Question

найти уравнение касательной к параболе в ее вершине. уравнение параболы: y=8x-2x^2 если возможно (рисунок)СРОЧНО

Answer 1

y=-2x^2+8x

-b/2a=-8/-4=2

y=2

Answer 2

Finding the Equation of the Tangent to the Parabola at its Vertex

The equation of the parabola is given as: y = 8x - 2x^2.

To find the equation of the tangent to the parabola at its vertex, we need to find the derivative of the parabola's equation and then use it to find the equation of the tangent.

The derivative of the parabola's equation is the slope of the tangent at the vertex.

The general form of the equation of a tangent to a curve at a given point is: y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope of the tangent.

Let's proceed with finding the equation of the tangent to the parabola at its vertex.

Solution

The first step is to find the derivative of the parabola's equation to get the slope of the tangent at the vertex.

The derivative of the parabola's equation y = 8x - 2x^2 is given by: y' = 8 - 4x.

At the vertex of the parabola, the slope of the tangent is given by the value of the derivative at that point. Since the vertex of a parabola is at the point where the derivative is equal to 0, we can find the x-coordinate of the vertex by setting the derivative equal to 0 and solving for x.

Setting y' = 0, we get: 8 - 4x = 0 4x = 8 x = 2.

Now that we have the x-coordinate of the vertex, we can find the corresponding y-coordinate by substituting x = 2 into the original parabola equation: y = 8*2 - 2*2^2 y = 16 - 8 y = 8.

So, the vertex of the parabola is at the point (2, 8).

Now, we have the point of tangency (2, 8) and the slope of the tangent m = 8 - 4x.

Substituting these values into the general form of the equation of a tangent, we get: y - 8 = (8 - 4*2)(x - 2).

Simplifying this, we get the equation of the tangent to the parabola at its vertex.

Therefore, the equation of the tangent to the parabola at its vertex is: y - 8 = -8(x - 2).

This is the equation of the tangent to the parabola at its vertex.

I hope this helps! If you have further questions or need additional assistance, feel free to ask.