Overview
This lecture introduces the foundational ideas of calculus, focusing on limits. It explains the two main goals of calculus—finding the slope of a curve at a point (the tangent problem) and finding the area under a curve—and walks through the concept of limits using examples, tables, and graphical reasoning.
Goals of Calculus
- The first goal is to find the slope (tangent) of a curve at a specific point.
- The second goal is to calculate the area under a curve between two points.
- Calculus provides solutions to problems that cannot be solved using algebra or geometry alone.
The Tangent Problem and Limits
- The tangent at a point is the slope of the curve at that point, which cannot be found algebraically.
- To approximate the tangent, use a secant line connecting point P (fixed) and point Q (movable) on the curve.
- As point Q approaches point P, the secant line more closely approximates the tangent line.
- The process of bringing Q infinitely close to P without coinciding is the concept of a limit.
Calculating Slopes Using Limits
- The slope of a secant line is (y₂ - y₁) / (x₂ - x₁).
- Using points P (1,1) and Q (x, x²) on y = x², the slope formula becomes (x² - 1) / (x - 1).
- Direct substitution x = 1 gives 0/0 (undefined), so we factor and simplify before taking the limit.
- As x approaches 1, the slope approaches 2, so the tangent line at (1,1) has slope 2.
- The tangent line equation is y - 1 = 2(x - 1), or y = 2x - 1.
The Area Problem and Limits
- The area under a curve can be approximated using rectangles.
- Increasing the number and decreasing the width of rectangles (using limits) yields a better approximation.
- With an infinite number of infinitesimally thin rectangles, the area calculation becomes exact.
Formal Definition and Notation of Limits
- A limit describes what value a function approaches as the input approaches a specific number.
- The actual value at that point can be undefined; only the approach matters.
- Limits are written as: lim(x→a) f(x) = L.
Using Tables and Graphs to Evaluate Limits
- Create tables of x-values approaching the target number from left and right.
- If function values approach the same number from both sides, the limit exists and equals that number.
- If the left- and right-side limits are different, the overall limit does not exist.
One-Sided Limits and Non-Existence of Limits
- A right-sided limit is written lim(x→a⁺) f(x); left-sided is lim(x→a⁻) f(x).
- The overall limit exists only if both one-sided limits are equal.
- If one or both sides approach positive/negative infinity, a vertical asymptote exists.
Key Terms & Definitions
- Limit — The value a function approaches as the input gets arbitrarily close to a specific number.
- Tangent Line — A line that touches a curve at one point and has the same slope as the curve at that point.
- Secant Line — A line that intersects a curve at two points.
- One-Sided Limit — The limit of a function as the input approaches from only one side (left or right).
- Asymptote — A line that a graph approaches but never touches.
- 0/0 (Indeterminate Form) — An undefined expression, often encountered when evaluating limits directly.
Action Items / Next Steps
- Practice constructing tables for limits and evaluating them from both sides.
- Be able to explain and compute one-sided limits.
- Prepare to learn faster algebraic techniques for computing limits in the next session.