Calculus Course Outlines

math-1109 MATH-1109 Differential and Integral Calculus

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Content Outline of the Two Comprehensive Notes

Note 1: Differentiation and Calculus Concepts

I. Foundations: Limit, Continuity, and Differentiability A. Limit 1. Definition of Limit 2. Mathematical Notation and Condition for Existence (LHL = RHL) B. Continuity 1. Definition and Mathematical Condition (${\lim }_{x\to a}f(x)=f(a)$) C. Differentiation / Derivative / Differentiability 1. Definition of Differential Coefficient 2. Fundamental Formula for the Derivative 3. Existence Condition (LHD = RHD) 4. Formulas for Left Hand Derivative (LHD) and Right Hand Derivative (RHD)

II. Advanced Differentiation Techniques A. Successive Differentiation 1. General concept of finding higher-order derivatives 2. Leibnitz’s Theorem (for the $n^{th}$ derivative of a product $UV$)

B. Partial Differentiation and Total Differentials 1. Definition of Partial Differentiation (differentiating with respect to one variable while holding others constant) 2. Total Differentials ($du$) (for functions of two and three variables) 3. Formula for $\frac{du}{dt}$ when $x$ and $y$ are functions of $t$ (Parametric Equation)

III. Homogeneous Functions and Euler’s Theorem A. Homogeneous Function Definition and Form ($x^n\Phi (y/x)$) B. Euler’s Theorem Formulas 1. For two variables 2. For three variables 3. For Composite Functions ($V=f(u)$) 4. Property of Derivatives of Homogeneous Functions

IV. Applications of Derivatives A. Maximum and Minimum 1. Definitions of Maximum and Minimum values 2. Determination Rules (Second Derivative Test) B. Indeterminate Forms and L’Hospital’s Theorem 1. Method for $\frac{0}{0}$ form 2. Methods for other indeterminate forms (e.g., $0\times \infty$) C. Tangent and Normal 1. Definitions of Tangent and Normal 2. Equations of Tangent and Normal for Explicit ($y=f(x)$) and Implicit ($f(x,y)=0$) Functions D. Subtangent and Subnormal 1. Definitions and Formulas for Subtangent and Subnormal E. Angle of Intersection 1. Formula for $\tan \alpha$ between two implicit curves 2. Conditions for Orthogonal Cut ($\alpha =90^{\circ }$) and Curves Touching ($\alpha =0^{\circ }$) F. Asymptotes 1. Definition of Asymptote 2. Finding Oblique Asymptotes ($y=mx+c$) (for distinct and repeated roots)

V. Expansion Functions and Theorems A. Rolle’s Theorem (Conditions and Conclusion) B. Mean Value Theorem (Conditions and Conclusion) C. Taylor’s Theorem (Conditions and Expansion Formula)

VI. Curvature A. Definitions: Angle of Contignance ($\delta \Psi$), Arc Length ($\delta s$), Average Curvature, and Curvature ($\kappa$) B. Radius of Curvature ($\rho$) Definition ($\rho =1/\kappa$) C. Formulas for $\rho$ (Explicit, Implicit, Parametric, and Polar Functions)

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Note 2: Integration and Its Applications

I. Indefinite Integration Techniques A. Standard Substitutions for Integrals Involving Linear and Quadratic Terms Under Radicals B. Substitutions for Trigonometric Integrals 1. Tangent Half-Angle Substitution ($\tan (x/2)=z$) 2. Derivative-Based Decomposition (Numerator = $m$(Denominator) + $n$(Derivative))

II. Definite Integration A. General Properties of Definite Integral (Five fundamental properties, including change of variable and periodic functions) B. Wallis’s Theorem 1. Formula for ${\int }_0^{\pi /2}{\sin }^{n}xdx$ (for $n$ even and $n$ odd)

III. Reduction Formulas A. Formula for $\int {\sin }^{n}xdx$ ($I_n$) B. Formula for $\int {\tan }^{n}xdx$ ($I_n$) C. Formula for $\int e^{ax}{\cos }^{n}xdx$ ($I_n$) D. Formula for $\int {\sin }^{m}x{\cos }^{n}xdx$ ($I_{m,n}$)

IV. Beta and Gamma Functions (Eulerian Integrals) A. Beta Function ($\text{B}(m,n)$) 1. Definition (First Eulerian Integral) 2. Alternative Forms (Integral from 0 to $\infty$, Trigonometric form) B. Gamma Function ($\Gamma (n)$) 1. Definition (Second Eulerian Integral) 2. Properties ($\Gamma (n+1)=n!$, $\Gamma (n+1)=n\Gamma (n)$) C. Relation between Beta and Gamma Functions 1. $\text{B}(m,n)=\frac{\Gamma (m)\Gamma (n)}{\Gamma (m+n)}$ 2. Derived formula for ${\int }_0^{\pi /2}{\sin }^p\theta {\cos }^q\theta d\theta$

V. Applications of Integration A. Area (Rectangular Coordinates) 1. Area bounded by $y=f(x)$ and the $x$-axis 2. Area bounded by $x=f(y)$ and the $y$-axis B. Length (Arc Length of Curves) 1. Formula for Explicit ($y=f(x)$ and $x=f(y)$) 2. Formula for Parametric ($x=\Phi (t),y=\Psi (t)$) 3. Formula for Polar ($r=f(\theta )$) C. Volume of Revolution 1. Revolution about the $x$-axis ($\int \pi y^{2}dx$) 2. Revolution about the $y$-axis ($\int \pi x^{2}dy$) 3. Polar Coordinates (about initial line and $\theta =\pi /2$) D. Surface Area of Revolution 1. Revolution about the $x$-axis ($\int 2\pi yds$) 2. Revolution about the $y$-axis ($\int 2\pi xds$) 3. Polar Coordinates (about the initial line) 4. Definitions of differential arc length ($ds$)