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

Section 1.1: Tangent Lines, Slopes, and Point-Slope Form

• 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: $y - y<em>0 = m(x - x</em>0)$
• Slope between two points: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• Perpendicular lines: if one line has slope $m<em>1$, a line perpendicular to it has slope $m</em>2 = -\frac{1}{m_1}$.
• Tangent lines are crucial for topics like optimization and related rates; their slope is a central study object.

Section 1.2: Functions, Graphs, Domain, and the Vertical Line Test

• 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: $f(x) = \begin{cases} x^2, &amp; x \le 1,\ x, &amp; x &gt; 1.\end{cases}$
• The graph of a function is the set of all points $(x, f(x))$.

Section 1.3: Higher-Dimensional Extensions and Course Themes

• 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.