculations<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Integrate functions and calculate areas<\/div><\/div><\/div><div class=\"unit\" id=\"unit-8\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 8<\/div><h2>Series and Applications<\/h2><p>Sequences and Their Applications<\/p><\/div><div class=\"unit-body\"><h3>Sequences<\/h3><ul><li><strong>Arithmetic:<\/strong> a\u2099 = a\u2081 + (n-1)d<\/li><li><strong>Geometric:<\/strong> a\u2099 = a\u2081 \u00b7 r\u207f\u207b\u00b9<\/li><\/ul><h3>Series Sums<\/h3><ul><li><strong>Arithmetic Series:<\/strong> S\u2099 = n\/2(a\u2081 + a\u2099)<\/li><li><strong>Geometric Series:<\/strong> S\u2099 = a\u2081(1-r\u207f)\/(1-r)<\/li><\/ul><h3>Infinite Series<\/h3><ul><li>Convergent series: Sum approaches a limit<\/li><li>Divergent series: Sum does not converge<\/li><\/ul><h3>Taylor Series<\/h3><p>f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)\u00b2\/2! + ...<\/p><h3>Applications<\/h3><ul><li>Physics problems involving motion<\/li><li>Engineering calculations<\/li><li>Finance and economics<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Work with series and apply to real problems<\/div><\/div><\/div><div class=\"congrats\"><h2>\u2728 Congratulations!<\/h2><p>You've completed Mathematics! Master numbers and solve the world.<\/p><\/div><\/main><\/div>\r\n<button id=\"back-top\" onclick=\"window.scrollTo({top:0,behavior:'smooth'})\">\u2191<\/button>\r\n<script>const progress=document.getElementById('progress');window.addEventListener('scroll',()=>{const scrollPercent=(window.scrollY\/(document.documentElement.scrollHeight-window.innerHeight))*100;progress.style.width=scrollPercent+'%';document.getElementById('back-top').classList.toggle('visible',window.scrollY>400);});document.querySelectorAll('.unit').forEach(unit=>{const observer=new IntersectionObserver(entries=>{entries.forEach(entry=>{if(entry.isIntersecting){const id=entry.target.id;document.querySelectorAll('.toc-list a').forEach(a=>{a.classList.toggle('active',a.href.endsWith(id));});}});},{rootMargin:'-20% 0px -70% 0px'});observer.observe(unit);});<\/script>\r\n<\/body>\r\n<\/html>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Mathematics (113) \u2013 DAE Mechanical 1st Year \u2013 Pak Notes Hub Pak Notes HubAlgebraTrigonometryCalculus \u2211 DAE Mechanical \u2014 1st Year \u2014 Subject Code 113 MathematicsComplete Notes \u2014 Easy English Algebra \u00b7 Trigonometry \u00b7 Calculus \u00b7 Vectors \u00b7 Complete Curriculum Algebra & FunctionsTrigonometryCalculus \ud83d\udcd0 Table of Contents 1 Algebra 2 Functions 3 Trigonometry 4 Vectors 5 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"page_builder":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-645","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts\/645","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=645"}],"version-history":[{"count":4,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts\/645\/revisions"}],"predecessor-version":[{"id":649,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts\/645\/revisions\/649"}],"wp:attachment":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=645"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=645"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=645"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

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\n\t\t\t\t\t\r\n\r\n\r\n Mathematics (113) \u2013 DAE Mechanical 1st Year \u2013 Pak Notes Hub<\/title>\r\n <link href=\"https:\/\/fonts.googleapis.com\/css2?family=Inter:wght@400;500;600;700&family=Fira+Code:wght@400;500&display=swap\" rel=\"stylesheet\"\/><style>*,*::before,*::after{box-sizing:border-box;margin:0;padding:0;}:root{--green:#1a7a4a;--green-dark:#145f38;--green-light:#e8f5ee;--teal:#17a589;--accent:#f0b127;--dark:#1c2833;--mid:#566573;--light:#f4f6f7;--code-bg:#f0f4f8;--code-border:#2e86c1;--note-bg:#fef9e7;--note-border:#f0b127;--white:#ffffff;--radius:10px;--shadow:0 4px 24px rgba(26,122,74,.10);}html{scroll-behavior:smooth;}body{font-family:'Inter',sans-serif;background:#f2f6f3;color:var(--dark);line-height:1.7;font-size:15px;}nav{position:sticky;top:0;z-index:100;background:var(--green-dark);padding:0 2rem;display:flex;align-items:center;justify-content:space-between;height:56px;box-shadow:0 2px 12px 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1rem;text-align:left;font-weight:600;font-size:.82rem;}.data-table tbody tr:nth-child(even){background:var(--green-light);}.data-table tbody tr:hover{background:#d4efdf;}.data-table td{padding:.6rem 1rem;border-bottom:1px solid #dce8e0;vertical-align:top;}.data-table td:first-child{font-weight:600;color:var(--green-dark);}.info-box{background:var(--note-bg);border:1px solid var(--note-border);border-radius:7px;padding:.85rem 1.1rem;margin:.9rem 0;font-size:.88rem;line-height:1.6;color:#7d6608;display:flex;align-items:flex-start;gap:.7rem;}.practice{background:var(--green-light);border:1.5px dashed var(--green);border-radius:8px;padding:.85rem 1.1rem;margin-top:1.4rem;color:var(--green-dark);font-size:.88rem;font-weight:500;}.congrats{background:linear-gradient(135deg,var(--green) 0%,#145f38 100%);border-radius:var(--radius);padding:2.5rem 2rem;text-align:center;color:#fff;margin-bottom:2.5rem;box-shadow:var(--shadow);}.congrats h2{font-size:1.8rem;margin-bottom:.5rem;}.congrats p{color:#d5f5e3;font-size:1rem;}.progress-bar{position:fixed;top:0;left:0;height:3px;background:var(--accent);z-index:200;transition:width .1s linear;width:0%;}#back-top{position:fixed;bottom:2rem;right:2rem;background:var(--green);color:#fff;width:42px;height:42px;border-radius:50%;border:none;cursor:pointer;font-size:1.1rem;box-shadow:0 4px 14px rgba(26,122,74,.35);display:flex;align-items:center;justify-content:center;opacity:0;transition:opacity .25s,transform .25s;pointer-events:none;}#back-top.visible{opacity:1;pointer-events:auto;}#back-top:hover{transform:translateY(-2px);}@media(max-width:720px){.page-wrap{grid-template-columns:1fr;}.sidebar{position:static;display:none;}.hero::before{display:none;}nav .nav-links{display:none;}}<\/style>\r\n<\/head>\r\n<body>\r\n<div class=\"progress-bar\" id=\"progress\"><\/div>\r\n<nav><div class=\"nav-brand\">Pak <span>Notes Hub<\/span><\/div><div class=\"nav-links\"><a href=\"#unit-1\">Algebra<\/a><a href=\"#unit-3\">Trigonometry<\/a><a href=\"#unit-5\">Calculus<\/a><\/div><\/nav>\r\n<section class=\"hero\"><div class=\"hero-tag\">\u2211 DAE Mechanical \u2014 1st Year \u2014 Subject Code 113<\/div><h1>Mathematics<br\/><span>Complete Notes \u2014 Easy English<\/span><\/h1><p>Algebra \u00b7 Trigonometry \u00b7 Calculus \u00b7 Vectors \u00b7 Complete Curriculum<\/p><div class=\"hero-pills\"><div class=\"pill\">Algebra & Functions<\/div><div class=\"pill\">Trigonometry<\/div><div class=\"pill\">Calculus<\/div><\/div><\/section>\r\n<div class=\"page-wrap\"><aside class=\"sidebar\"><div class=\"sidebar-title\">\ud83d\udcd0 Table of Contents<\/div><ul class=\"toc-list\"><li><a href=\"#unit-1\"><span class=\"toc-num\">1<\/span> Algebra<\/a><\/li><li><a href=\"#unit-2\"><span class=\"toc-num\">2<\/span> Functions<\/a><\/li><li><a href=\"#unit-3\"><span class=\"toc-num\">3<\/span> Trigonometry<\/a><\/li><li><a href=\"#unit-4\"><span class=\"toc-num\">4<\/span> Vectors<\/a><\/li><li><a href=\"#unit-5\"><span class=\"toc-num\">5<\/span> Limits & Continuity<\/a><\/li><li><a href=\"#unit-6\"><span class=\"toc-num\">6<\/span> Differentiation<\/a><\/li><li><a href=\"#unit-7\"><span class=\"toc-num\">7<\/span> Integration<\/a><\/li><li><a href=\"#unit-8\"><span class=\"toc-num\">8<\/span> Series & Applications<\/a><\/li><\/ul><\/aside><main><div class=\"unit\" id=\"unit-1\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 1<\/div><h2>Algebra Fundamentals<\/h2><p>Equations and Inequalities<\/p><\/div><div class=\"unit-body\"><h3>Linear Equations<\/h3><p>Form: ax + b = 0<br>Solution: x = -b\/a<\/p><h3>Quadratic Equations<\/h3><p>Form: ax\u00b2 + bx + c = 0<br>Using quadratic formula: x = [-b \u00b1 \u221a(b\u00b2 - 4ac)] \/ 2a<\/p>\r\n\r\n<svg width=\"100%\" height=\"180\" viewBox=\"0 0 600 180\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"30\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Quadratic Formula Visualization<\/text>\r\n <rect x=\"50\" y=\"50\" width=\"500\" height=\"110\" fill=\"#e8f5ee\" stroke=\"#1a7a4a\" stroke-width=\"2\" rx=\"8\"\/>\r\n <text x=\"300\" y=\"85\" font-size=\"20\" font-weight=\"700\" fill=\"#1a7a4a\" text-anchor=\"middle\">x = [-b \u00b1 \u221a(b\u00b2 - 4ac)] \/ 2a<\/text>\r\n <text x=\"130\" y=\"120\" font-size=\"14\" fill=\"#566573\" font-weight=\"600\">Discriminant: \u0394 = b\u00b2 - 4ac<\/text>\r\n <circle cx=\"100\" cy=\"145\" r=\"4\" fill=\"#17a589\"\/>\r\n <text x=\"110\" y=\"150\" font-size=\"13\" fill=\"#1c2833\">\u0394 > 0: Two real roots<\/text>\r\n <circle cx=\"280\" cy=\"145\" r=\"4\" fill=\"#17a589\"\/>\r\n <text x=\"290\" y=\"150\" font-size=\"13\" fill=\"#1c2833\">\u0394 = 0: One root<\/text>\r\n <circle cx=\"450\" cy=\"145\" r=\"4\" fill=\"#17a589\"\/>\r\n <text x=\"460\" y=\"150\" font-size=\"13\" fill=\"#1c2833\">\u0394 < 0: Complex roots<\/text>\r\n<\/svg>\r\n\r\n<h3>Types of Solutions<\/h3><table class=\"data-table\"><thead><tr><th>Discriminant (\u0394)<\/th><th>Roots<\/th><\/tr><\/thead><tbody><tr><td>\u0394 > 0<\/td><td>Two real distinct roots<\/td><\/tr><tr><td>\u0394 = 0<\/td><td>One repeated real root<\/td><\/tr><tr><td>\u0394 < 0<\/td><td>Complex conjugate roots<\/td><\/tr><\/tbody><\/table><h3>Inequalities<\/h3><ul><li>Linear inequalities: ax + b > c<\/li><li>Quadratic inequalities: ax\u00b2 + bx + c > 0<\/li><li>Solution represented on number line<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Solve linear and quadratic equations with applications<\/div><\/div><\/div><div class=\"unit\" id=\"unit-2\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 2<\/div><h2>Functions and Relations<\/h2><p>Function Concepts<\/p><\/div><div class=\"unit-body\"><h3>Function Definition<\/h3><p>Relation where each input has exactly one output. Notation: f(x) = y<\/p><h3>Types of Functions<\/h3><ul><li><strong>Linear:<\/strong> f(x) = mx + c<\/li><li><strong>Quadratic:<\/strong> f(x) = ax\u00b2 + bx + c<\/li><li><strong>Polynomial:<\/strong> f(x) = a\u2099x\u207f + ... + a\u2081x + a\u2080<\/li><li><strong>Exponential:<\/strong> f(x) = a\u02e3<\/li><li><strong>Logarithmic:<\/strong> f(x) = log\u2090(x)<\/li><\/ul><h3>Function Operations<\/h3><ul><li>Addition: (f + g)(x) = f(x) + g(x)<\/li><li>Multiplication: (f\u00b7g)(x) = f(x)\u00b7g(x)<\/li><li>Composition: (f\u2218g)(x) = f(g(x))<\/li><\/ul><h3>Inverse Functions<\/h3><p>If f(a) = b, then f\u207b\u00b9(b) = a<\/p><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Analyze functions and find their properties<\/div><\/div><\/div><div class=\"unit\" id=\"unit-3\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 3<\/div><h2>Trigonometry<\/h2><p>Trigonometric Ratios and Identities<\/p><\/div><div class=\"unit-body\"><h3>Trigonometric Ratios<\/h3>\r\n\r\n<svg width=\"100%\" height=\"280\" viewBox=\"0 0 600 280\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"25\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Right Triangle - Trigonometric Ratios<\/text>\r\n \r\n <line x1=\"150\" y1=\"220\" x2=\"450\" y2=\"220\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n <line x1=\"150\" y1=\"70\" x2=\"150\" y2=\"220\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n <line x1=\"150\" y1=\"70\" x2=\"450\" y2=\"220\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n \r\n <text x=\"300\" y=\"245\" font-size=\"15\" font-weight=\"600\" fill=\"#1a7a4a\" text-anchor=\"middle\">Adjacent<\/text>\r\n <text x=\"115\" y=\"150\" font-size=\"15\" font-weight=\"600\" fill=\"#1a7a4a\" text-anchor=\"middle\">Opposite<\/text>\r\n <text x=\"320\" y=\"135\" font-size=\"15\" font-weight=\"600\" fill=\"#1a7a4a\" text-anchor=\"middle\">Hypotenuse<\/text>\r\n \r\n <path d=\"M 180 220 Q 185 210 195 210\" fill=\"none\" stroke=\"#f0b127\" stroke-width=\"2\"\/>\r\n <text x=\"200\" y=\"215\" font-size=\"14\" font-weight=\"700\" fill=\"#f0b127\">\u03b8<\/text>\r\n \r\n <rect x=\"140\" y=\"210\" width=\"10\" height=\"10\" fill=\"none\" stroke=\"#17a589\" stroke-width=\"2\"\/>\r\n \r\n <rect x=\"20\" y=\"50\" width=\"110\" height=\"20\" fill=\"#e8f5ee\" rx=\"4\"\/>\r\n <text x=\"75\" y=\"64\" font-size=\"13\" font-weight=\"600\" fill=\"#145f38\" text-anchor=\"middle\">sin \u03b8 = opp\/hyp<\/text>\r\n <rect x=\"20\" y=\"80\" width=\"110\" height=\"20\" fill=\"#e8f5ee\" rx=\"4\"\/>\r\n <text x=\"75\" y=\"94\" font-size=\"13\" font-weight=\"600\" fill=\"#145f38\" text-anchor=\"middle\">cos \u03b8 = adj\/hyp<\/text>\r\n <rect x=\"20\" y=\"110\" width=\"110\" height=\"20\" fill=\"#e8f5ee\" rx=\"4\"\/>\r\n <text x=\"75\" y=\"124\" font-size=\"13\" font-weight=\"600\" fill=\"#145f38\" text-anchor=\"middle\">tan \u03b8 = opp\/adj<\/text>\r\n<\/svg>\r\n\r\n<table class=\"data-table\"><thead><tr><th>Ratio<\/th><th>Definition<\/th><\/tr><\/thead><tbody><tr><td>sin \u03b8<\/td><td>opposite \/ hypotenuse<\/td><\/tr><tr><td>cos \u03b8<\/td><td>adjacent \/ hypotenuse<\/td><\/tr><tr><td>tan \u03b8<\/td><td>opposite \/ adjacent<\/td><\/tr><\/tbody><\/table><h3>Special Angles<\/h3><ul><li>sin 30\u00b0 = 1\/2, cos 30\u00b0 = \u221a3\/2<\/li><li>sin 45\u00b0 = \u221a2\/2, cos 45\u00b0 = \u221a2\/2<\/li><li>sin 60\u00b0 = \u221a3\/2, cos 60\u00b0 = 1\/2<\/li><\/ul><h3>Trigonometric Identities<\/h3><ul><li>sin\u00b2\u03b8 + cos\u00b2\u03b8 = 1<\/li><li>1 + tan\u00b2\u03b8 = sec\u00b2\u03b8<\/li><li>sin(A \u00b1 B) = sinA cosB \u00b1 cosA sinB<\/li><li>cos(A \u00b1 B) = cosA cosB \u2213 sinA sinB<\/li><\/ul><h3>Law of Sines and Cosines<\/h3><p>Sine: a\/sinA = b\/sinB = c\/sinC<br>Cosine: c\u00b2 = a\u00b2 + b\u00b2 - 2ab cosC<\/p><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Solve trigonometric equations and triangle problems<\/div><\/div><\/div><div class=\"unit\" id=\"unit-4\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 4<\/div><h2>Vectors<\/h2><p>Vector Algebra<\/p><\/div><div class=\"unit-body\"><h3>Vector Representation<\/h3><p>v = (x, y, z) or v = x\u00ee + y\u0135 + zk\u0302<\/p>\r\n\r\n<svg width=\"100%\" height=\"280\" viewBox=\"0 0 600 280\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"25\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Vector Representation in 2D Space<\/text>\r\n <defs>\r\n <marker id=\"arrowhead2\" markerWidth=\"10\" markerHeight=\"10\" refX=\"9\" refY=\"3\" orient=\"auto\">\r\n <polygon points=\"0 0, 10 3, 0 6\" fill=\"#1a7a4a\"\/>\r\n <\/marker>\r\n <\/defs>\r\n \r\n <line x1=\"100\" y1=\"220\" x2=\"550\" y2=\"220\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead2)\"\/>\r\n <line x1=\"100\" y1=\"250\" x2=\"100\" y2=\"50\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead2)\"\/>\r\n <text x=\"560\" y=\"225\" font-size=\"14\" font-weight=\"600\" fill=\"#566573\">x<\/text>\r\n <text x=\"105\" y=\"45\" font-size=\"14\" font-weight=\"600\" fill=\"#566573\">y<\/text>\r\n \r\n <line x1=\"100\" y1=\"220\" x2=\"380\" y2=\"100\" stroke=\"#1a7a4a\" stroke-width=\"4\" marker-end=\"url(#arrowhead2)\"\/>\r\n <text x=\"250\" y=\"150\" font-size=\"16\" font-weight=\"700\" fill=\"#1a7a4a\">v<\/text>\r\n \r\n <line x1=\"100\" y1=\"220\" x2=\"380\" y2=\"220\" stroke=\"#17a589\" stroke-width=\"2\" stroke-dasharray=\"5,5\"\/>\r\n <line x1=\"380\" y1=\"220\" x2=\"380\" y2=\"100\" stroke=\"#17a589\" stroke-width=\"2\" stroke-dasharray=\"5,5\"\/>\r\n <text x=\"240\" y=\"245\" font-size=\"14\" font-weight=\"600\" fill=\"#17a589\">v\u2093 (x-component)<\/text>\r\n <text x=\"395\" y=\"160\" font-size=\"14\" font-weight=\"600\" fill=\"#17a589\">v\u1d67 (y-component)<\/text>\r\n \r\n <text x=\"30\" y=\"140\" font-size=\"13\" fill=\"#1c2833\" font-weight=\"600\">|v| = \u221a(x\u00b2 + y\u00b2)<\/text>\r\n \r\n <circle cx=\"100\" cy=\"220\" r=\"4\" fill=\"#f0b127\"\/>\r\n <text x=\"85\" y=\"245\" font-size=\"13\" font-weight=\"600\" fill=\"#1c2833\">O(0,0)<\/text>\r\n<\/svg><h3>Vector Operations<\/h3><ul><li><strong>Addition:<\/strong> a + b = (a\u2081+b\u2081, a\u2082+b\u2082, a\u2083+b\u2083)<\/li><li><strong>Scalar Multiplication:<\/strong> k\u00b7a = (ka\u2081, ka\u2082, ka\u2083)<\/li><li><strong>Dot Product:<\/strong> a\u00b7b = |a||b|cos \u03b8<\/li><li><strong>Cross Product:<\/strong> a \u00d7 b (perpendicular to both)<\/li><\/ul><h3>Vector Properties<\/h3><ul><li>Magnitude: |v| = \u221a(x\u00b2 + y\u00b2 + z\u00b2)<\/li><li>Unit Vector: \u00fb = v\/|v|<\/li><li>Parallel vectors: a \u00d7 b = 0<\/li><\/ul><h3>Applications<\/h3><ul><li>Displacement and velocity<\/li><li>Force resolution<\/li><li>Work calculation<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Perform vector operations and solve applied problems<\/div><\/div><\/div><div class=\"unit\" id=\"unit-5\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 5<\/div><h2>Limits and Continuity<\/h2><p>Foundations of Calculus<\/p><\/div><div class=\"unit-body\"><h3>Limit Concept<\/h3><p>lim(x\u2192a) f(x) = L means f(x) approaches L as x approaches a<\/p><h3>Limit Laws<\/h3><ul><li>Sum: lim(f + g) = lim f + lim g<\/li><li>Product: lim(f\u00b7g) = (lim f)\u00b7(lim g)<\/li><li>Quotient: lim(f\/g) = (lim f)\/(lim g), if lim g \u2260 0<\/li><\/ul><h3>Continuity Definition<\/h3><p>Function f is continuous at x = a if:<\/p><ul><li>f(a) is defined<\/li><li>lim(x\u2192a) f(x) exists<\/li><li>lim(x\u2192a) f(x) = f(a)<\/li><\/ul><h3>Discontinuities<\/h3><ul><li><strong>Removable:<\/strong> Can be fixed<\/li><li><strong>Jump:<\/strong> Left and right limits differ<\/li><li><strong>Infinite:<\/strong> Function approaches infinity<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Calculate limits and analyze continuity<\/div><\/div><\/div><div class=\"unit\" id=\"unit-6\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 6<\/div><h2>Differentiation<\/h2><p>Rates of Change<\/p><\/div><div class=\"unit-body\"><h3>Derivative Definition<\/h3><p>f'(x) = lim(h\u21920) [f(x+h) - f(x)] \/ h<\/p>\r\n\r\n<svg width=\"100%\" height=\"260\" viewBox=\"0 0 600 260\" style=\"margin:1rem 0;background:#f8fffe;border-radius:8px;border:2px solid #a2d9b5;\">\r\n <text x=\"300\" y=\"25\" font-size=\"16\" font-weight=\"700\" fill=\"#145f38\" text-anchor=\"middle\">Derivative - Rate of Change (Slope of Tangent)<\/text>\r\n <defs>\r\n <marker id=\"arrowhead3\" markerWidth=\"10\" markerHeight=\"10\" refX=\"9\" refY=\"3\" orient=\"auto\">\r\n <polygon points=\"0 0, 10 3, 0 6\" fill=\"#566573\"\/>\r\n <\/marker>\r\n <\/defs>\r\n \r\n <line x1=\"80\" y1=\"220\" x2=\"550\" y2=\"220\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead3)\"\/>\r\n <line x1=\"80\" y1=\"230\" x2=\"80\" y2=\"50\" stroke=\"#566573\" stroke-width=\"2\" marker-end=\"url(#arrowhead3)\"\/>\r\n <text x=\"560\" y=\"225\" font-size=\"13\" font-weight=\"600\" fill=\"#566573\">x<\/text>\r\n <text x=\"85\" y=\"45\" font-size=\"13\" font-weight=\"600\" fill=\"#566573\">y<\/text>\r\n \r\n <path d=\"M 100 200 Q 200 180 300 130 T 500 80\" fill=\"none\" stroke=\"#1a7a4a\" stroke-width=\"3\"\/>\r\n \r\n <line x1=\"150\" y1=\"210\" x2=\"450\" y2=\"90\" stroke=\"#f0b127\" stroke-width=\"2\" stroke-dasharray=\"5,5\"\/>\r\n \r\n <circle cx=\"300\" cy=\"130\" r=\"5\" fill=\"#17a589\"\/>\r\n <text x=\"310\" y=\"125\" font-size=\"14\" font-weight=\"600\" fill=\"#17a589\">(x, f(x))<\/text>\r\n \r\n <text x=\"380\" y=\"110\" font-size=\"14\" font-weight=\"600\" fill=\"#f0b127\">Tangent line<\/text>\r\n <text x=\"350\" y=\"160\" font-size=\"13\" fill=\"#1c2833\">Slope = f'(x)<\/text>\r\n <text x=\"200\" y=\"210\" font-size=\"15\" font-weight=\"700\" fill=\"#1a7a4a\">y = f(x)<\/text>\r\n<\/svg><h3>Differentiation Rules<\/h3><ul><li><strong>Power Rule:<\/strong> d\/dx(x\u207f) = nx\u207f\u207b\u00b9<\/li><li><strong>Product Rule:<\/strong> d\/dx(fg) = f'g + fg'<\/li><li><strong>Quotient Rule:<\/strong> d\/dx(f\/g) = (f'g - fg')\/g\u00b2<\/li><li><strong>Chain Rule:<\/strong> d\/dx(f(g(x))) = f'(g(x))\u00b7g'(x)<\/li><\/ul><h3>Derivatives of Special Functions<\/h3><ul><li>d\/dx(e\u02e3) = e\u02e3<\/li><li>d\/dx(ln x) = 1\/x<\/li><li>d\/dx(sin x) = cos x<\/li><li>d\/dx(cos x) = -sin x<\/li><\/ul><h3>Applications<\/h3><ul><li>Finding maximum and minimum values<\/li><li>Determining increasing\/decreasing intervals<\/li><li>Analyzing motion (velocity, acceleration)<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Differentiate functions and solve optimization problems<\/div><\/div><\/div><div class=\"unit\" id=\"unit-7\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 7<\/div><h2>Integration<\/h2><p>Antiderivatives and Area<\/p><\/div><div class=\"unit-body\"><h3>Indefinite Integral<\/h3><p>\u222bf(x)dx = F(x) + C, where F'(x) = f(x)<\/p><h3>Integration Rules<\/h3><ul><li>\u222bx\u207f dx = x\u207f\u207a\u00b9\/(n+1) + C<\/li><li>\u222be\u02e3 dx = e\u02e3 + C<\/li><li>\u222b(1\/x) dx = ln|x| + C<\/li><li>\u222bsin x dx = -cos x + C<\/li><li>\u222bcos x dx = sin x + C<\/li><\/ul><h3>Definite Integral<\/h3><p>\u222b\u2090\u1d47 f(x)dx = F(b) - F(a)<\/p><h3>Integration Techniques<\/h3><ul><li><strong>Substitution:<\/strong> u-substitution<\/li><li><strong>Parts:<\/strong> \u222bu dv = uv - \u222bv du<\/li><\/ul><h3>Applications<\/h3><ul><li>Finding area under curves<\/li><li>Volume calculations<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Integrate functions and calculate areas<\/div><\/div><\/div><div class=\"unit\" id=\"unit-8\"><div class=\"unit-header\"><div class=\"unit-num-badge\">Unit 8<\/div><h2>Series and Applications<\/h2><p>Sequences and Their Applications<\/p><\/div><div class=\"unit-body\"><h3>Sequences<\/h3><ul><li><strong>Arithmetic:<\/strong> a\u2099 = a\u2081 + (n-1)d<\/li><li><strong>Geometric:<\/strong> a\u2099 = a\u2081 \u00b7 r\u207f\u207b\u00b9<\/li><\/ul><h3>Series Sums<\/h3><ul><li><strong>Arithmetic Series:<\/strong> S\u2099 = n\/2(a\u2081 + a\u2099)<\/li><li><strong>Geometric Series:<\/strong> S\u2099 = a\u2081(1-r\u207f)\/(1-r)<\/li><\/ul><h3>Infinite Series<\/h3><ul><li>Convergent series: Sum approaches a limit<\/li><li>Divergent series: Sum does not converge<\/li><\/ul><h3>Taylor Series<\/h3><p>f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)\u00b2\/2! + ...<\/p><h3>Applications<\/h3><ul><li>Physics problems involving motion<\/li><li>Engineering calculations<\/li><li>Finance and economics<\/li><\/ul><div class=\"practice\"><strong>\u270f\ufe0f Practice:<\/strong> Work with series and apply to real problems<\/div><\/div><\/div><div class=\"congrats\"><h2>\u2728 Congratulations!<\/h2><p>You've completed Mathematics! Master numbers and solve the world.<\/p><\/div><\/main><\/div>\r\n<button id=\"back-top\" onclick=\"window.scrollTo({top:0,behavior:'smooth'})\">\u2191<\/button>\r\n<script>const progress=document.getElementById('progress');window.addEventListener('scroll',()=>{const scrollPercent=(window.scrollY\/(document.documentElement.scrollHeight-window.innerHeight))*100;progress.style.width=scrollPercent+'%';document.getElementById('back-top').classList.toggle('visible',window.scrollY>400);});document.querySelectorAll('.unit').forEach(unit=>{const observer=new IntersectionObserver(entries=>{entries.forEach(entry=>{if(entry.isIntersecting){const id=entry.target.id;document.querySelectorAll('.toc-list a').forEach(a=>{a.classList.toggle('active',a.href.endsWith(id));});}});},{rootMargin:'-20% 0px -70% 0px'});observer.observe(unit);});<\/script>\r\n<\/body>\r\n<\/html>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Mathematics (113) \u2013 DAE Mechanical 1st Year \u2013 Pak Notes Hub Pak Notes HubAlgebraTrigonometryCalculus \u2211 DAE Mechanical \u2014 1st Year \u2014 Subject Code 113 MathematicsComplete Notes \u2014 Easy English Algebra \u00b7 Trigonometry \u00b7 Calculus \u00b7 Vectors \u00b7 Complete Curriculum Algebra & FunctionsTrigonometryCalculus \ud83d\udcd0 Table of Contents 1 Algebra 2 Functions 3 Trigonometry 4 Vectors 5 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"page_builder":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-645","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts\/645","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=645"}],"version-history":[{"count":4,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts\/645\/revisions"}],"predecessor-version":[{"id":649,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=\/wp\/v2\/posts\/645\/revisions\/649"}],"wp:attachment":[{"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=645"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=645"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/paknoteshub.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=645"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}