Finding Tangent And Normal Line Equations For Y = 2x³ - 3x² + 6 At (2, 10)
In the realm of calculus, determining the equations of tangent and normal lines to a curve at a specific point is a fundamental concept with significant applications in various fields, including physics, engineering, and computer graphics. This article delves into the process of finding these equations for the curve defined by the equation y = 2x³ - 3x² + 6 at the point (2, 10). We will explore the underlying principles, step-by-step calculations, and the significance of these lines in understanding the behavior of the curve at the given point.
1. Understanding Tangent and Normal Lines
Before embarking on the calculations, it is crucial to grasp the essence of tangent and normal lines.
- The tangent line to a curve at a point is a straight line that touches the curve at that point and has the same slope as the curve at that point. It represents the instantaneous rate of change of the curve at that specific location. Imagine zooming in on the curve at the point of tangency; the tangent line would appear to perfectly align with the curve's direction.
- The normal line, on the other hand, is a straight line that is perpendicular to the tangent line at the same point on the curve. It represents the direction of the greatest change in the curve's value at that point. The normal line is orthogonal to the tangent line, forming a 90-degree angle at their intersection.
These lines provide valuable insights into the curve's behavior. The tangent line approximates the curve's direction in the immediate vicinity of the point, while the normal line indicates the direction of the curve's steepest ascent or descent. Their equations are essential tools in analyzing and manipulating curves in various mathematical and scientific contexts.
2. Finding the Slope of the Tangent Line
To determine the equation of the tangent line, we first need to find its slope. The slope of the tangent line at a point on a curve is given by the derivative of the curve's equation evaluated at that point. In other words, we need to find dy/dx for the given curve and then substitute the x-coordinate of the point (2, 10) into the derivative.
The given equation of the curve is:
y = 2x³ - 3x² + 6
To find the derivative, we apply the power rule of differentiation, which states that if y = axⁿ, then dy/dx = naxⁿ⁻¹. Differentiating each term in the equation, we get:
dy/dx = d(2x³)/dx - d(3x²)/dx + d(6)/dx
Applying the power rule:
dy/dx = 6x² - 6x + 0
Simplifying, we obtain the derivative:
dy/dx = 6x² - 6x
This expression represents the slope of the tangent line at any point x on the curve. To find the slope at the specific point (2, 10), we substitute x = 2 into the derivative:
m_tangent = 6(2)² - 6(2) = 6(4) - 12 = 24 - 12 = 12
Therefore, the slope of the tangent line at the point (2, 10) is 12. This value signifies the steepness of the curve at that point, indicating how much the y-value changes for a small change in the x-value along the tangent line.
3. Equation of the Tangent Line
Now that we have the slope of the tangent line (m_tangent = 12) and a point on the line (2, 10), we can use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form is given by:
y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line and m is the slope.
Substituting the values we have:
y - 10 = 12(x - 2)
Now, we simplify the equation to the slope-intercept form (y = mx + b):
y - 10 = 12x - 24
y = 12x - 24 + 10
y = 12x - 14
Thus, the equation of the tangent line to the curve y = 2x³ - 3x² + 6 at the point (2, 10) is y = 12x - 14. This equation represents a straight line that touches the curve at the given point and has a slope of 12. It provides a linear approximation of the curve's behavior in the vicinity of the point.
4. Finding the Slope of the Normal Line
The normal line is perpendicular to the tangent line. The slopes of perpendicular lines are negative reciprocals of each other. Therefore, if the slope of the tangent line is m_tangent, then the slope of the normal line, m_normal, is given by:
m_normal = -1 / m_tangent
We found the slope of the tangent line to be 12. Substituting this value into the formula:
m_normal = -1 / 12
So, the slope of the normal line at the point (2, 10) is -1/12. This negative reciprocal slope indicates that the normal line is significantly less steep than the tangent line and slopes in the opposite direction.
5. Equation of the Normal Line
Similar to finding the equation of the tangent line, we can use the point-slope form to find the equation of the normal line. We have the slope of the normal line, m_normal = -1/12, and the same point on the line, (2, 10). Substituting these values into the point-slope form:
y - y₁ = m(x - x₁)
y - 10 = (-1/12)(x - 2)
Now, we simplify the equation to the slope-intercept form:
y - 10 = (-1/12)x + 1/6
y = (-1/12)x + 1/6 + 10
To combine the constant terms, we find a common denominator:
y = (-1/12)x + 1/6 + 60/6
y = (-1/12)x + 61/6
Therefore, the equation of the normal line to the curve y = 2x³ - 3x² + 6 at the point (2, 10) is y = (-1/12)x + 61/6. This equation represents a straight line perpendicular to the tangent line at the given point, indicating the direction of the curve's steepest change.
6. Summary and Conclusion
In this comprehensive exploration, we successfully determined the equations of the tangent and normal lines to the curve y = 2x³ - 3x² + 6 at the point (2, 10). We achieved this by:
- Understanding the concepts of tangent and normal lines and their significance in analyzing curves.
- Finding the derivative of the curve's equation to determine the slope of the tangent line.
- Using the point-slope form to find the equation of the tangent line.
- Calculating the slope of the normal line as the negative reciprocal of the tangent line's slope.
- Applying the point-slope form again to find the equation of the normal line.
The equation of the tangent line was found to be y = 12x - 14, and the equation of the normal line was found to be y = (-1/12)x + 61/6. These equations provide valuable information about the curve's behavior at the specific point (2, 10). The tangent line approximates the curve's direction, while the normal line indicates the direction of the greatest change in the curve's value.
The concepts and techniques discussed in this article are fundamental in calculus and have wide-ranging applications in various fields. Mastering these concepts allows for a deeper understanding of curves and their properties, paving the way for more advanced mathematical explorations and problem-solving.
This detailed explanation provides a solid foundation for understanding tangent and normal lines, their equations, and their significance in calculus and related disciplines. Further exploration of these concepts can lead to a more profound appreciation of the power and elegance of mathematical analysis.