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Math

I've been studying higher math on my own, and also at local colleges, and I spend a lot of time on Khan Academy (my profile is here).  These pages show a proof of the work I've done.


Update:  25 June 2015 (Completed Calculus III, Differential Equations, and Linear Algebra)

From January to June 2015, I completed Calculus III during the Spring semester and Differential Equations and Linear Algebra during the Summer semester.  I really wanted to get the math requirements behind me.

24 December 2014 (Completed Calculus II)

https://sites.google.com/site/athenaroberts1/home/education/math/41OZOp61dQL.jpg


Calculus II went well :-)  I found it very difficult and stressful but, I managed to pass now I'm moving on to Calculus III.

Concepts I practiced are:
Volumes Using Cross-Sections
Volumes Using Cylindrical Shells
Arc Length
Areas of Surfaces of Revolution
Work and Fluid Forces
Moments and Centers of Mass
The Logarithm Defined as an Integral
Exponential Change and Separable Differential Equations
Integration by Parts
Trigonometric Integrals
Trigonometric Substitutions
Integration of Rational Functions by Partial Fractions
Integral Tables and Computer Algebra Systems
Improper Integrals
Solutions, Slope Fields, and Euler's Method
Indeterminate Forms and L'Hopital's Rule
First-Order Linear Equations
Sequences
Infinite Series
The Integral Test
Comparison Tests
The Ratio and Root Tests
Alternating Series, Absolute and Conditional Convergence
Power Series
Taylor and Maclaurin Series
Convergence of Taylor Series
The Binomial Series and Applications of Taylor Series
Polar Coordinates
Graphing in Polar Coordinates
Areas and Lengths in Polar Coordinates
Vectors
The Dot Product

22 August 2014 (Completed Sections 7.1 to 7.4 "Integration Techniques" - Calculus: Early Transcendental Functions by Roland Larson)



Since this has been going so well, I decided to enroll in Calculus II at a local community college so I can get college credit for what I've learned.  Being officially recognized as an aerospace engineering student has some advantages compared with going it alone (eg. access to internships),  Section 7.4 was as far as I got before my class started.

Concepts I practiced are:
Basic Integration Rules
Fitting Integrands to Basic Rules
Integration by Parts
Integrals Involving Powers of Sine and Cosine
Integrals Involving Powers of Secant and Tangent
Integrals Involving Sine-Cosine Products with Different Angles
Trigonometric Substitution

12 August 2014 (Completed Chapter 6 "Applications of Integration" - Calculus: Early Transcendental Functions by Roland Larson)



Concepts I practiced are:
Area of a Region Between Two Curves
Area of a Region Between Intersecting Curves
The Disc Method
The Washer Method
Solids with Known Cross-Sections
The Shell Method
Comparison of Disc and Shell Methods
Arc Length
Area of a Surface of Revolution
Work Done by a Constant Force
Work Done by a Variable Force
Fluid Pressure and Fluid Force
Mass
Center of Mass in a One-Dimensional System
Center of Mass in a Two-Dimensional System
Center of Mass of a Planar Lamina



30 July 2014 (Completed Chapter 5 "Transcendental Functions & Differential Equations" - Calculus: Early Transcendental Functions by Roland Larson)



Concepts I practiced are:
Using the Log Rule
Integrals of Trigonometric Functions
Bases Other than e
Differentiation and Integration
Applications of Exponential Functions
Differential Equations
Growth & Decay Models
General & Particular Solutions
Separation of Variables
Homogeneous Differential Equations
Derivative of an Inverse Function
Derivatives of Inverse Trigonometric Functions
Review of Basic Differentiation Rules
Integrals Involving Inverse Trigonometric Functions
Completing the Square
Review of Basic Integration Rules


22 July 2014 (Completed Chapter 4 "Integration" - Calculus: Early Transcendental Functions by Roland Larson)


Since Khan Academy doesn't have their Integral Calculus course complete yet, I pulled out my old Calculus book from college.  The last time I attempted to learn Calculus, learning Integration was what stopped me.  Just really didn't get it and certainly couldn't apply it to anything.  This time I was happy to note that it all made much more sense to me this time :-) 


Concepts I practiced are:
Antiderivatives
Notation for antiderivatives
Basic integration rules
Initial conditions and particular solutions
Sigma notation
Area
The area of a plane region
Upper and lower sums
Riemann sums
Definite integrals
properties of definite integrals
The fundamental theorem of calculus
The mean value theorem for integrals
Average value of a function
The second fundamental theorem of calculus
Pattern recognition
Change of variables
The general power rule for integration
Change of variables for definite integrals
Integration of even and odd functions
The natural logarithmic function
The number e
The natural logarithmic function and differentiation


30 June 2014 (Completed parts of the Integral Calculus course on Khan Academy)


The Integral Calculus course on Khan Academy isn't quite fleshed out yet.  Many parts are just videos.  If a section only had videos, I skipped over it. 


Concepts I worked on are:
Antiderivatives
Using rectangles to approximate area under a curve,
Area under a rate function equals net change,
Evaluating definite integrals
Functions defined by integrals
The fundamental theorem of calculus
Integration by parts
Integration by the reverse chain rule
Integration by u-substitution
Understanding sequences
Geometric sequences
Convergence and divergence of sequences
Arithmetic series
Understanding series
Convergence and divergence of series
Geometric series of constants
Creating power series from geometric series using algebra
Creating power series from geometric series using differentiation and integration
Maclaurin series for sin x, cos x, and e^x


6 June 2014 (Completed the Differential Calculus course on Khan Academy)



This is a screenshot of when I finally mastered everything in their course.
 Another screenshot.


Concepts I mastered on Khan Academy:
Finding limits numerically One-sided limits from graphs Two-sided limits from graphs Local LinearizationExtreme value theorem
Two-sided limits using algebra Two-sided limits using advanced algebra Continuity L'Hôpital's rule
Recognizing concavity
Limits at infinity where f(x) is unbounded Limits at infinity where x is unbounded Squeeze theorem  Mean value theoremExtreme values from graphs
Slope of secant lines Tangent slope is limiting value of secant slope The formal and alternate form of the derivative  Related ratesCritical numbers
Derivative intuition Recognizing slope of curves Graphs of functions and their derivatives Applications of differentiation in biology, economics, physics, etc.
Applications of derivatives: Motion along a line
Visualizing derivatives Symbolic differentiation Power rule    Concavity and the second derivative
Special derivatives Chain rule on two functions Chain rule on three functions  OptimizationApplications of derivatives: Tangent and normal lines
Product rule Quotient rule Implicit differentiation  Second derivative test 
Derivatives of inverse functions Logarithmic differentiation

 







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