Write the equation of the curve passing through the point (3; 4) if the slope of the tangent

Write the equation of the curve passing through the point (3; 4) if the slope of the tangent to this curve at any point is -2x.

Known point A (3; 4) and the slope of the tangent to the curve at any, at any point equal to (-2 * x).

The slope of the tangent is the value of the derivative of the function (in this case, our curve) at a particular point. But since by the condition it is said that the slope of the tangent at any point is the same, we need to find the general form of the antiderivatives for the function f (x) = – 2 * x.

F (x) = – x ^ 2 + C; the square of the variable X with a minus sign at the beginning is the antiderivative of our expression, and C complements the general view of the antiderivatives for the function f (x).

We know point A (3; 4). Substitute the coordinates of point A into the function F (x) and define its final form:

4 = – 3 ^ 2 + C;

C = 13; We get:

F (x) = – x ^ 2 +13.