find the arc length from (x1,y2) to (x2,y2) on the graph F(x)=mx+b

F(x) is a straight line, so arc length simply refers to the straight line distance between two points, P1(x1,y1) and P2(x2,y2) which is given by:

distance = √((x2-x1)²+(y2-y1)²)

To find the arc length from the point (x1, y2) to (x2, y2) on the graph of the equation F(x) = mx + b, you need to integrate the square root of the sum of the squares of the differentials dx and dy.

Step 1: Find the expression for dy/dx.
Differentiate F(x) = mx + b with respect to x to find dy/dx:
dy/dx = m

Step 2: Find the indefinite integral of √(1 + (dy/dx)^2) dx.
The arc length formula is given by:
L = ∫√(1 + (dy/dx)^2) dx
L = ∫√(1 + m^2) dx
L = √(1 + m^2) ∫dx
L = √(1 + m^2)x + C

Step 3: Evaluate the arc length between the limits (x1, x2).
To find the specific arc length between the points (x1, y2) and (x2, y2), substitute the limits into the expression obtained in Step 2:
L = √(1 + m^2)(x2 - x1)

Hence, the arc length from (x1, y2) to (x2, y2) on the graph F(x) = mx + b is given by √(1 + m^2)(x2 - x1).

To find the arc length between two points on a graph defined by the function F(x) = mx + b, we need to use the formula for arc length of a curve. The formula is given by:

L = ∫√(1 + (dy/dx)^2) dx

where L represents the arc length, dy/dx is the derivative of the function F(x), and the integral is taken over the interval between the x-coordinates of the two given points.

1. First, find the derivative of F(x) = mx + b. The derivative of a linear function is simply the slope of the line, which is equal to m in this case.

2. Next, calculate the arc length by integrating the formula using the derivative we found in step 1. Since F(x) = mx + b, the integral will be taken over the interval [x1, x2], where (x1, y1) and (x2, y2) are the given points. Thus, the integral becomes:

L = ∫[x1,x2] √(1 + m^2) dx

3. Integrate the expression with respect to x between the limits of integration x1 and x2:

L = √(1 + m^2) ∫[x1,x2] dx

4. Evaluate the integral to get the final expression for the arc length:

L = √(1 + m^2) * (x2 - x1)