Oswaldo Jiménez

Université Bordeaux I

Biography

I am a mathematician graduated at Central University of Venezuela (1981-1986). At this same University, I got a MSc degree in Mathematics (1986-1989). Then, I got a PhD (Doctorate) in Mechanical Engineering at Université de Bordeaux I, in France (1991-1995). I have worked at the Venezuelan Petroleum Company for almost 15 years (1989-2003), doing applied mathematical and numerical modelling, both for petroleum reservoirs and sedimentary basins. I have also worked for more than 16 years at the University, teaching numerical analysis, differential equations and scientific computing. I have been involved intensively in the mathematical and numerical modelling of porous media phenomena, through the numerical solution of ordinary and partial differential equations, using several numerical methods such as finite differences, finite elements, finite volumes, mimetics, among others. I have also been greatly involved in the process of scientific programming using some computer languages such as C, C++, Fortran, Python, Visual Basic, Matlab, Java, and others. I like solving complex problems by using mathematical, numerical and algorithmic tools.

Education

Phd Mechanical Engineering

Université Bordeaux I

MS Mathematics

Universidad Central de Venezuela

BS Mathematics

Universidad Central de Venezuela

Topics Covered

Functions

Derivatives

Differentiation

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Limits

Sequences

Series

Series Tests

Differential Equations

Applications of the Derivative

Integrals

Integration

Integration Techniques

Power Series

Powers and Polynomial

Trig Integrals

Trig Substitution

Matrices

Introduction to Matrices

Taylor Series

Equations and Inequalities

Functions

Linear Functions

Polar Coordinates

An Introduction to Geometry

Introduction to Conic Section

Continuous Functions

The Language of Algebra

Complex Numbers

Exponential and Logarithmic Functions

Quadratic Functions

Polynomials

Rational Functions

Trigonometry

Graphs and Statistics

Parametric Equations

Applications of Integration

Introduction to Trigonometry

Systems of Equations and Inequalities

Algebra Topics That are Reviewed at the Start of the Semester

Functions of Several Variables

Foundations for Geometry

Geometry Basics

Right Triangles

Circles

Introduction to Conic Sections

Introduction to Sequences and Series

Introduction to Combinatorics and Probability

Volume

Vectors

Oswaldo's Textbook Answer Videos

15:13

Calculus: Early Transcendentals

The point $ P(2, -1) $ lies on the curve $ y = 1/(1-x) $.

(a) If $ Q $ is the point $ (x, 1/(1-x)) $, use your calculator to find the slope of the secant line $ PQ $ (correct to six decimal places) for the following values of $ x $:
(i) $ 1.5 $ (ii) $ 1.9 $ (iii) $ 1.99 $ (iv) $ 1.999 $
(v) $ 2.5 $ (vi) $ 2.1 $ (vii) $ 2.01 $ (viii) $ 2.001 $

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $ P(2, -1) $.

Chapter 2: Limits and Derivatives

Section 1: The Tangent and Velocity Problems

13:09

Calculus: Early Transcendentals

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).

$ \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} $,
$ h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001 $

Chapter 2: Limits and Derivatives

Section 2: The Limit of a Function

26:16

Calculus: Early Transcendentals

(a) Evaluate $ h(x) = (\tan x - x)/x^3 $ for $ x $ = 1, 0.5, 0.1, 0.05, 0.01, and 0.005.

(b) Guess the value of $ \displaystyle \lim_{x \to 0}\frac{\tan x - x}{x^3} $.

(c) Evaluate $ h(x) $ for successively smaller values of $ x $ until you finally reach a value of 0 for $ h(x) $. Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 a method of evaluating this limit will be explained.)

(d) Graph the function $ h $ in the viewing rectangle $ [-1, 1] $ by $ [0, 1] $. Then zoom in toward the point where the graph crosses the $ y $ -axis to estimate the limit of $ h(x) $ as $ x $ approaches 0. Continue to zoom in until you observe distortions in the graph of $ h $. Compare with the results of part (c).

Chapter 2: Limits and Derivatives

Section 2: The Limit of a Function

14:04

Calculus: Early Transcendentals

Let $ g(x) = \text{sgn}(\sin x) $.

(a) Find each of the following limits or explain why it does not exist.
(i) $ \displaystyle \lim_{x \to 0^+}g(x) $
(ii) $ \displaystyle \lim_{x \to 0^-}g(x) $
(iii) $ \displaystyle \lim_{x \to 0}g(x) $
(iv) $ \displaystyle \lim_{x \to \pi^+}g(x) $
(v) $ \displaystyle \lim_{x \to \pi^-}g(x) $
(vi) $ \displaystyle \lim_{x \to \pi}g(x) $

(b) For which values of $ a $ does $ \displaystyle \lim_{x \to a}g(x) $ not exist?
(c) Sketch a graph of $ g $.

Chapter 2: Limits and Derivatives

Section 3: Calculating Limits Using the Limit Laws

06:45

Calculus: Early Transcendentals

(a) Find a number $ \delta $ such that if $ | x - 2 | < \delta $, then $ | 4x - 8 | < \varepsilon $, where $ \varepsilon = 0.1 $.

(b) Repeat part (a) with $ \varepsilon = 0.01 $.

Chapter 2: Limits and Derivatives

Section 4: The Precise Definition of a Limit

08:05

Calculus: Early Transcendentals

Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit.

$ \displaystyle \lim_{x \to 4}\frac{x^2 - 2x - 8}{x - 4} = 6 $

Chapter 2: Limits and Derivatives

Section 4: The Precise Definition of a Limit

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