Question

Найти все точки графика функции y = (x+2)/(x-2) в каждой из которых касательная к графику образует угол 135 с положительным направлением оси OX.

Answer 1

Применены формулы дифференцирования, свойство касательной

Answer 2

Finding Points on the Graph of the Function y = (x + 2)/(x - 2) with a Tangent Line at an Angle of 135° with the Positive Direction of the x-axis

To find the points on the graph of the function y = (x + 2)/(x - 2) where the tangent line forms an angle of 135° with the positive direction of the x-axis, we need to determine the x-coordinates of these points.

Let's start by finding the derivative of the function y = (x + 2)/(x - 2) to determine the slope of the tangent line at any given point on the graph.

The derivative of y with respect to x can be found using the quotient rule:

dy/dx = [(x - 2)(1) - (x + 2)(1)] / (x - 2)^2

Simplifying the expression, we get:

dy/dx = -4 / (x - 2)^2

Now, we know that the slope of the tangent line at any point on the graph is given by the derivative dy/dx. To find the points where the tangent line forms an angle of 135° with the positive direction of the x-axis, we need to find the x-coordinates where the tangent line has a slope of -1 (since the tangent of 135° is -1).

Setting dy/dx equal to -1, we have:

-4 / (x - 2)^2 = -1

Simplifying the equation, we get:

(x - 2)^2 = 4

Taking the square root of both sides, we have:

x - 2 = ±2

Solving for x, we get two possible values:

1. When x - 2 = 2, we have x = 4. 2. When x - 2 = -2, we have x = 0.

Therefore, the two x-coordinates where the tangent line forms an angle of 135° with the positive direction of the x-axis are x = 4 and x = 0.

To find the corresponding y-coordinates, we substitute these x-values back into the original function y = (x + 2)/(x - 2):

1. For x = 4: y = (4 + 2)/(4 - 2) = 6/2 = 3. Therefore, the point on the graph is (4, 3).

2. For x = 0: y = (0 + 2)/(0 - 2) = -2/-2 = 1. Therefore, the point on the graph is (0, 1).

In summary, the two points on the graph of the function y = (x + 2)/(x - 2) where the tangent line forms an angle of 135° with the positive direction of the x-axis are (4, 3) and (0, 1).