Section 3.9: Slopes and Tangents to Parametric Curves
Suppose that a curve C is given by the parametric equations x = x(t) and y = y(t).
~ 0 (t) = hx0 (t), y 0 (t)i is a vector that is tangent to the curve. The slope of
Then the vector R
the tangent vector and, hence, the slope of the tangent line to the curve is given by
Slope =
y 0 (t)
Rise
= 0 .
Run
x (t)
Alternatively, by the Chain Rule,
dy
dy dx
=
dt
dx dt
dy
y 0 (t)
= 0 .
dx
x (t)
Example: Find an equation of the tangent line to the curve defined by x = 1 − t3 and
y = t2 − 3t + 1 at t = 1.
Example: Find an equation of the tangent line to the curve defined by x = 2t + 3 and
y = t2 + 2t at (5, 3).
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Example: Find the points on the curve defined by x = t3 − 3t2 and y = t3 − 3t where the
tangent is horizontal or vertical.
Example: The curve defined by x = cos t and y = sin t cos t crosses itself at the origin and
therefore has two tangents at (0, 0). Find the equations of these tangent lines.
Example: At what points on the curve defined by x = t3 + 4t and y = 6t2 is the tangent
parallel to the line with parametric equations x = −7t and y = 12t − 5?
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