What is the 4 step process when finding the slope of the tangent line at any given point?
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The 4-step process for finding the slope of the tangent line at any given point is as follows:
Step 1: Determine the equation of the curve or function in which you want to find the slope of the tangent line.
Step 2: Choose a point on the curve at which you want to find the tangent line's slope. This point is typically denoted by a specific x-value.
Step 3: Find the derivative of the equation from step 1. This derivative represents the rate of change of the function at any given point.
Step 4: Substitute the x-value of the chosen point from step 2 into the derivative function obtained in step 3. The result is the slope of the tangent line at that point.
The 4-step process for finding the slope of the tangent line at any given point on a curve is as follows:
Step 1: Determine the equation for the curve. This can often be given to you, or you may need to find it using various methods such as graphing, algebraic manipulation, or calculus techniques like differentiation.
Step 2: Select a specific point on the curve where you want to find the slope of the tangent line. Identify the coordinates of this point, denoted as (x, y).
Step 3: Differentiate the equation of the curve with respect to x. In other words, find the derivative of the curve's equation. This will give you the equation for the slope of the curve at any point on the curve.
Step 4: Substitute the x-coordinate of the desired point into the derivative equation obtained in Step 3. This will give you the slope of the tangent line at that specific point.
To summarize, you need to know the equation of the curve, select a specific point, differentiate the curve's equation, and substitute the x-coordinate of the desired point into the derivative equation to find the slope of the tangent line.
Example of 4 step process
f(x) = -1/2(x^2)
Step 1
f(x + h)= -1/2(x + h)^2
f(x + h)= -1/2(x^2 + 2xh + h^2)
f(x + h)= -1/2 x^2 - xh - 1/2 h^2
Step 2
f(x + h)-f(x)= -1/2 x^2 - xh - 1/2 h^2 - (-1/2 x^2)
f(x + h)-f(x)= -1/2 x^2 - xh - 1/2 h^2 + 1/2 x^2
f(x + h) - f(x) = -xh - 1/2 h^2
f(x + h) - f(x) = h (-x - 1/2 h)
Step 3
(f(x + h) - f(x))/h = (h(-x - 1/2 h))/h
(f(x + h) - f(x))/h = -x - 1/2 h
Step 4
Evaluate lim h-->0
lim h-->0 = -x - 1/2 (0)
lim h-->0 = -x
Dx(-1/2 x^2) = -x