Question

Написать уравнение касательной к графику функции y=8x3-1 в точке пересечения его с осью абсцисс. Помогите пожалуйста =)

Answer 1

Finding the Point of Intersection

To find the point of intersection between the graph of the function y = 8x^3 - 1 and the x-axis (or the line y = 0), we set the function equal to zero and solve for x:

8x^3 - 1 = 0

Solving for x

To solve this equation, we can use various methods such as factoring, the rational root theorem, or numerical methods. Let's use the rational root theorem to find a possible rational root.

The rational root theorem states that if a polynomial equation has a rational root p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p is a factor of the constant term and q is a factor of the leading coefficient.

In our equation, the constant term is -1 and the leading coefficient is 8. The factors of -1 are ±1, and the factors of 8 are ±1, ±2, ±4, and ±8. Therefore, the possible rational roots are:

±1/1, ±1/2, ±1/4, ±1/8

By testing these values, we find that x = -1 is a root of the equation.

Finding the Slope of the Tangent Line

To find the slope of the tangent line at the point of intersection, we need to find the derivative of the function y = 8x^3 - 1 and evaluate it at x = -1.

The derivative of the function is:

dy/dx = 24x^2

Evaluating the derivative at x = -1, we get:

dy/dx = 24(-1)^2 = 24

So, the slope of the tangent line at the point of intersection is 24.

Writing the Equation of the Tangent Line

To write the equation of the tangent line, we use the point-slope form of a linear equation:

y - y1 = m(x - x1)

where (x1, y1) is the point of intersection and m is the slope of the tangent line.

Substituting the values x1 = -1, y1 = 0, and m = 24 into the equation, we get:

y - 0 = 24(x - (-1))

Simplifying, we have:

y = 24(x + 1)

Therefore, the equation of the tangent line to the graph of the function y = 8x^3 - 1 at the point of intersection with the x-axis is y = 24(x + 1).

I hope this helps! Let me know if you have any further questions.