Lines

- Overview

Lines are important in calculus because when you zoom in far enough on a curve, it looks and behaves like a line. 

Here are some equations for lines in calculus:

• Straight line: y = mx + c
• Line in two dimensions: ax + by = c
• Slope-intercept form: y = mx + b
• Vector equation: →ℓ(t)=→p+t→d
• Parametric equations: x=x0+at,y=y0+bt,z=z0+ct
• Symmetric equations: x−x0a=y−y0b=z−z0c

The derivative of a function has many applications to problems in calculus, including:

1. Curve sketching
2. Solving maximum and minimum problems
3. Solving distance, velocity, and acceleration problems
4. Solving related rate problems
5. Approximating function values

- Lines and Slopes

In calculus, the slope of a line is the change in y per unit change in x. The slope formula is m = (y2 – y1)/(x2 – x1). ∆x = x2 - x1, ∆y = y2 - y1. 

Let P₁(x₁, y₁) and P₂(x₂, y₂) be points on a nonvertical line L. The slope L is m = rise/run = ∆y/∆x. This can be interpreted as a measure of "sensitivity". 

For example, if y=100x+5, a small change in x corresponds to a change one hundred times as large in y. 

The first derivative of a function is the slope of the tangent line for any point on the function. This tells when the function is increasing, decreasing, or where it has a horizontal tangent. 

Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes. 

The equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.

- General Linear Equations

The equation Ax + By = C (A and B not both 0) is a general linear equation in x and y.

For example, 2x+3y=5 is a linear equation in standard form.  

- Regression Analysis

Regression analysis is a statistical method that examines the relationship between one or more independent variables and a dependent variable. It can be used to assess the strength of the relationship between variables and to model future relationships. 

Regression analysis is a type of predictive modeling technique. It can be used for forecasting, time series modeling, and finding causal relationships between variables. 

Regression analysis is often expressed in a graph. For example, a simple linear regression could be used to predict weight based on height. In this example, the equation would be Weight = 80 + 2 x (Height). 

Here are some other examples of regression analysis: 

• Longitudinal studies: Examine how early life circumstances relate to outcomes in adulthood.
• Business analytics: Find the equation representing the relationship between employee satisfaction and product sales.

Please refer to the following for more details;

Wikipedia: Regression Analysis

- Linear Regression

Linear regression is based on linear algebra and does not use calculus. The equation for linear regression for two variables is: y = a + bx. 

Here, a and b are constant numbers, x is the independent variable, and y is the dependent variable. 

In univariate linear regression, a linear regression model is created by estimating outputs with a straight line. The equation for this line is: y = mx + c. Here, m is the gradient of the line and c is the y-intercept

[More to come ...]