Overview
This lecture introduces the foundational concepts of calculus, focusing on limits, tangent lines, and area under curves, and explains how to compute limits using tables and algebraic techniques.
Goals of Calculus
- Calculus has two main goals: finding the slope of a curve (tangent) at a point and finding the area under a curve between two points.
- Finding the slope of a curve at a point involves constructing a tangent line.
- Finding the area under a curve relates to summing infinitely many small areas.
The Tangent Problem & Limits
- Slope requires two points, but the tangent is defined using one fixed point where we want the slope.
- A secant line connects two points on a curve; as the second point approaches the first, the secant approximates the tangent.
- The process of bringing the second point arbitrarily close without being identical introduces the concept of a limit.
- The limit allows us to define the slope of the tangent as the secant's slope as the points converge but never coincide.
Example: Finding the Slope with Limits
- For the curve ( y = x^2 ) at the point (1,1), fix P=(1,1) and let Q=(x, x^2).
- The slope of secant PQ is ( \frac{x^2 - 1}{x - 1} ).
- Factor and simplify to ( x + 1 ), avoiding x=1.
- As x approaches 1, the slope approaches 2; thus, the slope of the tangent at (1,1) is 2.
- The tangent line is ( y = 2x - 1 ).
The Area Problem & Rectangles
- To approximate area under a curve, divide the region into many rectangles and sum their areas.
- Making rectangles infinitely small and numerous yields an exact area, involving limits as the width approaches zero.
Defining Limits
- A limit describes what a function's value approaches as the input approaches a certain value.
- The actual value at that point may be undefined or irrelevant.
- The limit must be approached from both sides (left and right) and should match for the limit to exist.
Using Tables to Estimate Limits
- Create tables of values approaching the target x-value from both sides.
- Observe if the function values approach a common number.
One-Sided Limits and Non-Existence
- Right-sided limit: value approached as x increases to a point.
- Left-sided limit: value approached as x decreases to a point.
- If the left and right limits differ, the general limit does not exist.
Limits Leading to Infinity (Asymptotes)
- If a function grows without bound as x approaches a value, the limit is infinite.
- Vertical asymptotes occur when function values approach positive or negative infinity from one or both sides.
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a specific value.
- Tangent Line — A line touching a curve at only one point and matching its slope there.
- Secant Line — A line intersecting a curve at two points.
- One-Sided Limit — The limit of a function as the input approaches from only the left or right.
- Asymptote — A line that a function approaches but never touches or crosses.
Action Items / Next Steps
- Practice constructing tables and estimating limits for given functions.
- Complete homework on finding limits using both tables and algebraic simplification.
- Review definitions of tangent, secant, and asymptote for next class.