What are is the 4 step process when finding the slope of the tangent line at any given point?
To find the slope of the tangent line at any given point, you can follow the 4-step process:
Step 1: Choose a point on the curve
Start by selecting a specific point on the curve where you want to find the slope of the tangent line. This point will be denoted as (x₁, y₁), representing the x-coordinate and y-coordinate of the point.
Step 2: Determine an infinitesimally close second point
Choose a second point on the curve that is infinitesimally close to the first point. This second point will be denoted as (x₂, y₂). The x-coordinate of this point will be slightly different from the x-coordinate of the first point, such as x₁ + Δx, where Δx is a very small value close to zero.
Step 3: Calculate the slope using the two points
Next, calculate the slope between the two points using the formula:
Slope = (y₂ - y₁) / (x₂ - x₁)
By substituting the values of (x₁, y₁) and (x₂, y₂) into the slope formula, you can find the slope of the secant line passing through the two points.
Step 4: Take the limit as the second point approaches the first point
To find the slope of the tangent line, you need to take the limit as the second point (x₂, y₂) approaches the first point (x₁, y₁). This can be written as:
lim(x₂, y₂) → (x₁, y₁) (y₂ - y₁) / (x₂ - x₁)
Evaluating this limit will give you the slope of the tangent line at the chosen point.