Dick said:
Kinetica said:
Dick said:
A tangent line is a line that touches a curve at only one point and has the same slope as the curve at that point. It is important because it helps us understand the rate of change of a curve at a specific point and can be used to approximate the curve's behavior near that point.
The slope of a tangent line can be found using the derivative of the curve at the desired point. The derivative represents the rate of change of the curve and when evaluated at a specific point, it gives the slope of the tangent line at that point.
A perpendicular line is a line that forms a 90-degree angle with another line. In the context of a tangent line, a perpendicular line would intersect the tangent line at a right angle. This is important because a perpendicular line to a tangent line will have a slope that is the negative reciprocal of the tangent line's slope.
To find the equation of a line that is perpendicular to a tangent line, you will first need to find the slope of the tangent line using the derivative. Then, take the negative reciprocal of that slope and use it as the slope of the perpendicular line. Next, use the point of tangency and the new slope to find the equation of the perpendicular line using the point-slope form.
Yes, there can be more than one line perpendicular to a tangent line. This is because a tangent line can intersect a curve at multiple points, and each point will have a different slope and therefore a different perpendicular line.