Mission: Tangent & Normal Squad

Differentiation (Indices & Lines)
Group:
Date:
🔧 PART 1: THE INDEX "TUNE-UP" 15 Mins
⚠️ RULE: Rewrite fractions and roots as powers first. \( \frac{1}{x^n} = x^{-n} \) | \( \sqrt{x} = x^{0.5} \)
Function Rewrite as \( y = ax^n + bx^m \)
1. \( y = \frac{4}{x^2} \)\( y = 4x^{-2} \)
2. \( y = 3\sqrt{x} \)\( y = \)
3. \( y = \frac{1}{2x} + x^2 \)\( y = \)
4. \( y = \frac{6}{\sqrt{x}} - x \)\( y = \)
5. \( y = \frac{x^3 + 4}{x} \)\( y = \)
🚀 PART 2: THE GRADIENT HUNTER 25 Mins
Prob A: Curve \( y = \frac{8}{x} + x \) at \( x = 2 \)
(Rewrite: \( 8x^{-1} + x \))

1. \( \frac{dy}{dx} \):

2. Tangent Grad (\( m_T \)) at \( x=2 \):

3. Normal Grad (\( m_N = -1/m_T \)):

Prob B: Curve \( y = 2\sqrt{x} - 3 \) at \( x = 9 \)

1. \( \frac{dy}{dx} \):

2. Tangent Grad (\( m_T \)) at \( x=9 \):

3. Normal Grad (\( m_N \)):

Mission Phase 2

Page 2/2
🏔️ PART 3: THE FINAL MISSION 40 Mins

Formula: \( y - y_1 = m(x - x_1) \)

Scenario 1: Tangent Ramp. Curve \( y = \sqrt{x} + \frac{1}{x} \) at \( x = 1 \).

1. Coords: If \( x = 1 \), \( y_1 = \)?

2. Slope: \( \frac{dy}{dx} \) at \( x=1 \)?

3. Line Equation (Rearrange to \( ax+by+c=0 \)):

Scenario 2: Normal Anchor. Normal to \( y = x^2 - \frac{8}{x} \) at \( (2, 0) \).

1. Differentiate \( y = x^2 - 8x^{-1} \):

2. Tangent \( m_T \) at \( x=2 \):

3. Normal \( m_N \):

4. Equation of Normal passing through \( (2,0) \):

🏁 CHECKPOINT: AI VERIFICATION

Prompt: "Find the equation of the normal to \( y = x^2 - \frac{8}{x} \) at \( x=2 \)."