Calculus Lecture Notes

Introduction

• Start of the calculus course.
• Review of Math 2 concepts and basic algebra essential for calculus.

Topics Covered

1. Lines and Slopes

   • Definition and characteristics of lines.
   • Importance of slope in graphing lines.
   • Formula derivation for calculating slope.
   • Example of identifying slope from two points using coordinates (x1, y1) and (x2, y2).

2. Equation of a Line

   • Introduction to point-slope form and slope-intercept form.
   • Conversion between forms.
   • Example problems on finding the equation of a line given points and parallel/perpendicular conditions.

3. Special Lines

   • Horizontal lines (y = constant) and vertical lines (x = constant).
   • Characteristics and graphing techniques.

4. Parallel and Perpendicular Lines

   • Parallel lines: same slope.
   • Perpendicular lines: slopes as negative reciprocals.

5. Angles of Inclination

   • Angle between a line and the x-axis.
   • Relation of angle of inclination to slope using tangent function.
   • Finding slopes from angles and vice versa using trigonometry.

6. Trigonometry Refresher

   • Importance of tangent function in relating angles to slopes.
   • Example: Calculating the slope from an angle of inclination (e.g., 30°).
   • Reverse calculation: Finding angle given a slope using inverse tangent.

7. Distance Formula

   • Derivation using the Pythagorean theorem.
   • Formula: ( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
   • Application in calculating the distance between two points.

Important Concepts and Formulas

• Slope Formula: (m = \frac{y_2 - y_1}{x_2 - x_1})
• Point-Slope Form: (y - y_1 = m(x - x_1))
• Slope-Intercept Form: (y = mx + b)
• Tangent Relationship: (m = \tan(\theta))
• Distance Formula: (D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})

Key Takeaways

• Understanding and manipulating line equations are fundamental in calculus.
• Trigonometry plays a crucial role in relating angles with slopes.
• Proficiency in algebra and trigonometry is essential for mastering calculus.