Calculus serves as the foundation of advanced subjects in all areas of mathematics. The objective of this course is to introduce students to the fundamental concepts of limit, continuity, differential and integral calculus of functions of one variable, it focuses on techniques of integration and applications of integrals. The course also aims at introducing the students to infinite series, parametric curves and polar coordinates. The student would be introduced to vector calculus, the calculus of multivariable functions, and double and triple integrals along with their applications. Also providing a strong foundation and applications of Ordinary Differential Equations is the goal of the course.

What will i learn?

- Define and find the limits and continuity of the problems
- Apply the concept of derivatives to solve the problems
- Determine the slope, maxima and minima of functions
- Find the integration of various functions
- Discuss the convergence or divergence of series
- Evaluate the area between curves, volume and arc length
- Identify and apply the basic concepts of Vectors and Calculus
- Demonstrate various concepts regarding Multivariable functions and partial derivatives
- Evaluate the double and triple integrals of the functions
- Identify types of differential equations & their solutions
- Apply techniques to solve 1st order ordinary differential equations and various models depicted by 1st order ordinary differential equations
- Apply methods to solve 2nd and higher order ordinary differential equations and various models depicted by higher order ordinary differential equations

Curriculum for this course

15 Topics

Functions and graphs

0 Sub Topics

Differential equations

0 Sub Topics

Vector calculus

0 Sub Topics

Multiple integrals

0 Sub Topics

Multivariable functions and partial derivatives

0 Sub Topics

Vectors and analytic geometry in space

0 Sub Topics

Conic section

0 Sub Topics

Tests for convergence

0 Sub Topics

Infinite series

0 Sub Topics

Applications of derivatives

0 Sub Topics

Derivatives

0 Sub Topics

Limits and continuity

0 Sub Topics

Nonlinear systems

0 Sub Topics

Lines and systems of equations

0 Sub Topics

Application of differential equations

0 Sub Topics

Packages

Title | Lectures | Price |
---|---|---|

Inquiry Based Session-1 lesson | 1 | 25$ |

Regular Package/ 4 lesson | 4 | 90$ |

Standard Package/8 Lesson | 8 | 180$ |

Premium Package/12 | 12 | 270$ |

Fast Track/16 Lesson | 16 | 360$ |

Requirements

- Basic Calculus

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Description

**Functions
and graphs:** Domain and range
of a function. Examples: polynomial, rational, piecewise defined functions,
absolute value of functions and evaluation of such functions. Operations with
functions: sum, product, quotient and composition. Graphs of functions: linear,
quadratic, piecewise defined functions.

**Lines and systems of equations:** Equation of a straight line, slope and intercept of a line, parallel
and perpendicular lines. Systems of linear equations, solution of system of
linear equations.

**Nonlinear systems:** at least one quadratic equation.

**Limits and continuity:** Functions, limit of a function. Graphical approach. Properties of
limits. Theorems of limits. Limits of polynomials, rational and transcendental
functions. Limits at infinity, infinite limits, one-sided limits. Continuity.

**Derivatives: **Definition, techniques of differentiation. Derivatives of polynomials
and rational, exponential, logarithmic and trigonometric functions. The chain
rule. Implicit differentiation. Rates of change in natural and social sciences.
Related rates. Linear approximations and differentials. Higher derivatives,
Leibnitz's theorem.

**Applications of derivatives:** Increasing and decreasing functions. Relative extrema and optimization.
First derivative test for relative extrema. Convexity and point of inflection.
The second derivative test for extrema. Curve sketching. Mean value theorems.
Indeterminate forms and L'Hopitals rule. Inverse functions and their
derivatives.

**Infinite
series:** Sequences and series.
Convergence and absolute convergence.

**Tests
for convergence:**
divergence test, integral test, p-series test, comparison test, limit
comparison test, alternating series test, ratio test, root test. Power series.
Convergence of power series. Representation of functions as power series.
Differentiation and integration of power series. Taylor and MacLaurin series.
Approximations by Taylor polynomials.

**Conic
section:** Parameterized curves and
polar coordinates. Curves defined by parametric equations. Calculus with parametric
curves. Tangents, areas, arc length. Polar coordinates. Polar curves, tangents
to polar curves, Areas and arc length in polar coordinates. Gamma functions

**Vectors and analytic geometry in space:** Coordinate system. Rectangular, cylindrical and spherical coordinates. The
dot product, the cross product. Equations of lines and planes. Quadric
surfaces. Vector-valued functions: Vector-valued functions and space curves.
Derivatives and integrals of vector valued functions. Arc length. Curvature,
normal and Binormal vectors.

**Multivariable functions and partial
derivatives:** Functions of several variables. Limits
and Continuity. Partial derivatives, Composition and chain rule. Directional
derivatives and the gradient vector. Implicit function theorem for several
variables. Maximum and minimum values. Optimization problems. Lagrange
Multipliers.

**Multiple integrals:** Double integrals over rectangular domains and iterated integrals.
Non-rectangular domains. Double integrals in polar coordinates. Triple
integrals in rectangular, cylindrical and spherical coordinates. Applications
of double and triple integrals. Change of variables in multiple integrals.

**Vector calculus:** Vector fields. Line integrals. Green's theorem. Curl and divergence.
Surface integrals over scalar and vector fields. Divergence theorem. Stokes'
theorem.

**Differential equations:** Introduction to
differential equations. Techniques to solve various types of 1st order ordinary
differential equations. 2nd and higher order differential equations. Methods to
solve 2nd-order differential equations;

**Application
of differential equations:** Applications
of first-order differential equations. Applications of 2nd order differential
equations and its analysis.

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About the instructor

**0**Reviews**0**Students**6**Courses

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Hi, I am Dr. Muhammad Usman graduated from Peking University in the field of applied and computational mathematics with distinguished student. I also completed my postdoc in field of computational mathematics. I have numerous research articles published in well-reputed international journals. Therefore, I have rich experience in research and supervise project students. As for as my teaching career is concerned, I have 14 years’ experience in teaching. I have taught various courses at A-level, O-level, graduate and postgraduate level. In the recent past years, I involve extensively in teaching online at A-level and O-level. Positive feedback of my past students always encourage me to deliver more in the better way. I look forward to working together with you as partners in your child’s educational growth and development!!

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