Solutions to an autonomous, chaotic system of ordinary differential equations (Analysis of a novel three-dimensional chaotic system (Li et al., 2013)). 

Each dot represents particle.  Initially, they are close together, but after a short time their trajectories are completely different.  This sensitivity to initial conditions is what classifies it as a chaotic system.

📚📈Calculus may test my limits📉😜, but I want you to know that you are ❌NOT❌tangential to my life‼️‼️Send this to 🔟 homies 👬👯 who are INTEGRAL📊to your survival in math class✖️➕➖➗ or NEVER🙅get a good calculus grade💯AGAIN
calculus gothic

-The limits keep getting farther and farther away. Where are they going? Where did they start? Will they ever stop?

-The unit circle tells us to bow before it. All hail the unit circle. All hail.

-You have been scribbling the integral symbol and the summation symbol for so long. You can’t write 3′s or capital S’s normally anymore. It is a reflex, muscle memory.

-Piles of math homework surround you as you become a machine, cranking out more math problems as you hone your skills. You build your own castle out of math homework. It is never-ending.

-Trigonometry rids us of our sins. and cosines. and tangents.

6

Plots of the Exponential Function over the Complex Plane

The exponential function is more complicated in the complex plane. On the real axis, the real part follow the expected exponential shape, and the imaginary part is identically zero. However, as the imaginary part changes, the exponential varies sinusoidally, with a period of 2π in the imaginary direction.

In this plot, you can see the main branch on the real axis (y=0), and part of each branch on either side. This plot runs from -2π..2π in the y-direction and from -3..3 in the x-direction. Diagram of the real part of exponential function in the complex plane. The surface is given by:

The colour in the density plot (right) runs from red (large negative values), through green (nearly zero) to blue (large positive values) Please note that the this colour scheme is non-linear (it is based on the Arctan function and therefore emphaseses changes near zero more), and thus a given change in hue does NOT necessarily reflect a similar change in value, although at the same magnitude, the changes should be identical. Diagram of the real part of exponetial function in the complex plane, as the operand approaches infinity. The plot is given by:

The plot below shows the absolute value of the real part as the operand approaches infinity:

The colouring of the density plots is different to the graphs above. It runs from green (small), through blue and red to yellow (large). It is highly non-linear and changes near zero give a larger hue shift than large values.

The plot below shows the imaginary part of the exponetial function as the operand approaches infinity. This plot is given by:

Note that the orientation of the surface graph below is different to provide a better view of the structure of the function.

The plot below shows the absolute value of the imaginary part as the operand approaches infinity:

The colouring of the density plots is different to the graphs above. It runs from green (small), through blue and red to yellow (large). It is highly non-linear and changes near zero give a larger hue shift than large values.

AP tests are about a week away, so I thought I’d share some helpful resources everyone can use!

First off, my calculus teacher is AMAZING! He’s always so energetic about calculus and he loves teaching others about it as well. He’s so good that he has workshops throughout the year that other calculus teachers attend, so they can learn as well.

His website is Ask Mr. Calculus, which has helpful step by step solutions to FRQS + other resources. He’s also getting a hang of twitter, with #APTESTHELP
  
Youtube Channels:
PatrickJMT
integralCALC
ProfRobBob

Practice Tests:
2006 Practice Exams
2008 AB Practice Exam
2010-2011 5 Steps to a 5
Collection of Questions from 1969-1998
5+ Practice Exams

Other Resources:
AB Study Guide
AB & BC Final Notes
How to Justify Answers on FRQs
Things I Wish I Could Have Told Them ! Use these tips for FRQs

Honestly, after you get the concepts down, all there’s left to do is to practice practice practice! Good luck everyone!