Calculus: Tangent Lines
What is a Tangent Line?
A tangent line to a curve at a point is the straight line that touches the curve at exactly
one point and has the same instantaneous slope as the curve at that location.
To write the equation of a tangent line, you need:
1. The slope of the tangent line, given by the derivative f ′ (x).
2. A point on the curve, written as (x1 , y1 ).
Point–Slope Formula
Once you know the slope m and the point (x1 , y1 ), the tangent line is:
y − y1 = m(x − x1 )
Graph Example 1: Tangent Line Touching the Parabola
Here we show the parabola f (x) = 21 x2 and its tangent line at x = 1.
y
y = 12 x2
(1, 0.5)
x
Tangent Line
1
What is a Tangent Line?
A tangent line to a curve at a point is the straight line that touches the curve at exactly
one point and has the same instantaneous slope as the curve at that location.
To write the equation of a tangent line, you need:
1. The slope of the tangent line, given by the derivative f ′ (x).
2. A point on the curve, written as (x1 , y1 ).
Point–Slope Formula
Once you know the slope m and the point (x1 , y1 ), the tangent line is:
y − y1 = m(x − x1 )
Graph Example 1: Tangent Line Touching the Parabola
Here we show the parabola f (x) = 21 x2 and its tangent line at x = 1.
y
y = 12 x2
(1, 0.5)
x
Tangent Line
1
