Calculus Lecture Notes

Introduction to Calculus

• Official start of calculus course
• Review of essential concepts from Math 2 and algebra needed for calculus

Lines and Slope

Basic Properties of Lines

• Lines have infinite points
• Require at least two points to define
• Lines are straight and do not curve
• Essential properties: slope (rise/fall of the line)

Finding the Slope

• Slope indicates how a line rises or falls
• Formula: (\text{slope} (m) = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1})
• To find the slope, pick two points and use their coordinates in the formula

Example Calculation

1. Choose points ((x_1, y_1)) and ((x_2, y_2))
2. Plug into formula: (m = \frac{y_2 - y_1}{x_2 - x_1})

Point-Slope Form

• Starting from (m = \frac{y_2 - y_1}{x_2 - x_1})
• Fixing one point: ( m = \frac{y - y_1}{x - x_1})
• Rearrange: (y - y_1 = m(x - x_1))
• Called Point-Slope Form

Example Calculation

1. Identify points and slope
2. Plug into point-slope formula
3. Simplify to slope-intercept form (y = mx + b)

Slope-Intercept Form

• Standard form: (y = mx + b)
• (m) is the slope, (b) is the y-intercept
• Example: Convert from point-slope to slope-intercept for easy graphing

Special Lines

• Horizontal lines: ( y = c )
• Vertical lines: ( x = c )

Parallel and Perpendicular Lines

Parallel Lines

• Same slope
• E.g., Staircases

Perpendicular Lines

• Slopes: negative reciprocals (m_1 = -\frac{1}{m_2})
• Intersection at 90 degrees

Example: Finding Parallel/Perpendicular Lines

1. Identify slope of given line
2. For parallel: use same slope
3. For perpendicular: use negative reciprocal of given slope
4. Apply point-slope form with identified slope and given point

Angle of Inclination

• Angle a line makes with the x-axis
• Relationship: ( \tan(\theta) = \frac{y_2 - y_1}{x_2 - x_1} = m ) (slope)

Example Calculation

1. Given angle, find tangent to get slope: ( \text{slope} = \tan(\theta) )
2. Given slope, find angle using inverse tangent: ( \theta = \tan^{-1}(m) )

Distance Formula

• Used to find distance between two points
• Derived from Pythagorean theorem
• Formula: ( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
• Steps: Identify points' coordinates, plug into formula, solve

Conclusion

• Ensure familiarity with algebra and trigonometry for calculus success
• Review and practice basic forms and formulas