Section 9.2

Calculus with Parametric Curves

How do you find the slope when isn't directly a function of ? We use the Chain Rule: The change in over time divided by the change in over time.

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The Parametric Derivative

The Key Formula

(provided )

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Why This Works

Since isn't directly a function of , we use the Chain Rule. The change in over time, divided by the change in over time, gives slope!

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Think of It As

At any moment, slope = how fast you're moving up vs. how fast you're moving right

Second Derivative (Concavity)

Important: First find as a function of , then take its derivative with respect to , and divide by again.

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Worked Example

Complete Analysis of a Parametric Curve

A curve C is defined by the parametric equations

Parametric curve with tangent lines at self-intersection

(a) Show that C has two tangents at (3, 0) and find their equations

Find t values: At (3, 0), we need and

From , we get or

Checking: ✓ and ✓

Find derivatives:

At :

At :

(b) Find where the tangent is horizontal or vertical

Horizontal tangent:

when

Points: (1, -2) at t = 1 and (1, 2) at t = -1

Vertical tangent:

when

Point: (0, 0) at t = 0

(c) Determine where the curve is concave upward or downward

Second derivative:

Using the quotient rule on :

So:

Concave upward: when (since numerator is always positive)

Concave downward: when

(d) Key features of the sketch

  • Vertical tangent at origin (0, 0)
  • Horizontal tangents at (1, 2) and (1, -2)
  • Self-intersection at (3, 0) with two distinct tangent lines
  • Concave down for t < 0 (upper loop), concave up for t > 0 (lower loop)
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Level Up Examples

Cycloid Tangent Lines

The cycloid is defined by

(a) Find the tangent to the cycloid at the point where θ = π/3

Find the point:

Find derivatives:

At :

(b) At what points is the tangent horizontal? When is it vertical?

Horizontal tangent:

for integer n

At : point is

(top of each arch)

Vertical tangent:

for integer n

At : cusps at

(where the curve touches the ground)

💡 Key Insight

At the cusps (θ = 0, 2π, ...), both and , so the formula is indeterminate. Using L'Hôpital's Rule or the limit as , we find the tangent approaches vertical.

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Practice Quiz

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