Calculus with parametric curves

Slope of the tangent to a parametric curve: dy/dx=(dy/dt)/(dx/dt) - derivation by using the Chain Rule;
rate of change of the slope of the tangent to a parametric curve (i.e., d²y/dx²);
example: Stewart, Exercise 10.2/7 - computing the first and second derivatives by treating the curve as a parametric curve and as a graph of a function, and checking that the results are the same;
area under a parametric curve - derivation by using the Substitution Rule;
example: area A of half circle of radius r (questions: how to compute the integral, why did we get a negative value);
arc length of a parametric curve - derivation by using the Pythagorean Theorem and the Chain Rule;
example: length of a circle of radius r - derivation by using two different parameterizations;
surface area of a surface obtained by rotating a parametric curve.
Useful tricks:
- deriving all of trigonometry from the $LaTeX: \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ , $LaTeX: \cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta$ ,
- use that sin is odd and cos is even to derive expressions for $LaTeX: \sin(\alpha - \beta)$ and $LaTeX: \cos(\alpha - \beta)$ ;
- use the basic formulas to derive expressions for $LaTeX: \sin 2\alpha$ , $LaTeX: \cos 2\alpha$ , $LaTeX: \cos^2 \alpha$ (expressed in terms of $LaTeX: \cos 2\alpha$ ), $LaTeX: \sin 3\alpha$ , $LaTeX: \tan(\alpha+ \beta)$ , and to obtain identities like $LaTeX: \sin \alpha \cos \beta = \frac12 \left[ \sin(\alpha + \beta) +\sin(\alpha - \beta) \right]$ , etc.
Discussion of Problem 5 of Gradescope HW 2:
- geometric intuition;
- examples of systems of linear equations with one solution, with no solutions, and with infinitely many solutions.

[Sec. 10.2]