Finding Equations of Normal Lines Parallel to a Given Line

In summary, to find the equation of each normal line to the curve y = x^3 - 4x that is parallel to the line x + 8y - 8 = 0, you need to differentiate y = x^3 - 4x and equate it to the slope of the given line, -8. Solve for x and substitute it back into dy/dx to find the slope of the tangent. The slope of the normal line is the negative reciprocal of this value. However, be careful to correctly calculate the gradient of the given line.
  • #1
SherlockOhms
310
0

Homework Statement


Find the equation of each of the normal lines to the curve y = x^3 - 4x that is parallel to the line x + 8y - 8 = 0.


Homework Equations


Differentiation, y - y1 = m(x - x1)


The Attempt at a Solution


Well, clearly I start by differentiating y = x^3 - 4x to gey dy/dx = 3x^2 - 4. Then, this is the slope of the tangent to the graph y. This slope is equal to the slope of the line x + 8y - 8 = 0. Which is -8. So, I then evaluate for x and sub back into dy/dx, giving me the slope of the tangent. The slope of the normal line is the negative reciprocal of this. If you solve for x using
3x^2 - 4 = -8, you get an imaginary value though. Where have I gone wrong?
 
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  • #2
You have the gradient of the given line wrong.
 
  • #3
CAF123 said:
You have the gradient of the given line wrong.
Cool. See where I went wrong now. Stupid slip.
 

Related to Finding Equations of Normal Lines Parallel to a Given Line

What is the equation of a normal line?

The equation of a normal line is a mathematical representation of a line that is perpendicular to a given curve at a specific point. It is written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.

How is the equation of a normal line different from the equation of a tangent line?

The equation of a normal line and the equation of a tangent line both represent lines that touch a curve at a specific point. However, the normal line is perpendicular to the curve at that point, while the tangent line is parallel to the curve at that point.

How do you find the equation of a normal line?

To find the equation of a normal line, you first need to find the slope of the tangent line at the given point. Then, you can use the negative reciprocal of that slope as the slope for the normal line. Finally, you can use the point-slope form of a line to find the equation of the normal line.

Why is the equation of a normal line important in mathematics?

The equation of a normal line is important because it allows us to find the slope of a curve at a specific point. This information is useful in various applications, such as finding the rate of change or the direction of motion of an object at a given point.

Can the equation of a normal line be used to find the equation of a curve?

No, the equation of a normal line only represents a single line that is perpendicular to a given curve at a specific point. It cannot be used to find the entire equation of the curve. However, it can be used to find the slope of the curve at that point, which can be useful in finding the equation of the curve using other methods.

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