Find the points on the graph of y = (x+1)/
(x+2) where the tangent line is parallel to
the line x - y = 2.
To find the points on the graph of y = (x+1)/(x+2) where the tangent line is parallel to the line x - y = 2, we need to follow these steps:
Step 1: Determine the slope of the line x - y = 2.
Step 2: Find the derivative of the function y = (x+1)/(x+2).
Step 3: Set the derivative equal to the slope of the line and solve for x.
Step 4: Substitute the values of x into the original function to find the corresponding y-coordinates.
Step 5: Formulate the points.
Let's go through these steps one by one.
Step 1: Determine the slope of the line x - y = 2.
To determine the slope of the line x - y = 2, we need to rewrite this equation in slope-intercept form (y = mx + b), where m is the slope. Rewriting the equation, we have:
y = x - 2
Comparing this with the slope-intercept form, we find that the slope of the line is 1.
Step 2: Find the derivative of the function y = (x+1)/(x+2).
To find the derivative, we can use the quotient rule. Let me calculate the derivative for you.
The derivative of y = (x+1)/(x+2) is dy/dx = [(x+2)(1) - (x+1)(1)] / (x+2)^2.
Simplifying this expression, we can write:
dy/dx = (x+2 - x - 1) / (x+2)^2.
Simplifying further, we get:
dy/dx = 1 / (x+2)^2.
Step 3: Set the derivative equal to the slope of the line and solve for x.
Since the slope of the tangent line is parallel to the line x - y = 2, which has a slope of 1, we set the derivative equal to 1:
1 / (x+2)^2 = 1.
To solve for x, we can cross-multiply:
(x+2)^2 = 1.
Taking the square root of both sides, we get:
x + 2 = ±1.
Solving for x, we have two cases:
Case 1: x + 2 = 1.
x = 1 - 2 = -1.
Case 2: x + 2 = -1.
x = -1 - 2 = -3.
Step 4: Substitute the values of x into the original function to find the corresponding y-coordinates.
Using the original function y = (x+1)/(x+2), we can substitute the values of x we obtained into the equation to find the corresponding y-coordinates.
For x = -1, y = (-1+1)/(-1+2) = 0/1 = 0.
For x = -3, y = (-3+1)/(-3+2) = -2/-1 = 2.
Step 5: Formulate the points.
The points on the graph of y = (x+1)/(x+2) where the tangent line is parallel to the line x - y = 2 are (-1, 0) and (-3, 2).
Thus, the points on the graph of y = (x+1)/(x+2) where the tangent line is parallel to the line x - y = 2 are (-1, 0) and (-3, 2).