Calculus theory

1. Functions
   1. Mistakes regarding functions
   2. Defining a function
   3. Operations with functions
   4. Visualising the domain of a function
   5. Graph of single variable functions
   6. Transforming the graph of functions
   7. Linear algebra and deforming graphs of functions
   8. Guessing the graphs of single variable functions
   9. Guessing the graphs of multivariable functions
   10. Level curves and level surfaces
2. Polar coordinates and parametric curves
   1. Mistakes regarding polar coordinates
   2. Mistakes regarding parametric curves
   3. Polar coordinates
   4. Parametric curves
   5. Parametrization of curves (add graphs with vectors?)
3. Limits and continuity
   1. Mistakes regarding limits
   2. Informal limit and continuity of a single variable function
   3. Formal limit and continuity of a single variable function
   4. Limit and continuity of a multivariable function
   5. Properties of limits
   6. Limits at or with infinity
   7. Theorems covering limits and continuity of functions
   8. Informal discussion of the Euler's constant
4. Differentiation and derivatives
   1. Mistakes regarding derivatives
   2. Defining the derivative
   3. Partial derivatives and direction
   4. Defining the gradient (the discussion about perpendicularity is still confusing. df/dx and df/dy mean that there is no Z component, after all, the gradient is a 2D vector for a function of two variables.)
   5. Conditions for differentiability for a single variable
   6. Conditions for differentiability for many variables
   7. Linear approximation for one variable
   8. Linear approximation for two variables
   9. Derivative formulas
   10. Derivative of logarithm and exponential
   11. Derivative of trigonometric functions
   12. Derivative of inverse functions
   13. Chain rule for single variable functions
   14. Chain rule for multivariable functions
   15. Implicit differentiation
5. Applications of differentiation
   1. Increasing and decreasing functions
   2. Extreme values of a function
   3. Finding extreme values of a single variable function
   4. Finding critical points of a single variable function
   5. l'Hospital rule
   6. Finding extreme values of a multivariable function
   7. Finding critical points of a multivariable function
   8. Lagrange's multipliers
6. Integration and total change