Calculus Lecture Notes

Introduction to Calculus

• Calculus is built upon the concept of limits, which is fundamental to understanding calculus.
• The main goals of calculus are:
  1. Find the slope of a curve at a point (tangent line problem).
  2. Find the area under a curve between two points.

Goals of Calculus

1. Slope of a Curve at a Point

• Given any curve (not a straight line), determine the slope at a specific point.
• Tangent Line: A line that touches the curve at exactly one point.
  • Finding the tangent involves finding the slope of the curve at a point.
• Approximation using Secant Lines: Connect two points on the curve with a straight line (secant line) as an approximation to the tangent.
  • As one point on the secant line gets closer to the point of interest, the secant line approaches the tangent line.
  • Limit: Describes this process of getting closer without actually touching.

2. Area Under a Curve

• Determine the area under a curve between two points.
• Not feasible with basic geometry due to curvature.
• Calculus allows approximation by breaking down the area into rectangles and refining this approximation through limits.

Limits

• Fundamental Concept: How close one point (Q) can get to another point (P) without actually being the same point.
  • This forms the basis of defining the slope of a tangent.
• Limit Notation: Describes the behavior of a function as it approaches a certain value.
  • lim (x -> a) f(x) = L signifies the value that f(x) approaches as x approaches a.
• Importance: Limits allow us to define instantaneous rates and areas under curves, crucial for derivative and integral calculus.

Finding the Slope of the Tangent Line

• Example Problem: Find the tangent line to the curve y = x² at point (1,1).
  • Use a movable point Q (x, x²) to establish a secant line.
  • Calculate the slope of the secant and use limits to determine the slope at the tangent point.
  • Factor and simplify the slope formula to find the limit.
  • Final Result: Slope of tangent = 2, Tangent line equation = y = 2x - 1

Area Problem and Limits

• Approximating area under a curve using rectangles.
  • Make rectangles smaller and increase their number to refine approximation.
  • Use the concept of limits to define precision.

Defining a Limit

• Limit Definition: Describes the function's behavior as the variable approaches a given value, not necessarily reaching it.
• One-Sided Limits: Limits from the left or the right.
  • Right-Sided Limit: lim (x -> a^+) f(x)
  • Left-Sided Limit: lim (x -> a^-) f(x)
• A general limit exists if the left-sided and right-sided limits are equal.

Limits and Asymptotes

• Infinite Limits: When a function's value grows towards infinity as the variable approaches a point.
  • Indicates vertical asymptotes.
• Examples: Various forms of approaching and diverging behavior indicating the presence of asymptotes.

Conclusion

• Understanding limits is essential for calculus, setting the foundation for derivatives and integrals.
• The lecture covered the basic introduction to limits, tangent lines, and area problems.