Why do I need to learn about calculus?

Calculus is a fundamental tool for understanding modern theories and techniques to create software such as artificial intelligence, machine learning, deep learning, data mining, security, digital imagine processing and natural language processing.

What can I do after finishing learning about calculus?

You will then be prepared to be able to learn modern theories and techniques to create security, data mining, image processing or natural language processing software.

What should I do now?

First, please audit the course and read its lecture notes to grasp the core concepts of single-variable calculus: MIT 18.01 – Single Variable Calculus, Fall 2007 (Lecture Notes).

When you audit this course, refer to this book whenever you have difficulty understanding any of the lectures: George F. Simmons (1996). Calculus With Analytic Geometry. McGraw-Hill.

Alternatively, you can read one of the books below:

• George F. Simmons (1996). Calculus With Analytic Geometry. McGraw-Hill
• C. Henry Edwards David E. Penney (2008). Calculus – Early Transcendentals. Pearson
• George B. Thomas et al. (2018). Thomas’ Calculus: Early Transcendentals. Pearson Education
• James Stewart et al. (2020). Calculus: Early Transcendentals. Cengage Learning

After that, please audit the course and read its lecture notes to grasp the core concepts of multivariable calculus: MIT 18.02 – Multivariable Calculus, Fall 2007 (Lecture Notes).

You will need some Linear Algebra knowledge (specifically Inverse Matrix and Determinant) to understand Multivariable Calculus. You will need some knowledge of linear algebra (specifically inverse matrices and determinants) to understand multivariable calculus. Therefore, please learn the basics of linear algebra at the same time.

After that, please watch the Highlights of Calculus videos, to review many core concepts of calculus. Most mathematical concepts should be learned several times using different approaches in order to fully understand their problems, solutions, and applications.

After that, please audit the course and read its readings to grasp the core concepts of differential equations: MIT 18.03 – Differential Equations, Spring 2006 (Readings).

When you audit this course, refer to this book whenever you have difficulty understanding any of the lectures: C. Henry Edwards and David E. Penney (2013). Elementary Differential Equations with Boundary Value Problems. Pearson Education.

What is the difference between calculus and analysis?

Calculus means a method of calculation. Calculus is about differentiation and integration.

Real analysis includes calculus, and other topics that may not be of interest to engineers but of interest to pure mathematicians such as measure theory, lebesgue integral, topology, functional analysis, complex analysis, PDE, ODE, proofs of theorems.

What does early transcendentals mean?

Transcendentals in this context refers to functions like the exponential, logarithmic, and trigonometric functions.

The early transcendentals approach means that the book introduces polynomial, rational functions, exponential, logarithmic, and trigonometric functions at the beginning, then use them as examples when developing differential calculus. This approach is good for students who do not need to take much rigorous math.

The classical approach is the late transcendentals. It means that the book develops differential calculus using only polynomials and rational functions as examples, then introduces the other functions afterwards. This approach is good for students who need to understand more rigorous definitions of the transcendental functions.

Single Variable Calculus Terminology Review:

• Slope.
• Derivative.
• Rate of Change.
• Limit.
• Continuity.
• Chain Rule.
• Implicit Differentiation.
• Linear Approximations.
• Quadratic Approximations.
• Critical Point.
• Newton’s Method.
• Mean Value Theorem.
• Differentials.
• Antiderivatives.
• Differential Equations.
• Separation of Variables.
• First Fundamental Theorem of Calculus.
• Indeterminate Forms.
• L’Hospital’s Rule.
• Improper Integrals.
• Infinite Series.
• Taylor’s Series.
• Taylor’s Formula.
• Power Series.
• Geometric Series.
• Euler’s Formula.

Multivariable Calculus Terminology Review:

• Vectors.
• Dot Product.
• Cross Product.
• Inverse Matrix.
• Determinant.
• Equations of Planes: ax + by + cz = d
• Parametric Equations = as trajectory of a moving point.
• Velocity Vector.
• Acceleration Vector.
• Level Curve.
• Tangent Plane.
• Saddle Point.
• Functions of Several Variables.
• Partial Derivatives.
• Second Derivatives.
• Second Derivative Test.
• Differentials.
• Gradients.
• Directional Derivatives.
• Lagrange Multipliers.

Differential Equation Terminology Review:

• Isocline (equal slope): a line which joins neighboring points with the same gradient.
• Direction Fields.
• Integral Curve: The graph of a particular solution of a differential equation.
• IVP: Initial Value Problem.
• Euler’s Numerical Method.
• Linear First Order ODE Standard Form: y′ + p(x)y = q(x)
• Integrating Factor or Euler Multiplier: The method is based on (ux)’ = ux’ + u’x.
• Substitution: to change variables to end up with a simpler equation to solve.
• Bernoulli Equations: y′ + p(x)y = q(x)yⁿ.
• Homogeneous Equations: y′ = F (y/x)
• Autonomous Equations: dx/dt = f(x). If we think of as time, the naming comes from the fact that the equation is independent of time.

Matrix Calculus Terminology Review:

• Matrix Derivatives.

After finishing calculus, please click on Topic 18 – Linear Algebra to continue.