Question

Как найти точки на графике y=x^3-3x^2+x+1,в которых касательная образует с положительным направлением оси абсцисс угол 45. Объясните как составить уравнение

Answer 1

Производная это и есть угол который она образует между касательной к графику и самим графиков то есть тангенс угла. 
$f'(x)=tga=k\\ y'=3x^2-6x+1=tga\\ tg45=1\\ 3x^2-6x+1=1\\ 3x^2-6x=0\\ 3x(x-2)=0\\ x=0 \\ x=2$
то есть две точки $0$ и $2$  точки ординаты можно найти подставив эти значение в уравнение 
  

Answer 2

Finding Points on the Graph

To find the points on the graph of the function \( y = x^3 - 3x^2 + x + 1 \) where the tangent line forms a 45-degree angle with the positive direction of the x-axis, we can follow these steps:

1. Derive the Function - First, we need to find the derivative of the function \( y = x^3 - 3x^2 + x + 1 \) to get the slope of the tangent line at any point on the graph.

2. Set Up the Equation - Once we have the derivative, we can set up an equation to find the points where the tangent line forms a 45-degree angle with the positive x-axis.

3. Solve for Points - By solving the equation, we can find the x-coordinates of the points of interest, and then we can find the corresponding y-coordinates by substituting these x-values back into the original function.

Deriving the Function

The derivative of the function \( y = x^3 - 3x^2 + x + 1 \) is given by:

\[ y' = 3x^2 - 6x + 1 \]

This derivative represents the slope of the tangent line at any point on the graph.

Setting Up the Equation

To find the points where the tangent line forms a 45-degree angle with the positive x-axis, we can use the fact that the tangent of a 45-degree angle is 1. Therefore, we can set up the equation:

\[ 3x^2 - 6x + 1 = 1 \]

Solving for Points

Solving the equation \( 3x^2 - 6x + 1 = 1 \) gives us the x-coordinates of the points where the tangent line forms a 45-degree angle with the positive x-axis. Once we have the x-coordinates, we can find the corresponding y-coordinates by substituting these x-values back into the original function \( y = x^3 - 3x^2 + x + 1 \).

I hope this helps! Let me know if you need further assistance.