Tangent Lines, Functions, and Graphs – Section 1.x Notes

The tangent line to a curve at a point touches the curve and shares its slope at that point.
Point-slope form of a line: y - y0 = m(x - x0)
Slope between two points: m = \frac{y2 - y1}{x2 - x1}
Perpendicular lines: if one line has slope m1, a line perpendicular to it has slope m2 = -\frac{1}{m_1}.
Tangent lines are crucial for topics like optimization and related rates; their slope is a central study object.

A function assigns a unique output to each input, represented as a mapping f: X \to Y.
Domain: set of x-values where the function is defined.
Range: set of actual y-outputs.
The vertical line test: if any vertical line intersects a graph in more than one point, the graph does not represent a function of x.
Example: A circle defined by x^2 + y^2 = r^2 is not a function of x because most x-values have two y-values; it can be split into two semicircle functions: y = \sqrt{r^2 - x^2} and y = -\sqrt{r^2 - x^2}.
Functions can be piecewise, meaning different formulas apply over different subdomains. Example: f(x) = \begin{cases} x^2, & x \le 1,\ x, & x > 1.\end{cases}
The graph of a function is the set of all points (x, f(x)).

Ideas extend to higher dimensions: tangent lines generalize to tangent vectors/planes on curves/surfaces in 3D.
Core calculus questions in higher dimensions involve finding tangent objects and calculating areas/volumes.
Course Themes: Focus is on understanding slopes and tangent lines as a foundation for optimization, related rates, and area/volume problems.
Toolkit Reminder: Point-slope form, slope formula, perpendicular slope relationship, function notation, domain, range, and the vertical line test are fundamental.
Key Takeaway: Develop fluency in line equations, recognizing functions, and understanding how these concepts generalize.