Topic 9.7 Notes – Defining Polar Coordinates and Differentiating in Polar Form

1. What Polar Coordinates Are

A point in the plane can be written as $(r, \theta)$:

• $r$ = distance from the origin (the pole)
• $\theta$ = angle measured counterclockwise from the positive $x$-axis

A polar equation looks like:

$$r = f(\theta)$$

To use calculus, we convert to parametric form:

$$x = r\cos\theta, \quad y = r\sin\theta$$

So if $r = f(\theta)$, then

$$x(\theta) = f(\theta)\cos\theta, \quad y(\theta) = f(\theta)\sin\theta$$

That’s why this topic lives in Unit 9. A polar curve is just a parametric curve where the parameter is $\theta$.

Also remember the coordinate relationships:

$$r^2 = x^2 + y^2$$

Here’s what that setup looks like geometrically. Focus on the point labeled $(2, \pi/4)$ and the right triangle it forms with the axes.

Polar grid with point $(2, \pi/4)$

That right triangle is why $x = r\cos\theta$ and $y = r\sin\theta$.

2. Derivatives with Respect to θ

Suppose $r = f(\theta)$.

$\frac{dr}{d\theta}$

Differentiate normally.

This tells you how fast the distance from the origin is changing.

• Set $\frac{dr}{d\theta} = 0$ to find possible closest or farthest points from the origin.
• Always compare $r$-values (and check endpoints if there’s an interval).

Example:
If $r = 3 + 2\cos\theta$, then
$$\frac{dr}{d\theta} = -2\sin\theta$$ Set equal to 0 → $\sin\theta = 0$.

$\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$

Now use product rule.

Since
$x = r\cos\theta$,
$y = r\sin\theta$,

$$\frac{dx}{d\theta} = \frac{dr}{d\theta}\cos\theta - r\sin\theta$$

$$\frac{dy}{d\theta} = \frac{dr}{d\theta}\sin\theta + r\cos\theta$$

These describe the tangent vector.

If both derivatives equal 0 at the same $\theta$, the curve may have a cusp or special behavior.

This is exactly what you’d do with parametrics using $t$. Nothing new - just $\theta$.

3. Slope of the Tangent Line $\frac{dy}{dx}$

Since the curve is parametric in $\theta$,

$$\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$$

Substitute the formulas above:

$$\frac{dy}{dx} = \frac{ \frac{dr}{d\theta}\sin\theta + r\cos\theta }{ \frac{dr}{d\theta}\cos\theta - r\sin\theta }$$

You don’t have to memorize this monster. Many students just:

1. Find $dr/d\theta$
2. Find $dx/d\theta$ and $dy/d\theta$
3. Divide

That’s often safer on an FRQ.

Tangent Line Procedure

If asked for the equation of the tangent line at $\theta = \theta_0$:

1. Compute $r(\theta_0)$
2. Convert to Cartesian:
   • $x = r\cos\theta$
   • $y = r\sin\theta$
3. Compute slope using $\frac{dy}{dx}$
4. Use point-slope form
   $y - y_0 = m(x - x_0)$

Students often forget step 2. The tangent line must be written in x and y, not in terms of $\theta$.

Horizontal and Vertical Tangents

Think parametric logic.

If both are 0, you investigate further.

This shows up a lot in multiple choice where they want you to reason quickly without fully simplifying.

4. Second Derivative $\frac{d^2y}{dx^2}$

Same idea as parametrics:

$$\frac{d^2y}{dx^2} = \frac{d}{d\theta}\left(\frac{dy}{dx}\right) \div \frac{dx}{d\theta}$$

So you:

1. Differentiate $\frac{dy}{dx}$ with respect to $\theta$
2. Divide by $dx/d\theta$

Used for:

• Concavity
• Inflection behavior

Algebra gets messy fast. Keep work organized.

5. When These Derivatives Matter

• Closest/farthest from origin → solve $dr/d\theta = 0$
• Slope of tangent line → compute $dy/dx$
• Horizontal/vertical tangents → check numerator/denominator separately
• Concavity → use second derivative

On tests, they love mixing concepts. For example, finding where the curve is closest to the origin and also has a horizontal tangent. That forces you to understand what each derivative actually represents.