Find an equation of the tangent line to the curve at the given point. y = x3 − 3x + 2, (3, 20)

Sagot :

Answer:

[tex]y - 20 = 24(x - 3)[/tex]

Step-by-step explanation:

Equation of a line:

The equation of a line, in point-slope form, has the following format:

[tex]y - y_0 = m(x - x_0)[/tex]

In which the point is [tex](x_0,y_0)[/tex] and the slope is m.

(3, 20)

This means that [tex]x_0 = 3, y_0 = 20[/tex]. So

[tex]y - y_0 = m(x - x_0)[/tex]

[tex]y - 20 = m(x - 3)[/tex]

Slope:

The slope is the derivative of the function at the point:

The function is:

[tex]y = x^3 - 3x + 2[/tex]

The derivative is:

[tex]y^{\prime}(x) = 3x^2 - 3[/tex]

At the point, we have that [tex]x = 3[/tex]. So

[tex]m = y^{\prime}(3) = 3*3^2 - 3 = 27 - 3 = 24[/tex]

So the equation to the tangent line to the curve a the point is:

[tex]y - 20 = 24(x - 3)[/tex]