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Unit 9.2 – Second Derivatives of Parametric Equations BC ONLY

AP® Calculus BC | Concavity and Parametric Curves

Why This Matters: The second derivative for parametric equations tells us about the concavity of the curve and helps identify inflection points. While the formula looks complex, the concept is straightforward: differentiate \(\frac{dy}{dx}\) with respect to \(x\) by using the chain rule through the parameter \(t\). This is a guaranteed topic on BC exams!

📐 The Second Derivative Formula

Second Derivative: Parametric Form

THE FORMULA:

For parametric equations \(x = f(t)\) and \(y = g(t)\):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) \]

Using the chain rule:

\[ \frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt} \]

The Key Insight:

To find \(\frac{d^2y}{dx^2}\), we need to differentiate \(\frac{dy}{dx}\) with respect to \(x\). But since everything is in terms of \(t\), we use:

\[ \frac{d}{dx} = \frac{d/dt}{dx/dt} \]

🔄 The Complete Expanded Formula

Using the Quotient Rule

STEP-BY-STEP DERIVATION:

Step 1: Start with \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\)

Step 2: Differentiate with respect to \(t\) using quotient rule:

\[ \frac{d}{dt}\left(\frac{dy}{dx}\right) = \frac{d}{dt}\left(\frac{dy/dt}{dx/dt}\right) \]
\[ = \frac{(dx/dt) \cdot \frac{d^2y}{dt^2} - (dy/dt) \cdot \frac{d^2x}{dt^2}}{(dx/dt)^2} \]

Step 3: Divide by \(\frac{dx}{dt}\):

\[ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(dy/dx)}{dx/dt} = \frac{(dx/dt)(d^2y/dt^2) - (dy/dt)(d^2x/dt^2)}{(dx/dt)^3} \]

📝 Note: The denominator is \((dx/dt)^3\), not \((dx/dt)^2\)! This is because we divide the quotient rule result by \(dx/dt\) one more time.

✨ Simplified Approach (Often Easier!)

The Practical Method:

  1. Find \(\frac{dy}{dx}\): Calculate \(\frac{dy/dt}{dx/dt}\) and simplify
  2. Differentiate with respect to \(t\): Find \(\frac{d}{dt}\left(\frac{dy}{dx}\right)\)
  3. Divide by \(\frac{dx}{dt}\): Get \(\frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt}\)

💡 Pro Tip: Simplifying \(\frac{dy}{dx}\) before differentiating makes Step 2 much easier! Don't leave it as a complex fraction.

📊 Concavity and Inflection Points

INTERPRETING THE SECOND DERIVATIVE

Concavity Test:
  • If \(\frac{d^2y}{dx^2} > 0\): Curve is concave up (shaped like ∪)
  • If \(\frac{d^2y}{dx^2} < 0\): Curve is concave down (shaped like ∩)
  • If \(\frac{d^2y}{dx^2} = 0\): Possible inflection point
Inflection Points:

Occur where \(\frac{d^2y}{dx^2} = 0\) or is undefined, AND concavity changes

For parametric equations:

  • Set \(\frac{d}{dt}(dy/dx) = 0\) (numerator of second derivative)
  • Check that \(\frac{dx}{dt} \neq 0\) at that value of \(t\)
  • Verify concavity changes by testing values around \(t\)

📖 Comprehensive Worked Examples

Example 1: Basic Second Derivative

Problem: For \(x = t^2\) and \(y = t^3\), find \(\frac{d^2y}{dx^2}\).

Solution:

Step 1: Find \(\frac{dy}{dx}\)

\[ \frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2 \]
\[ \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2} \]

Step 2: Differentiate \(\frac{dy}{dx}\) with respect to \(t\)

\[ \frac{d}{dt}\left(\frac{3t}{2}\right) = \frac{3}{2} \]

Step 3: Divide by \(\frac{dx}{dt}\)

\[ \frac{d^2y}{dx^2} = \frac{3/2}{2t} = \frac{3}{4t} \]

ANSWER: \(\frac{d^2y}{dx^2} = \frac{3}{4t}\)

Example 2: Using Quotient Rule Method

Problem: For \(x = \cos t\) and \(y = \sin t\), find \(\frac{d^2y}{dx^2}\).

Step 1: Find first derivatives

\[ \frac{dx}{dt} = -\sin t, \quad \frac{dy}{dt} = \cos t \]
\[ \frac{dy}{dx} = \frac{\cos t}{-\sin t} = -\cot t \]

Step 2: Differentiate \(\frac{dy}{dx}\) with respect to \(t\)

\[ \frac{d}{dt}(-\cot t) = -(-\csc^2 t) = \csc^2 t \]

Step 3: Apply formula

\[ \frac{d^2y}{dx^2} = \frac{\csc^2 t}{-\sin t} = -\frac{1}{\sin^3 t} = -\csc^3 t \]

Example 3: Concavity Analysis

Problem: For \(x = t^2\), \(y = t^3 - 3t\), determine concavity at \(t = 1\).

Find second derivative:

\[ \frac{dx}{dt} = 2t, \quad \frac{dy}{dt} = 3t^2 - 3 \]
\[ \frac{dy}{dx} = \frac{3t^2 - 3}{2t} = \frac{3(t^2 - 1)}{2t} \]

Differentiate with respect to \(t\):

Using quotient rule on \(\frac{3(t^2-1)}{2t}\):

\[ \frac{d}{dt}\left(\frac{3(t^2-1)}{2t}\right) = \frac{2t(6t) - 3(t^2-1)(2)}{4t^2} = \frac{6t^2 + 6}{4t^2} = \frac{3(t^2+1)}{2t^2} \]

Complete the calculation:

\[ \frac{d^2y}{dx^2} = \frac{3(t^2+1)/(2t^2)}{2t} = \frac{3(t^2+1)}{4t^3} \]

At \(t = 1\):

\[ \frac{d^2y}{dx^2}\Big|_{t=1} = \frac{3(2)}{4} = \frac{3}{2} > 0 \]

The curve is concave up at \(t = 1\).

Example 4: Finding Inflection Points

Problem: Find inflection points for \(x = t - \sin t\), \(y = 1 - \cos t\) on \([0, 2\pi]\).

Setup:

\[ \frac{dx}{dt} = 1 - \cos t, \quad \frac{dy}{dt} = \sin t \]
\[ \frac{dy}{dx} = \frac{\sin t}{1 - \cos t} \]

Find where \(\frac{d}{dt}(dy/dx) = 0\):

This requires quotient rule and solving the resulting equation.

The inflection point occurs at \(t = \pi\) on this interval.

📊 Formula Comparison

Derivatives Summary
Type Regular Function Parametric
First Derivative \(\frac{dy}{dx}\) directly \(\frac{dy/dt}{dx/dt}\)
Second Derivative \(\frac{d^2y}{dx^2}\) directly \(\frac{d/dt(dy/dx)}{dx/dt}\)
Concavity Test Same: \(f'' > 0\) → concave up Same: \(\frac{d^2y}{dx^2} > 0\) → concave up

💡 Essential Tips & Strategies

✅ Success Strategies:

  • Simplify \(\frac{dy}{dx}\) first: Makes differentiation much easier
  • Use quotient rule carefully: Keep track of numerator and denominator
  • Remember the denominator: It's \((dx/dt)^3\) in expanded form
  • Don't forget the final division: By \(\frac{dx}{dt}\)
  • For trig functions: Know your trig derivatives cold
  • Check your signs: Negative signs are easy to lose
  • For concavity: Just check the sign at given \(t\)
  • Factor when possible: Makes the quotient rule cleaner

🔥 Common Shortcuts:

  • If \(\frac{dy}{dx}\) simplifies nicely: The quotient rule becomes much easier
  • For circles/ellipses: Expect trig functions in answer
  • For polynomials: Usually get rational functions
  • Symmetry: Can sometimes predict sign of second derivative

❌ Common Mistakes to Avoid

  • Mistake 1: Forgetting to simplify \(\frac{dy}{dx}\) before differentiating
  • Mistake 2: Using \((dx/dt)^2\) instead of \((dx/dt)^3\) in denominator
  • Mistake 3: Forgetting the final division by \(\frac{dx}{dt}\)
  • Mistake 4: Quotient rule errors (wrong sign, wrong order)
  • Mistake 5: Differentiating with respect to \(x\) instead of \(t\)
  • Mistake 6: Sign errors when simplifying
  • Mistake 7: Forgetting to evaluate at specific \(t\) when asked
  • Mistake 8: Not checking if \(\frac{dx}{dt} \neq 0\)
  • Mistake 9: Confusing concave up/down interpretation
  • Mistake 10: Algebraic errors when expanding quotient rule

📝 Practice Problems

Find \(\frac{d^2y}{dx^2}\) for these parametric equations:

  1. \(x = t^3\), \(y = t^2\)
  2. \(x = e^t\), \(y = e^{-t}\)
  3. \(x = 2t\), \(y = t^2 - 1\), and evaluate at \(t = 1\)
  4. \(x = \sin t\), \(y = \cos t\), and determine concavity at \(t = \pi/4\)

Answers:

  1. \(\frac{d^2y}{dx^2} = -\frac{2}{9t^4}\)
  2. \(\frac{d^2y}{dx^2} = \frac{2}{e^{3t}}\)
  3. \(\frac{d^2y}{dx^2} = \frac{1}{2}\) at \(t = 1\)
  4. Concave down (negative second derivative)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

  • Show \(\frac{dy}{dx}\) first: Don't skip to second derivative
  • Show simplification: Simplify \(\frac{dy}{dx}\) before differentiating
  • Show differentiation: Calculate \(\frac{d}{dt}(dy/dx)\)
  • Show division: Divide by \(\frac{dx}{dt}\)
  • For concavity: Evaluate and state conclusion
  • For inflection points: Show where second derivative is 0
  • Simplify final answer: Unless calculator problem
  • State units/context: If applicable

💯 Exam Strategy:

  1. Write \(\frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt}\)
  2. Find and simplify \(\frac{dy}{dx}\)
  3. Differentiate simplified form w.r.t. \(t\)
  4. Divide by \(\frac{dx}{dt}\)
  5. Simplify if reasonable
  6. Evaluate at specific \(t\) if asked
  7. Interpret (concavity) if asked

⚡ Quick Reference Guide

SECOND DERIVATIVE ESSENTIALS

The Formula:

\[ \frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt} \]

Expanded (using quotient rule):

\[ \frac{d^2y}{dx^2} = \frac{(dx/dt)(d^2y/dt^2) - (dy/dt)(d^2x/dt^2)}{(dx/dt)^3} \]

Concavity:

  • \(\frac{d^2y}{dx^2} > 0\) → concave up ∪
  • \(\frac{d^2y}{dx^2} < 0\) → concave down ∩

Process:

  1. Find and simplify \(\frac{dy}{dx}\)
  2. Differentiate w.r.t. \(t\)
  3. Divide by \(\frac{dx}{dt}\)

Master Second Derivatives of Parametric Equations! The fundamental formula: \(\frac{d^2y}{dx^2} = \frac{d/dt(dy/dx)}{dx/dt}\). Process: (1) find \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\) and SIMPLIFY, (2) differentiate this with respect to \(t\), (3) divide by \(\frac{dx}{dt}\). Expanded form using quotient rule: \(\frac{(dx/dt)(d^2y/dt^2) - (dy/dt)(d^2x/dt^2)}{(dx/dt)^3}\)—note the denominator is CUBED! The second derivative tells us about concavity: positive = concave up (∪), negative = concave down (∩). Inflection points where second derivative = 0 and concavity changes. Critical tip: simplify \(\frac{dy}{dx}\) BEFORE differentiating—makes the quotient rule much easier. Common errors: wrong denominator power, forgetting final division by \(dx/dt\), quotient rule mistakes. This is guaranteed BC content—appears on virtually every exam! Practice until automatic! 🎯✨