Tangent and Area in Intro Calculus

Jul 9, 2024

Introduction to Calculus: Tangent and Area Problems 📏

Introduction

  • Calculus begins with the concept of limits.
  • Limits are crucial for learning advanced calculus.

Objectives of Calculus

Objective 1: Find the Tangent to a Curve at a Point

  • Given any curve, find the slope at a specific point.
    • This refers to finding the tangent at a point, which is the line that intersects the curve at just one place in that area.
  • Key points:
    • The slope of a line is easy, but that of a curve is not.
    • The tangent and the slope are interrelated.

Objective 2: Find the Area Under a Curve

  • Between two points on any given curve.
    • Impossible with basic geometry (you can't use formulas for rectangles, triangles, etc.).
  • Example: It is not possible to calculate the area under a curve with geometry without integration.

Tangent Problem

Concept

  • Problem: Given an arbitrary curve and a point P, find the slope of the curve at P.
  • Difficulty: You cannot form the equation of a line with just one point.

Solution

  • Introduce a second point Q on the curve.
  • PQ is a secant line that touches the curve at two points.
  • We approximate the tangent by:
    1. Moving Q closer to P.
    2. The secant line becomes closer and closer to the tangent.
    3. Note: We can't make Q exactly coincide with P because we need two points to define a line.
    4. This approach uses the concept of limit.
  • Limit: What happens when two points come infinitely close to each other without being the same?

Procedure for Finding the Slope Using Limits

  1. Define points P and Q in terms of coordinates.
  2. Use the slope of the secant line as (Y2 - Y1) / (X2 - X1).
  3. Move Q infinitely close to P.
  4. Simplify the slope expression to avoid indeterminate forms (0/0).
  5. Evaluate the limit of the secant slope to find the tangent slope.
  6. Worked example: Find the tangent for the function y = x^2 at the point (1,1).

Area Problem

Concept

  • Second major problem of calculus: Find the area under a curve.
  • Difficulty: It cannot be solved with basic geometric methods due to the curved nature.

Solution

  • Approximation using rectangles:
    1. Divide the area into many small rectangles.
    2. The smaller and more numerous the rectangles, the better the approximation.
    3. Limit: Make the number of rectangles approach infinity.

Definition of Limit

  • In simple terms, a limit answers what the function does as a variable approaches a given value.
  • Example: Limit of x^2 as x approaches 2.

One-Sided and General Limits

  • One-Sided Limit: From one direction only (left or right).
    • Example: Approximation from the right or left for a function f(x) as x approaches 2.
    • If they differ, the overall limit does not exist.
  • General Limit: Requires that limits from both the left and right are equal.

Limits in Calculus

  • Limits are the foundation for solving advanced calculus problems.
  • Use of tables and graphs to evaluate limits.

Conclusion

  • Understanding the tangent and area problems is fundamental to progressing in calculus.
  • The methods described here are starting points for more advanced approaches.

Note: Worked examples detail practicing these essential concepts in limits. Familiarity with limits facilitates the understanding of more advanced calculus procedures.