Master derivatives, integrals, and limits
136
Sub Topics
240
MCQs
84
MCOs
165
True/False
99
Fill Blanks
44
Rearrange
81
Matching
44
Comprehensions
87
Flashcard Decks
Curriculum
What You'll Learn
01 Functions and Limits 4 topics
1 Review of Functions
- Domain and Range
- Function Notation
- Common Functions (Polynomial, Rational, Exponential, Logarithmic, Trigonometric)
- Composition of Functions
- Inverse Functions
2 The Concept of Limit
- Intuitive Understanding of Limits
- One-sided Limits
- Infinite Limits
- Limits at Infinity
3 Calculating Limits
- Direct Substitution
- Factoring Techniques
- Rationalization
- Using Trigonometric Identities
- Squeeze Theorem
4 Continuity
- Definition of Continuity
- Types of Discontinuities
- Intermediate Value Theorem
- Extreme Value Theorem
02 Derivatives 5 topics
1 Definition of the Derivative
- Rate of Change
- Slope of a Tangent Line
- Formal Definition (Limit of Difference Quotient)
- One-sided Derivatives
2 Differentiation Rules
- Constant Rule
- Power Rule
- Sum and Difference Rules
- Product Rule
- Quotient Rule
- Chain Rule
3 Derivatives of Specific Functions
- Polynomial Functions
- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions
- Inverse Trigonometric Functions
4 Implicit Differentiation
- Definition and Technique
- Applications to Related Rates
5 Logarithmic Differentiation
- Definition and Technique
- Applications to Complex Functions
03 Applications of Derivatives 8 topics
1 Linear Approximation and Differentials
- Tangent Line Approximation
- Differentials
- Error Estimation
2 Mean Value Theorem
- Rolle's Theorem
- Mean Value Theorem
- Applications
3 Monotonicity and Concavity
- Increasing and Decreasing Functions
- First Derivative Test
- Concavity and Inflection Points
- Second Derivative Test
4 Optimization Problems
- Absolute Extrema
- Applied Optimization
- Business and Economics Applications
- Geometry Applications
5 Related Rates
- Setting Up Related Rates Problems
- Solving Related Rates Problems
6 L'Hôpital's Rule
- Indeterminate Forms 0/0 and ∞/∞
- Other Indeterminate Forms
7 Newton's Method
- Algorithm and Implementation
- Error Analysis and Convergence
8 Curve Sketching
- Complete Analysis of Functions
- Asymptotes (Vertical, Horizontal, Oblique)
04 Integration 5 topics
1 Antiderivatives and Indefinite Integrals
- Definition and Notation
- Basic Integration Rules
- Initial Value Problems
2 Area Under a Curve
- Riemann Sums
- Definition of the Definite Integral
- Properties of Definite Integrals
3 Fundamental Theorem of Calculus
- First Fundamental Theorem
- Second Fundamental Theorem
- Applications
4 Integration Techniques
- Substitution Method
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Partial Fractions
5 Numerical Integration
- Trapezoidal Rule
- Simpson's Rule
- Error Estimation
05 Applications of Integration 6 topics
1 Area Between Curves
- Finding Areas in the Plane
- Areas with Respect to y
2 Volume
- Disk Method
- Washer Method
- Shell Method
- Volumes of Known Geometric Solids
3 Arc Length and Surface Area
- Arc Length Formula
- Surface Area of Revolution
4 Work and Fluid Pressure
- Work Done by a Variable Force
- Hydrostatic Pressure and Force
5 Center of Mass and Moments
- Moments and Center of Mass for Discrete Systems
- Moments and Center of Mass for Continuous Systems
- Theorems of Pappus and Guldinus
6 Improper Integrals
- Infinite Limits of Integration
- Discontinuous Integrands
- Convergence Tests
06 Differential Equations 4 topics
1 Basic Concepts
- Definition and Classification
- Solutions and Initial Value Problems
2 Separable Equations
- Method of Separation of Variables
- Applications
3 First-Order Linear Differential Equations
- Integrating Factor Method
- Applications
4 Models Using First-Order Differential Equations
- Exponential Growth and Decay
- Newton's Law of Cooling
- Mixture Problems
- Population Models
07 Parametric Equations and Polar Coordinates 4 topics
1 Parametric Curves
- Definition and Graphing
- Eliminating the Parameter
2 Calculus with Parametric Curves
- Derivatives of Parametric Equations
- Arc Length for Parametric Curves
- Area Enclosed by Parametric Curves
3 Polar Coordinates
- Definition and Conversion
- Graphing in Polar Coordinates
- Common Polar Curves
4 Calculus in Polar Coordinates
- Derivatives in Polar Form
- Area in Polar Coordinates
- Arc Length in Polar Coordinates
08 Sequences and Series 6 topics
1 Sequences
- Definition and Notation
- Limits of Sequences
- Monotonic and Bounded Sequences
2 Infinite Series
- Definition and Notation
- Geometric Series
- Telescoping Series
- Harmonic Series
3 Tests for Convergence
- Divergence Test
- Integral Test
- Comparison Tests
- Ratio Test
- Root Test
- Alternating Series Test
4 Absolute and Conditional Convergence
- Definition and Properties
- Rearrangements of Series
5 Power Series
- Definition and Radius of Convergence
- Differentiation and Integration of Power Series
- Taylor and Maclaurin Series
- Common Maclaurin Series
6 Applications of Taylor Series
- Approximating Functions
- Error Estimation
- Computing Limits
- Evaluating Definite Integrals
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Calculus I
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