Question

найдите площадь треугольника ограниченного осями координат и касательной к графику функции у=3/х в точке с абсциссой х0=5

Answer 1

y(5)=3/5

y'=-3/x^2

y'(x0)(x-x0)=y-y0

y'(5)=-3/25

-3/25(x-5)=y-3/5

-3/25x+3/5=y-3/5

y=-3/25x+6/5

x=0 y=6/5

y=0 6/5=3/25x  x=(6/5)*(25/3)=10

S=(10*6/5)/2=6

Answer 2

Finding the Area of the Triangle

To find the area of the triangle bounded by the coordinate axes and the tangent to the graph of the function y = 3/x at the point with x-coordinate x0 = 5, we can use the formula for the area of a triangle formed by the x-axis, y-axis, and a tangent line.

The formula for the area of a triangle formed by the x-axis, y-axis, and a tangent line is given by:

Area = (1/2) * base * height

In this case, the base of the triangle is the x-coordinate x0 = 5, and the height is the y-coordinate of the point on the graph of the function y = 3/x at x = 5.

Calculating the Height of the Triangle

To calculate the height of the triangle, we need to find the y-coordinate of the point on the graph of the function y = 3/x at x = 5.

The function y = 3/x can be used to find the y-coordinate at x = 5.

Substitute x = 5 into the function y = 3/x to find the y-coordinate:

y = 3/5

Calculating the Area of the Triangle

Now that we have the base (x0 = 5) and the height (y = 3/5), we can calculate the area of the triangle using the formula:

Area = (1/2) * base * height

Substitute the values of the base and height into the formula:

Area = (1/2) * 5 * (3/5)

Area = (1/2) * 5 * 3/5

Area = (1/2) * 3

Area = 3/2 square units

So, the area of the triangle bounded by the coordinate axes and the tangent to the graph of the function y = 3/x at the point with x-coordinate x0 = 5 is 3/2 square units.