Calculus Lecture Notes

Introduction

• Official start of Calculus class.
• Focus on reviewing Math 2 concepts and basic algebra needed for success in calculus.

Section 0.1: Lines

Key Concepts

• Definition of Lines: Infinite points, require two points to define a line.
• Graphing a Line: Need two points or a point and a slope.
• Slope: Represents how a line rises or falls.
  • Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} )
  • Derived from choosing two points ((x_1, y_1)) and ((x_2, y_2)).

Deriving the Slope Formula

• Use coordinates ((x_1, y_1)) and ((x_2, y_2)) for generality.
  • Rise: ( y_2 - y_1 )
  • Run: ( x_2 - x_1 )
• Slope ( m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} )

Point-Slope Form

• Fix one point to derive point-slope form.
  • Equation: ( y - y_1 = m(x - x_1) )
• Allows plug-and-play for ( x ) to solve for ( y ).

Graphing Lines

Using Point-Slope and Slope-Intercept

• Example: Given two points, find slope and apply point-slope form.
• Convert to slope-intercept form for easy graphing: ( y = mx + b )

Horizontal and Vertical Lines

• Horizontal Line: ( y = c ), crosses the y-axis.
• Vertical Line: ( x = c ), crosses the x-axis.

Parallel and Perpendicular Lines

Parallel Lines

• Have the same slope.
• Example: Finding an equation parallel to a given line.

Perpendicular Lines

• Slopes are negative reciprocals of each other.
• Example: Finding an equation perpendicular to a given line.

Angle of Inclination

• Angle a line makes with the x-axis.
• Relation with Slope: ( \tan(\theta) = \text{slope} )
  • Example: Given angle, find slope using tangent.
  • Reverse: Given slope, find angle using inverse tangent.

Distance Formula

• Derived using the Pythagorean theorem.
• Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
• Used to calculate distance between two points.

Conclusion

• Emphasis on understanding algebra and trigonometry foundations for calculus success.
• Upcoming topics include derivatives and integrals involving trig functions.