Advanced Calculus

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.

Advanced Level 0(0 Ratings) 0 Students enrolled
Created by Muhammad Usman Last updated Wed, 02-Aug-2023 English
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
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Differential equations
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Vector calculus
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Multiple integrals
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Multivariable functions and partial derivatives
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Vectors and analytic geometry in space
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Conic section
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Tests for convergence
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Infinite series
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Applications of derivatives
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Limits and continuity
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Nonlinear systems
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Lines and systems of equations
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Application of differential equations
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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$
  • Basic Calculus
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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
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  • 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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