Advanced calculus with series and sequences
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Calculus II is available on both platforms
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128
Sub Topics
122
MCQs
47
MCOs
80
True/False
55
Fill Blanks
26
Rearrange
43
Matching
21
Comprehensions
41
Flashcard Decks
Curriculum
What You'll Learn
01 Applications of Integration 6 topics
1 Area Between Curves
- Finding areas in Cartesian coordinates
- Finding areas in polar coordinates
2 Volumes of Solids
- Disk method
- Washer method
- Shell method
3 Volumes by Cross-sections
- Known cross-sectional area
- Slicing technique
4 Arc Length and Surface Area
- Arc length of parametric curves
- Arc length of polar curves
- Surface area of revolution
5 Work and Fluid Pressure/Force
- Work done by a variable force
- Fluid pressure and force
6 Centers of Mass and Moments
- Center of mass for a system of particles
- Center of mass for a plane lamina
- Theorems of Pappus and Guldinus
02 Integration Techniques 6 topics
1 Basic Integration Formulas (Review)
- Power rule
- Exponential and logarithmic functions
- Trigonometric functions
2 Integration by Parts
- The formula and its derivation
- Tabular integration (Tabular method)
- Recursive applications
3 Trigonometric Integrals
- Powers of sine and cosine
- Products of sines and cosines
- Powers of secant and tangent
- Products of secants and tangents
4 Trigonometric Substitutions
- Substitution for expressions with √(a²-x²)
- Substitution for expressions with √(a²+x²)
- Substitution for expressions with √(x²-a²)
5 Partial Fractions
- Linear factors
- Repeated linear factors
- Irreducible quadratic factors
- Repeated irreducible quadratic factors
6 Integration by Substitution (Advanced Cases)
- Rational functions of sine and cosine
- Rational functions of exponentials
- Miscellaneous substitutions
03 Improper Integrals 3 topics
1 Integrals with Infinite Limits
- Type I improper integrals
- Convergence and divergence tests
2 Integrals with Discontinuous Integrands
- Type II improper integrals
- Comparison tests
3 Applications of Improper Integrals
- Probability distributions
- Laplace transforms
- Infinite series through improper integrals
04 Infinite Sequences and Series 7 topics
1 Sequences
- Definition and notation
- Limit of a sequence
- Bounded and monotonic sequences
2 Infinite Series
- Definition and notation
- Partial sums
- Convergence and divergence
3 The Integral Test
- Statement and proof
- Estimating sums with the integral test
4 Comparison Tests
- Direct comparison test
- Limit comparison test
5 Alternating Series
- Alternating series test
- Absolute and conditional convergence
- Estimating sums of alternating series
6 Ratio and Root Tests
- Ratio test
- Root test
- When to use each test
7 Power Series
- Definition and basic properties
- Radius and interval of convergence
- Term-by-term differentiation and integration
05 Taylor and Maclaurin Series 5 topics
1 Taylor Polynomials
- Definition and computation
- Remainder term
- Error estimation
2 Taylor Series
- Definition and examples
- Taylor's theorem
- Convergence of Taylor series
3 Maclaurin Series
- Common Maclaurin series
- Exponential, logarithmic, and trigonometric functions
- Binomial series
4 Working with Taylor Series
- Addition and subtraction
- Multiplication and division
- Composition of series
5 Applications of Taylor Series
- Approximating definite integrals
- Limits using Taylor series
- Indeterminate forms
06 Parametric Equations and Polar Coordinates 4 topics
1 Parametric Curves
- Parametric equations and their graphs
- Eliminating the parameter
- Smooth curves and cusps
2 Calculus with Parametric Curves
- Derivatives of parametric equations
- Second derivatives and concavity
- Arc length of parametric curves
- Area of surface of revolution
3 Polar Coordinates
- The polar coordinate system
- Converting between polar and Cartesian coordinates
- Graphs of polar equations
4 Calculus in Polar Coordinates
- Derivatives in polar form
- Areas in polar coordinates
- Arc length of polar curves
- Slope and tangent lines
07 Vectors and the Geometry of Space 4 topics
1 Vectors in the Plane
- Vector operations
- The dot product
- Projections and work
2 Three-Dimensional Coordinate Systems
- The 3D Cartesian coordinate system
- Distance and midpoint formulas
- Equations of spheres
3 Vectors in Space
- 3D vector operations
- Dot product in 3D
- Cross product
- Triple scalar product and volume
4 Lines and Planes in Space
- Vector, parametric, and symmetric equations of lines
- Vector and scalar equations of planes
- Distances between points, lines, and planes
- Angles between lines and planes
08 Vector-Valued Functions 4 topics
1 Vector Functions and Space Curves
- Definition and examples
- Limits and continuity
- Sketching space curves
2 Derivatives and Integrals of Vector Functions
- Differentiation rules
- Integration of vector functions
- Fundamental theorem for vector functions
3 Arc Length and Curvature
- Arc length parameter
- Curvature and radius of curvature
- Normal and binormal vectors
4 Motion in Space
- Position, velocity, and acceleration vectors
- Tangential and normal components
- Projectile motion
- Planetary motion and Kepler's laws
09 Differential Equations (Introduction) 4 topics
1 Basic Concepts of Differential Equations
- Classification of differential equations
- Solutions and initial value problems
- Direction fields
2 Separable Equations
- Method of separation of variables
- Applications to growth and decay
- Mixing problems
3 First-Order Linear Equations
- Integrating factor method
- Applications to circuits and mechanics
4 Applications of Differential Equations
- Population models
- Newton's law of cooling
- Logistic growth
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