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# Calculus

- Introduction to Limits
- Determining One Sided Limits
- Determining When a Limit does not Exist
- Determining Limits Algebraically
- Infinite Limits and Vertical Asymptotes
- Limits at Infinity and Horizontal Asymptotes
- Definition of Continuity at a Point
- Classifying Topics of Discontinuity (removable vs. non-removable)
- Determining Limits Graphically
- Formal Definition of a Limit at a Point
- Continuous Functions
- Intemediate Value Theorem

- Tangent Line to a Curve
- Normal Line to a Tangent
- Slope of a Curve at a Point
- Average Velocity
- Instantaneous Velocity
- Limit Definition of Derivative
- First Principles Example 1: x²
- First Principles Example 2: x³
- First Principles Example 3: square root of x
- Standard Notation and Terminology
- Differentiable vs. Non-differentiable Functions
- Rate of Change of a Function
- Average Rate of Change Over an Interval
- Instantaneous Rate of Change at a Point

- Limits Involving Trigonometric Functions
- Intuitive Approach to the derivative of y=sin(x)
- Derivative Rules for y=cos(x) and y=tan(x)
- Differentiating sin(x) from First Principles
- Special Limits Involving sin(x), x, and tan(x)
- Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure
- Derivatives of y=sec(x), y=cot(x), y= csc(x)
- Differentiating Inverse Trigonometric Functions

- Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)
- Identifying Stationary Points (Critical Points) for a Function
- Identifying Turning Points (Local Extrema) for a Function
- Classifying Critical Points and Extreme Values for a Function
- Mean Value Theorem for Continuous Functions

- Relationship between First and Second Derivatives of a Function
- Analyzing Concavity of a Function
- Notation for the Second Derivative
- Determining Points of Inflection for a Function
- First Derivative Test vs Second Derivative Test for Local Extrema
- The special case of x⁴
- Critical Points of Inflection
- Application of the Second Derivative (Acceleration)
- Examples of Curve Sketching

- Sigma Notation
- Integration: the Area Problem
- Formal Definition of the Definite Integral
- Definite and indefinite integrals
- Integrals of Polynomial functions
- Determining Basic Rates of Change Using Integrals
- Integrals of Trigonometric Functions
- Integrals of Exponential Functions
- Integrals of Rational Functions
- The Fundamental Theorem of Calculus
- Basic Properties of Definite Integrals

- Solving Separable Differential Equations
- Slope Fields
- Exponential Growth and Decay Models
- Logistic Growth Models
- Net Change: Motion on a Line
- Determining the Surface Area of a Solid of Revolution
- Determining the Length of a Curve
- Determining the Volume of a Solid of Revolution
- Determining Work and Fluid Force
- The Average Value of a Function

- Introduction to Power Series
- Differentiating and Integrating Power Series
- Constructing a Taylor Series
- Constructing a Maclaurin Series
- Lagrange Form of the Remainder Term in a Taylor Series
- Determining the Radius and Interval of Convergence for a Power Series
- Applications of Power Series
- Power Series Representations of Functions
- Power Series and Exact Values of Numerical Series
- Power Series and Estimation of Integrals
- Power Series and Limits
- Product of Power Series
- Binomial Series
- Power Series Solutions of Differential Equations

- Geometric Series
- Nth Term Test for Divergence of an Infinite Series
- Direct Comparison Test for Convergence of an Infinite Series
- Ratio Test for Convergence of an Infinite Series
- Integral Test for Convergence of an Infinite Series
- Limit Comparison Test for Convergence of an Infinite Series
- Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
- Infinite Sequences
- Root Test for for Convergence of an Infinite Series
- Infinite Series
- Strategies to Test an Infinite Series for Convergence
- Harmonic Series
- Indeterminate Forms and de L'hospital's Rule
- Partial Sums of Infinite Series

## Delving into the Essence of Calculus

Calculus stands as a foundational pillar of modern mathematics, offering a systematic means for calculating the dynamics of change and accumulation. It bifurcates into differential calculus, concerned with understanding the instantaneous rate of change, and integral calculus, focused on the aggregation of quantities and the computations of areas under curves.

This mathematical discipline is instrumental in a multitude of fields, from physics and engineering to economics and biology, facilitating the modeling of continuous change and enabling the exploration of systems and processes that evolve over time.

## Why You Should Study Calculus?

Studying calculus goes beyond academic duty, offering insights into the mechanics of nature and technology. It enhances critical thinking and problem-solving skills, which are crucial for tackling complex challenges.

With applications in physics, engineering, economics, and biology, calculus models dynamic systems, fostering intellectual growth and a deeper appreciation for mathematics. Proficiency in calculus opens doors in technology, science, and finance, sectors where analytical skills are highly prized. Mastering calculus equips individuals with a versatile toolset for success, fostering innovation and continuous learning. Ultimately, calculus serves as a gateway to diverse opportunities, empowering individuals to understand and shape the world around them.

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