August 29, 20XX
The loyal aides and allies to the heroes often correspond to each other. Tails frequently meets Luigi multiple times. Yoshi was often seen with Knuckles. The Mushroom Kingdom royalty and governance with its Toadstool inhabitants often meet Acorn Kingdom and usual pink hedgehog hammer wielder with her rabbit friends. Professor Gadd gains certain assistance by Big the Cat. The Mushroom Kingdom gives form of military assistance with their allies such as Sir Swoop, Admiral Bobbery, General White, and Goldbob. Along with civilians such as the Yoshis, Bombette, Goombella the Goomba, Koops the Koopa, Madame Flurrie the wind spirit, a shadow being named Vivian, and Mowz the Squeek.
Gifted with the power of speed, he would known as the "fastest thing alive".
A nephew to Sir Charles the Hedgehog who is a dear friend to Dr. Kintobor as both are part of the Mobius royal court, and military.
They worked as a team with Dr. Sun, Doctor Chiller, a feline bio-chemist, Gazebo Boobowski, Lucas, Hacker the Mole, Dr. Bob Bobble, William Le Duck, Zonshen, and certain professors (similar to those from Mobius or Yamato Universities) such as Caninestein, Von Schlemmer, Turtle, Neinstein, Cheddermund, Agnes Hopkins (with her family consisting Donald Hopkins, Miriam Kintobor, Colin Kintobor, Bertha and Geraldo, Meredith Sanders, and Jennifer Vasquez), Clarke, and Dillon Pickle.
Along with Nate Morgan, Chuck Thorndyke, and Dr. Ellidy whereas they are helped by Cucumber the Assistant, Federica, Franco, and Raimondo.
Notable internships under Dr. Niven and Dr. Pin include Sorders, Quack, Bin, and Stork.
Both of his parents have served in the Mobian Great War.
As becoming "the fastest being alive", he soon worked with Dr. Kintobor to learn more about this world's "power-ups", rings, and items that can contain unimaginable abilities to physically change everything.
Sonic, in the future would also learn about the colonization of Corenia which faced threats whereas they allied with intergalactic Bem High Council against Andross Empire which benefited later from the league with the Shroob, Black Arms, the Xordas, Koopa Kingdom (known for black magic, and military assistance such as Razor Blade Brigadier), tribe of Goombas, X-Nauts and the Shadow Queen, the Kremlings, and Ganon's kingdom.
Within Kintobor's knowledge (such as making brash debatable decisions in his works with Professor Kingsford), he gained controversial help and financial backing from his certain family members (such as Colin Kintobor Jr and Dr. Warpnik), Grimer, Dr. Ian Droid, Dr. Brandon Quark, Professor Zed, Alf Garment and Sydney Bland, Arnem Abacus, Dr. Genius, Thomas and Miller, and other ambitious scientists and Mobian magical alchemists.
It somewhat mirrors his grandfather's studies of this world; he would try to unravel the secrets of Oliga, Illumia, and Solaris with Professors Pickle and Boobowski, the Witchcarts, and Dr. Fukurokov in Soleanna.
Math Notes: (filled with academic musings)
Differential and integral calculus of functions of one variable: analytic geometry, limits, continuity, derivatives, analysis of curves, integrals, and applications; algebraic, trigonometric, logarithmic, and exponential functions; and historical perspectives.
Analytic Geometry: Analytic geometry combines geometry and algebra, allowing you to study geometric shapes using algebraic techniques. Points, lines, curves, and other geometric objects can be described using equations, and their properties can be analyzed algebraically. This is the foundation for much of calculus.
Limits: Limits are fundamental in calculus. They describe the behavior of a function as the input (or variable) approaches a certain value. For instance, the limit of a function at a specific point defines its behavior as the input approaches that point, even if the function isn't defined at that point itself.
Continuity: A function is continuous at a point if it has no breaks, jumps, or gaps there. This means that its graph can be drawn without lifting the pen. Continuity is closely related to limits and is an important concept in understanding the behavior of functions.
Derivatives: The derivative of a function measures how it changes at each point. Geometrically, it represents the slope of the tangent line to the curve of the function at that point. Derivatives have many applications, such as finding rates of change, analyzing curves, and optimization problems.
Analysis of Curves: This involves using derivatives to analyze the behavior of curves. You can find critical points, where the derivative is zero or undefined, and use them to determine maximum, minimum, or inflection points on a curve.
Integrals: Integrals are used to find the area under a curve, among other things. They are closely related to the concept of an antiderivative, which is the reverse process of differentiation. Integrals have applications in calculating areas, volumes, and solving differential equations.
Applications of Calculus: Calculus has wide-ranging applications in various fields. It's used in physics, engineering, economics, biology, and many other areas to model and solve real-world problems involving rates of change and accumulation.
Algebraic, Trigonometric, Logarithmic, and Exponential Functions: These are various types of functions that appear frequently in calculus. Algebraic functions involve polynomial expressions, while trigonometric functions involve ratios of the sides of triangles. Logarithmic and exponential functions are important for modeling growth and decay processes.
Historical Perspectives: Learning about the history of calculus can provide insights into how these concepts were developed over time. Mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz independently invented calculus in the 17th century, and their approaches led to the modern framework of calculus.
Compute the limit of a function at a real number:
• Determine if a function is continuous at a real number
• Find the derivative of a function as a limit
• Find the equation of a tangent line to a function
• Compute derivatives using differentiation formulas
• Use differentiation to solve applications such as related rate problems and optimization problems
• Use implicit differentiation
• Graph functions using methods of calculus
• Evaluate a definite integral as a limit
• Evaluate integrals using the Fundamental Theorem of Calculus
• Apply integration to find area
Determining Continuity: A function is continuous at a real number if it doesn't have any "jumps," "holes," or "breaks" at that point. This means that the limit of the function as it approaches that point is equal to the value of the function at that point.
Finding Derivatives as a Limit: The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. It can be defined using a limit as the change in the function's output divided by the change in the input as the change approaches zero.
Equation of a Tangent Line: The equation of a tangent line to a function at a given point can be found using the derivative. The equation is in the form of y = mx + b, where "m" is the slope (which is the derivative at the given point) and "b" is the y-intercept.
Differentiation Formulas: There are various rules and formulas to differentiate different types of functions, such as power rule, product rule, quotient rule, chain rule, and more. These rules help you find the derivative of more complex functions by breaking them down into simpler parts.
Related Rate Problems: These are real-world problems where multiple variables are changing with respect to time, and you need to find how they are related. Calculus is used to determine how the rates of change of different variables are connected.
Optimization Problems: Optimization involves finding the maximum or minimum value of a function. This is often applied to real-world scenarios, such as finding the optimal dimensions of a container to hold a certain volume.
Implicit Differentiation: This technique is used when you have an equation involving both x and y, and you want to find the derivative of y with respect to x. It involves differentiating both sides of the equation with respect to x while treating y as a function of x.
Graphing Functions: Calculus can help you analyze and graph functions. You can find critical points, intervals of increase or decrease, concavity, inflection points, and more using calculus concepts.
Definite Integral as a Limit: The definite integral represents the area under a curve between two given points. It can be defined as a limit of Riemann sums, where the area is approximated by summing up areas of rectangles.
Fundamental Theorem of Calculus: This theorem states that differentiation and integration are inverse operations. The first part connects the derivative and integral, while the second part provides a way to evaluate definite integrals using antiderivatives.
Applying Integration to Find Area: Integration is used to find the area under a curve. This can be applied to finding areas between curves, areas bounded by curves and the x-axis or y-axis, and more.