Tangent Line
a line that intersects a circle at exactly one point (forming a right angle)
In calculus, a tangent line is a straight line that touches a curve at a single point and has the same slope as the curve at that point. It is also called a tangent and represents the instantaneous rate of change in the slope of the function at that point.
To find the equation of the tangent line at a point on a curve, we need to use the derivative of the function evaluated at that point. This will give us the slope of the curve at that point. Once we have the slope, 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 m is the slope of the line and (x1,y1) is a point on the line (in this case, the point of tangency).
For example, let’s say we want to find the equation of the tangent line to the function f(x) = x^2 – 3x + 2 at the point (2,-2). We start by finding the derivative of the function:
f'(x) = 2x – 3
Then, we evaluate the derivative at x = 2 to find the slope of the tangent line:
f'(2) = 2(2) – 3 = 1
So the slope of the tangent line is 1. Now we use the point-slope form of the linear equation:
y – (-2) = 1(x – 2)
Simplifying, we get:
y = x – 4
Therefore, the equation of the tangent line to the function at the point (2,-2) is y = x – 4.
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