Overview
This lecture introduces the foundational concepts of calculus, focusing on limits, their role in finding slopes (tangent lines) and areas under curves, and how to evaluate limits using tables and graphs.
Goals of Calculus
- The two main goals of calculus are: (1) finding the slope (tangent) of a curve at a point, and (2) finding the area under a curve between two points.
- Algebra cannot solve these problems for arbitrary curves; calculus is necessary.
The Tangent Problem and Limits
- To find the tangent to a curve at point P, use another point Q on the curve and draw a secant line through P and Q.
- As Q moves closer to P, the secant line better approximates the tangent line.
- The slope of the tangent is found by taking the limit as Q approaches P, but Q cannot actually be P (need two points to define a line).
- This process leads to the definition of a limit: getting arbitrarily close without being equal.
Example: Finding a Tangent Line Using Limits
- To find the tangent to y = x² at point (1, 1), select a general point Q = (x, x²) and compute the slope between P and Q: [x² - 1] / [x - 1].
- Factor and simplify to get x + 1.
- As x approaches 1 (but does not equal 1), the expression approaches 2.
- The equation of the tangent line at (1, 1) is y – 1 = 2(x – 1), or y = 2x – 1.
The Area Problem
- To approximate the area under a curve, use rectangles and sum their areas.
- As the number of rectangles increases and their width approaches zero, the approximation becomes exact (integral).
- Limits are used to formalize this process by letting the number of rectangles approach infinity.
Definition and Evaluation of Limits
- A limit asks: What does the function approach as the variable gets arbitrarily close to a specific value?
- The actual value at that point does not matter, only the behavior as you approach it.
- Use tables of values approaching from both left and right to estimate limits.
- The limit exists if both sides approach the same value.
One-sided Limits and Limit Existence
- Right-sided limits: approach from values greater than the target value (indicated by a "+" superscript).
- Left-sided limits: approach from values less than the target value (indicated by a "–" superscript).
- For a limit to exist at a point, left and right-sided limits must be equal.
- If they are not equal, the (two-sided) limit does not exist.
Limits and Infinite Behavior (Asymptotes)
- If the function approaches ±infinity as x approaches a value from one or both sides, there is a vertical asymptote at that point.
- Four cases exist for approaching infinity (from left/right, to positive/negative infinity).
- The limit exists only if both one-sided limits agree (both ±infinity).
Key Terms & Definitions
- Limit — The value a function approaches as the input approaches a given point.
- Tangent Line — A line that touches a curve at one point and represents the instantaneous slope at that point.
- Secant Line — A line passing through two points on a curve.
- One-sided Limit — The value the function approaches from one side (left or right) of a point.
- Vertical Asymptote — A line x = a where the function increases/decreases without bound as x approaches a.
- Undefined — An expression (like division by zero) not assigned a value in mathematics.
Action Items / Next Steps
- Practice constructing tables to estimate limits from both sides of a point.
- Complete homework problems on finding limits and identifying when they exist.
- Prepare for techniques to compute limits algebraically in the next section.