# Topic 17 – Calculus

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?

Please watch this MIT 18.01 – Single Variable Calculus, Fall 2007 course (Lecture Notes).  When you watch this course please refer to this George F. Simmons (1996). Calculus With Analytic Geometry. McGraw-Hill book when you have difficulties with understanding some lectures.

After that please watch this MIT 18.02 – Multivariable Calculus, Fall 2007 course (Lecture Notes). You will need some Linear Algebra knowledge (specifically Inverse Matrix and Determinant) to understand Multivariable Calculus.

After that please watch this Highlights of Calculus course to review many core concepts of Calculus.

After that please watch this MIT 18.03 – Differential Equations, Spring 2006 course (Readings). When you watch this course please refer to this C. Henry Edwards and David E. Penney (2013). Elementary Differential Equations with Boundary Value Problems. Pearson Education book when you have difficulties with understanding some lectures.

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.
• 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.

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.
• Functions of Several Variables.
• Partial Derivatives.
• Second Derivatives.
• Second Derivative Test.
• Differentials.
• Directional Derivatives.
• Lagrange Multipliers.
• Power Series.
• Geometric Series.
• Euler’s Formula.

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.

After finishing learning about calculus please click Topic 18 – Linear Algebra to continue.

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