Expert Online Calculus Tutors for K-12 Students

Precalculus

This is the most important course in highschool Mathematics because it connects the student's knowledge of Algebra and Geometry to start learning Calculus. It ends at Limits and Continuity which are building blocks for understanding the need for Calculus. Remember that it is very important to keep practicing Pre-Algebra, Algebra-I and Agebra-II concepts continuously to do good in Precalculus.

Functions

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Domain and Range of Functions by analyzing equations
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Composite Functions and their Domain and Range
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Inverse Functions
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Reducing Rational Polynomial functions
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End Behavior, Discontinuities and Graphs of Rational Polynomials
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Arithmetics with Rational Expressions

Trigonometry

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Quick recap of Algebra-II Trigonometry
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Trigonometric Identities based on Unit Circle Quadrants
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Law of sines and cosines
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Real world problems modeled using sinuisoidal models and solving sinusoidal equations
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Angle Addition Identities

Complex Numbers

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Complex Plane, Distance and Midpoints of Complex Numbers
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Complex Conjugates and Division of Complex Numbers
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Modulus and Argument of Complex Numbers
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Polar Form of Complex Numbers, Multiplication and Division

Vectors

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Introduction to Vectors, Direction, Magnitude and Components of Vectors
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Vector Addition, Subtraction and Scalar Multiplication
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Adding and Subtracting Vectors in Magnitude Direction Form
Vector Word Problems

Matrices

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Introduction to Matrices for storing data
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Addition, Subtraction and Scalar Multiplication of Matrices and their Properties
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Matrices as transformation of the Plane
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Matrix to Matrix Multiplication
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Solving System of Equations using Matrices
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Matrix Inverse and Invertibility of Matrices

Series and Sequences

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Arithmetic Series and related formulas, Word Problems
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Geometric Series and related formulas, Word Problems
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Binomial Theorem and related problems

Probability and Combinatorics

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Introduction to Probability using real world examples
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Introduction to Venn Diagrams and concepts of Unions and Intersections
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Multiplication Rule in Probability
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Formula of Combinations and Word Problems
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Word Problems with both Probability and Combinations
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Introduction to Probability Distributions, Decisions in Probability using Expected Value

Limits and Continuity

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Introduction to Concept of Limits and need for study of continuity of functions, the limit notation
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Estimating limits from Tables and Graphs
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Determining Limits using Algebraic Properties, Squeeze Theorem
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Introduction to discontinuities, its types and how they can be removed
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Horizontal and Vertical Asymptotes and their connection to Rational Functions
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Immediate Value Theorem

Derivatives

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Difference between Average and Instantaneous Rate of Change in Functions
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Definitation of Derivatives and their estimation
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Idea of differentiability v/s continuity
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Various rules for differentiation functions including Chain Rule, Differentiations of important functions
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Implicit Differentiation Problems
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Second derivative problems

Applications of Derivatives

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Graphical Understanding of derivative of a function
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Introduction to Related Rates
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L'Hopital's Rule and problems related to it
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Analyzing functions using Mean Value Theorem, Extreme Value Theorem and Critical Points like Relative and Absolute Extrema
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Concavity and Infection Points, Second Derivative Test
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Sketching graphs, connections between first, second and third derivatives
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Calculator based practice

Integrals

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Introduction to idea of Integration, Approximations using Reimann Sums and definition of Integrals
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Fundamental Theorem of Calculus and Interpreting behavior of accumulating functions
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Indefinite and Definite Integrals, their properties and common functions
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Integrating using different methods like substitution, long division, completing the square and trigonometric identities

Differential Equations

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Introduction and being able to verify solutions of given differential equations
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Sketching slope fields and reasoning questions
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Exponential models based on real world problems

Applications of Integrals

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Average value based problems
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Area between different curves and X Axis or Y Axis
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Volume of different 3D objects using double integration and Washer Method
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Calculator based practice

Parametric Equations, Polar Coordinates

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Introduction to Parametric Equations and their differentiation
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Finding arc lengths of curves given by parametric equations
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Introduction to Vector Valued Functions and their differentiation
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Defining Polar Coordinates and their differentiation
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Finding the area of a polar region
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Calculator based Practice

Infinite Sequence and Series

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Convergent and Divergent Series
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Tests for divergence and convergence like nth-term test, Integral test, Comparison test, Alternating series test, Ratio test
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Absolute and Conditional Convergence
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Lagrange error bound, Radius and Interval of convergence of Power series
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Representing functions as power series

Calculus

Precalculus is an important pre-requisite for Calculus. This course dives into new concepts of Derivatives and Integrals based on Limits. A good understanding of discontinuities, limits and functions in general is very good to have before delving into Calculus. A good sense of graphical representations of different functions is a handy tool at this point in Mathematics.

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