Polar Slope Slides POLAR SLOPE
Navigating Rates of Change on a Circular Grid
CALCULUS SEQUENCE // UNIT 4.1
The Spiral Challenge
\(r = \theta\)
"If the radius increases as I spin, is the slope of my path always positive?"
The Question
What is the difference between "getting farther away" and "going uphill"?
How do we define "Slope" (\(\frac{dy}{dx}\)) on a circular coordinate system?
The Rate Distinction
1
Radial Rate
\[ \frac{dr}{d\theta} \]
How fast \(r\) changes as \(\theta\) changes.
Measures "Radial Expansion".
2
Tangent Slope
\[ \frac{dy}{dx} \]
Actual slope of the curve in \(xy\)-plane.
Required for Tangent Line equations.
Radial rate \(\neq\) Slope of graph
The Master Formula
Step 1: Parametrize
\(x = r \cos \theta\)
\(y = r \sin \theta\)
Step 2: Chain Rule
From Parametrics:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]
THE POLAR DERIVATIVE
\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta} \]
Numerator:
Product Rule on \(r \sin\theta\)
Denominator:
Product Rule on \(r \cos\theta\)
Mission Brief: Cardioids
Find the slope of \(r = 1 + \cos\theta\) at \(\theta = \frac{\pi}{2}\)
Radial Info
\(r = 1\)
\(r' = -1\)
Trig Values
\(\sin\theta = 1\)
\(\cos\theta = 0\)
Calculation
\[ \frac{(-1)(1) + (1)(0)}{(-1)(0) - (1)(1)} \]
\[ \text{Slope} = 1 \]
Watch your signs! The product rule for \(x = r \cos \theta\) results in a subtraction.
Polar Foundation Worksheet Polar Slope Foundations
Calculus: Differentiation in Polar Coordinates
Name:
Date:
The Polar Derivative
Derived by treating the polar equation as a parametric curve: \(x = r\cos\theta, y = r\sin\theta\).
\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta} \]
1
Concept Check
1.1 If \(r = 3\) (a circle), what is \(\frac{dr}{d\theta}\)?
Does this mean the slope of the curve is zero everywhere? Explain.
1.2 Explain why the denominator of the \(\frac{dy}{dx}\) formula involves a subtraction sign.
2
Slope Calculations
Problem 2.1: The Limacon
Find the slope of the tangent line to the curve \(r = 2 - \sin\theta\) at \(\theta = \frac{\pi}{3}\).
Target
\(\frac{dy}{dx}\)
I. Find \(r\) and \(\frac{dr}{d\theta}\) at \(\theta = \frac{\pi}{3}\)
II. Evaluate \(\sin\theta\) and \(\cos\theta\) at \(\theta = \frac{\pi}{3}\)
III. Substitute into the formula and simplify
Problem 2.2: The Three-Leaved Rose
Determine the equation of the tangent line to \(r = \sin(3\theta)\) at \(\theta = \frac{\pi}{4}\).
I. Show your work to find \(\frac{dy}{dx}\)
II. Find \(x\) and \(y\) coordinates
III. Final Equation (Point-Slope Form)
3
Synthesis
A student claims that since \(r = \theta^2\) is always increasing for \(\theta > 0\), the slope of the curve \(\frac{dy}{dx}\) must also be always positive.
Challenge:
Using the polar derivative formula or a quick mental sketch, refute or support this claim. Consider the behavior of the curve as it passes through the quadrants of the Cartesian plane.
Tangent Extremes Slides TANGENT EXTREMES
Horizontal and Vertical Polar Analysis
CALCULUS SEQUENCE // UNIT 4.2
The Bounding Box
"How do we find the exact rectangle that perfectly contains this rose?"
The top/bottom edges occur where the tangent is horizontal .
The side edges occur where the tangent is vertical .
Anatomy of the Slope
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]
Horizontal
Slope is zero when the numerator is zero.
\[ \frac{dy}{d\theta} = 0 \]
Vertical
Slope is undefined when the denominator is zero.
\[ \frac{dx}{d\theta} = 0 \]
CRITICAL: If both are zero simultaneously, the slope is indeterminate (\(0/0\)). We must use limits!
Strategy: The Workflow
Step 01
The D-Parts
Find expressions for:
\(dy/d\theta = r' \sin\theta + r \cos\theta\)
\(dx/d\theta = r' \cos\theta - r \sin\theta\)
Step 02
Trig Combat
Set the desired expression to zero. Use identities like:
\(\sin^2\theta + \cos^2\theta = 1\)
\(\cos(2\theta) = 2\cos^2\theta - 1\)
Step 03
Location
Solve for \(\theta\). Plug \(\theta\) back into original equation to find points \((r, \theta)\).
Quick Check
For the circle \(r = \sin\theta\), find the values of \(\theta\) in \([0, \pi)\) where the tangent is horizontal.
Target Expression
\(dy/d\theta = 0\)
Work
\(\cos\theta\sin\theta + \sin\theta\cos\theta = 0\)
\(2\sin\theta\cos\theta = 0\)
\(\theta = 0, \pi/2\)
Tangent Extremes Worksheet Tangent Extremes
Calculus: Horizontal & Vertical Tangents in Polar
Name:
Date:
Horizontal Tangents
Set \( \frac{dy}{d\theta} = 0 \)
Check that \( \frac{dx}{d\theta} \neq 0 \)
Vertical Tangents
Set \( \frac{dx}{d\theta} = 0 \)
Check that \( \frac{dy}{d\theta} \neq 0 \)
1
Standard Analysis
1.1
Analyze the Cardioid: \(r = 1 + \sin\theta\) on the interval \( [0, 2\pi) \).
Part A: Horizontal Tangents
Solve \(\frac{dy}{d\theta} = 0\)
Part B: Vertical Tangents
Solve \(\frac{dx}{d\theta} = 0\)
2
The Bounding Box Challenge
A "Bounding Box" is the smallest rectangle with horizontal and vertical sides that completely encloses a polar curve. For the Rose Curve \( r = \cos(2\theta) \):
Step 1: Determine the y-coordinates of all horizontal tangents.
Step 2: Determine the x-coordinates of all vertical tangents.
Synthesis
State the dimensions of the bounding box (\(Width \times Height\)).
3
Limit Behavior
Consider the curve \(r = \theta\) at the pole (\(\theta = 0\)). Calculate both \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \) at this point.
Since both are zero, the slope is indeterminate. Using L'Hôpital's Rule or geometric intuition, what is the slope of the tangent line at the origin?
Pole Precision Slides POLE PRECISION
Analysis at the Origin
CALCULUS SEQUENCE // UNIT 4.3
The Infinite Zoom
"When we zoom into the origin of a rose curve, the loops vanish and we see straight intersecting lines."
The Question
What determines the slope of these lines? Why do they seem so simple compared to the rest of the curve?
The Indeterminate Origin
At the pole, \(r = 0\). Let's look at the numerator and denominator of our slope formula:
Numerator: \( \frac{dr}{d\theta}\sin\alpha + (0)\cos\alpha \)
Denominator: \( \frac{dr}{d\theta}\cos\alpha - (0)\sin\alpha \)
Slope = \( \frac{\frac{dr}{d\theta}\sin\alpha}{\frac{dr}{d\theta}\cos\alpha} \)
Requirement
This ONLY works if \(\frac{dr}{d\theta} \neq 0\)!
The Pole Shortcut
Master Rule
If a polar curve passes through the pole at \(\theta = \alpha\):
\[ \frac{dy}{dx} = \tan \alpha \]
The tangent line at the pole is simply the line \(\theta = \alpha\).
No complex product rules needed—just solve \(r(\theta) = 0\).
Rose Curves & Nodes
Example: \(r = \sin(2\theta)\)
The curve passes through the pole when \(\sin(2\theta) = 0\).
\(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\)
This means there are two distinct tangent lines intersecting at the origin:
Line 1: \(\text{Slope} = \tan(0) = 0\) (Horizontal)
Line 2: \(\text{Slope} = \tan(\frac{\pi}{2})\) (Vertical)
Nodal Intersection Visualization
Pole Precision Worksheet Pole Precision
Inquiry: Tangent Behavior at the Origin
Name:
The "Pole Shortcut" Discovery
If a curve passes through the origin (\(r=0\)) at an angle \(\theta = \alpha\), and \(\frac{dr}{d\theta} \neq 0\) at that point, the slope of the tangent line is simply: \( \frac{dy}{dx} = \tan \alpha \)
1
Testing the Shortcut
Consider the spiral \( r = \theta \). This curve passes through the pole at \(\theta = 0\).
1.1 Use the Full Formula
Calculate \(\frac{dy}{dx}\) using the product rule formula at \(\theta = 0\).
1.2 Use the Shortcut
Apply \(\tan(0)\). Compare the results.
2
Nodal Precision
Problem: \( r = \cos(3\theta) \) (The 3-Petal Rose)
Step A: Solve \(r = 0\) to find all angles \(\theta \in [0, \pi)\) where the curve passes through the pole.
Step B: Use the shortcut to find the slope of the tangent line for each angle found above.
\(\theta_1\):
\(\theta_2\):
\(\theta_3\):
Step C: Write the equations of the tangent lines in Cartesian form (\(y = mx\)).
3
Visual Proof
Zoomed Visualization
Explain why the tangent line at the pole is always a line passing through the origin. How does this connect the polar equation \(\theta = \alpha\) to the Cartesian derivative?
Distance Optimization Slides DISTANCE OPTIMIZATION
Proximity and Range in Polar Space
CALCULUS SEQUENCE // UNIT 4.4
Radar Tracking
Target ID: L-239
"When is the target closest to our station? When is it farthest away?"
In Polar Coordinates, "Distance from Origin" is simply the radius \(r\).
The Mission
Optimize \(r(\theta)\) to find critical distances.
Finding Extrema of \(r\)
The Goal
Find where the radius is at an absolute maximum or minimum over an interval \(\theta \in [a, b]\).
Strategy
Set \( \frac{dr}{d\theta} = 0 \)
Evaluate at critical points AND endpoints.
Interpretation
Local Max of \(|r|\)
Points furthest from origin (tips of petals).
Local Min of \(|r|\)
Points closest to origin (the pole or inner loops).
Case Study: \(r = 2 + 2\sin\theta\)
Differentiate
\[ \frac{dr}{d\theta} = 2\cos\theta \]
Solve
\[ 2\cos\theta = 0 \]
\[ \theta = \pi/2, 3\pi/2 \]
Compare Values:
At \(\theta = \pi/2\):
\(r = 2 + 2(1) = 4\)
Global Max
At \(\theta = 3\pi/2\):
\(r = 2 + 2(-1) = 0\)
Global Min (Pole)
Pro-Tips
Absolute Value
Remember that distance is \(|r|\) . If your critical point yields \(r = -5\), its distance from the origin is 5. Compare this carefully with positive values!
Trig Symmetry
Use symmetry to your advantage. Max/min values often occur on the axes or at the "tips" of petals in rose curves.
Optimization is Just Calculus 1 in Circular Motion
Radar Tracker Worksheet Radar Tracker
Optimization in Polar Coordinates
Station ID:
Flight Data Alert
"A target is moving along a path defined by the polar equation \( r = 4 - 3\cos\theta \). Your mission is to determine the exact moments of proximity and distance to ensure safe tracking parameters."
1
Radial Differentiation
1.1 Find the radial derivative \(\frac{dr}{d\theta}\).
1.2 Set \(\frac{dr}{d\theta} = 0\) and solve for \(\theta\) in \([0, 2\pi)\).
1.3 Extrema Table
Evaluate the radius at your critical points and interval endpoints.
\(\theta = \_\_\_\_\_\_\_\)
\(r = \_\_\_\_\_\)
\(\theta = \_\_\_\_\_\_\_\)
\(r = \_\_\_\_\_\)
\(\theta = \_\_\_\_\_\_\_\)
\(r = \_\_\_\_\_\)
2
Flight Analysis
A. The Danger Zone
Identify the point \((r, \theta)\) where the target is closest to the origin. If the station has a safety radius of 2 units, does the target enter the danger zone?
B. Maximum Range
Identify the point \((r, \theta)\) where the target is farthest from the origin.
3
Optimization Synthesis
"A student believes that since \(\frac{dr}{d\theta} = 0\) at a point, the tangent line to the curve at that point must be horizontal."
Prove or disprove this statement using your calculations from Problem 1.1 at the maximum distance point.
HINT
Recall: \(\frac{dy}{dx} = \frac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}\)
Calculus Logic Check
Polar Mastery Cheat Sheet Polar Mastery
CALCULUS QUICK-REFERENCE GUIDE
Topic 4.5
Differentiation Synthesis
1. Tangent Slope (\(dy/dx\))
Master Formula
\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin\theta + r\cos\theta}{\frac{dr}{d\theta}\cos\theta - r\sin\theta} \]
Numerator
\( \frac{dy}{d\theta} \)
Denominator
\( \frac{dx}{d\theta} \)
2. Special Tangents
Horizontal
Numerator = 0
\( \frac{dy}{d\theta} = 0 \)
Vertical
Denominator = 0
\( \frac{dx}{d\theta} = 0 \)
3. Slopes at the Pole
Shortcut
When \(r = 0\) at angle \(\theta = \alpha\), provided \(\frac{dr}{d\theta} \neq 0\):
\[ \frac{dy}{dx} = \tan \alpha \]
This means the tangent line is simply the line \(\theta = \alpha\).
4. Distance Optimization
To find points closest to or farthest from the origin:
Set \( \frac{dr}{d\theta} = 0 \)
Distance is \(|r|\). Always check endpoints of the interval!
Common Mistake Alert
Do not confuse \(\frac{dr}{d\theta}\) with the slope \(\frac{dy}{dx}\). The radial derivative measures how far the curve moves from the center, while the slope measures its "steepness" in the Cartesian grid.
Mystery Curve Workshop Worksheet Classified // Workshop
Mystery Curve Workshop
Student Investigator:
Case Clues: The Hidden Equation
A mystery polar curve \(r = f(\theta)\) has been detected with the following calculus properties. Your job is to analyze its behavior and identify the "suspect" curve.
Clue 01: The Radial Change
\( \frac{dr}{d\theta} = -2\sin\theta \)
Clue 02: Initial Position
At \(\theta = 0\), \(r = 3\)
1
Identify the mystery equation \(r(\theta)\).
Hint: Integrate the radial change and use Clue 02 to find the constant.
2
Locate the Vertical Tangents.
Show the calculus required to find the values of \(\theta\) in \([0, 2\pi)\) where the tangent line is vertical.
3
Behavior at the Pole.
Inquiry
Does this curve pass through the pole? If so, what is the slope of the tangent line at the pole? Show your derivation.
4
Point of Maximum Distance.
At what angle \(\theta\) is the curve furthest from the origin? Provide the Cartesian coordinates \((x, y)\) of this point.
Final Identification
Based on your analysis, name the shape of this curve (e.g., Cardioid, Limacon, Rose Curve, Circle).
