2.3 Analytic Geometry & Differential Calculus

2.3 Analytic Geometry & Differential Calculus

Analytic geometry provides geometric interpretations of algebraic equations, while differential calculus allows electronics engineers to calculate instantaneous rates of change, optimize system performance, and analyze dynamic waveforms.

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1. Conic Sections in ECE Applications

Conic sections are curves formed by intersecting a cone with a plane. Their reflective and geometric properties make them central to antenna design and radio navigation.

Summary of Conic Equations & ECE Uses

Worked ECE Board Example 1: Parabolic Reflector Focal Point

Problem: A satellite dish antenna has a parabolic cross-section. The opening diameter of the dish is $2\text{ meters}$ wide, and its depth at the center vertex $(0,0)$ is $0.5\text{ meters}$. Find the parabolic equation $y^2 = 4px$ and determine the exact position of the feed horn receiver at the focus $(p,0)$.

Solution:

• Step 1: Set vertex at $(0,0)$. The outer rim edge point lies at $x = 0.5\text{ m}$ and $y = \frac{2\text{ m}}{2} = 1.0\text{ m}$.
• Step 2: Substitute point $(0.5, 1.0)$ into $y^2 = 4px$: $$(1.0)^2 = 4p(0.5) \implies 1.0 = 2p \implies p = 0.5\text{ meters}$$
• Step 3: Parabola equation is $y^2 = 2x$. The feed horn receiver must be placed at the focal point $(0.5\text{ m}, 0)$ directly inside the dish.

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2. Limits, Indeterminate Forms & L'Hôpital's Rule

When evaluating limits $\lim_{x \to a} \frac{f(x)}{g(x)}$ that result in indeterminate forms $\left[\frac{0}{0}\right]$ or $\left[\frac{\infty}{\infty}\right]$, L'Hôpital's Rule applies:

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

provided the limit of the derivatives exists.

Worked ECE Board Example 2: Sinc Function Limit in Signal Processing

Problem: Evaluate the zero-frequency spectral magnitude limit $L = \lim_{t \to 0} \frac{1 - \cos(3t)}{t^2}$.

Solution:

• Step 1: Direct substitution yields $\frac{1 - \cos(0)}{0} = \frac{0}{0}$ (Indeterminate).
• Step 2: Apply L'Hôpital's Rule (First Derivative): $$L = \lim_{t \to 0} \frac{\frac{d}{dt}(1 - \cos(3t))}{\frac{d}{dt}(t^2)} = \lim_{t \to 0} \frac{3\sin(3t)}{2t}$$
• Step 3: Direct substitution still yields $\frac{0}{0}$. Apply L'Hôpital's Rule a second time: $$L = \lim_{t \to 0} \frac{\frac{d}{dt}(3\sin(3t))}{\frac{d}{dt}(2t)} = \lim_{t \to 0} \frac{9\cos(3t)}{2} = \frac{9\cos(0)}{2} = \frac{9}{2} = 4.5$$

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3. Derivative Rules & Transcendental Derivatives

Differentiation Rules Summary Table

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4. Optimization & Maximum Power Transfer Theorem Proof

Differential calculus is used to find maximum or minimum values of physical variables by setting the first derivative to zero ($f'(x) = 0$) and confirming concavity using the second derivative test ($f''(x) < 0$ for maximum, $f''(x) > 0$ for minimum).

Worked ECE Board Example 3: Maximum Power Transfer Proof

Problem: A DC source has a Thevenin equivalent voltage $V_{\text{th}}$ and internal resistance $R_{\text{th}}$. When connected to a variable load resistor $R_L$, the power delivered to the load is:

$$P(R_L) = I^2 R_L = \left(\frac{V_{\text{th}}}{R_{\text{th}} + R_L}\right)^2 R_L = \frac{V_{\text{th}}^2 R_L}{(R_{\text{th}} + R_L)^2}$$

Use calculus to prove that maximum power transfer occurs when $R_L = R_{\text{th}}$, and compute $P_{\text{max}}$.

Solution:

• Step 1: Take the derivative of $P$ with respect to $R_L$ using the quotient rule, treating $V_{\text{th}}$ and $R_{\text{th}}$ as constants: $$\frac{dP}{dR_L} = V_{\text{th}}^2 \left[ \frac{(1)(R_{\text{th}} + R_L)^2 - R_L \cdot 2(R_{\text{th}} + R_L)(1)}{(R_{\text{th}} + R_L)^4} \right]$$
• Step 2: Simplify the numerator: $$\frac{dP}{dR_L} = V_{\text{th}}^2 \left[ \frac{(R_{\text{th}} + R_L) - 2R_L}{(R_{\text{th}} + R_L)^3} \right] = V_{\text{th}}^2 \left[ \frac{R_{\text{th}} - R_L}{(R_{\text{th}} + R_L)^3} \right]$$
• Step 3: Set $\frac{dP}{dR_L} = 0$: $$R_{\text{th}} - R_L = 0 \implies R_L = R_{\text{th}}$$
• Step 4: Substitute $R_L = R_{\text{th}}$ back into the power equation: $$P_{\text{max}} = \frac{V_{\text{th}}^2 (R_{\text{th}})}{(R_{\text{th}} + R_{\text{th}})^2} = \frac{V_{\text{th}}^2 R_{\text{th}}}{4 R_{\text{th}}^2} = \frac{V_{\text{th}}^2}{4 R_{\text{th}}}$$

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5. Tangents & Normals to Curves

For a curve $y = f(x)$ at point $(x_0, y_0)$:

• Slope of Tangent Line: $m_t = f'(x_0)$
• Equation of Tangent Line: $y - y_0 = m_t(x - x_0)$
• Slope of Normal Line: $m_n = -\frac{1}{m_t} = -\frac{1}{f'(x_0)}$
• Equation of Normal Line: $y - y_0 = m_n(x - x_0)$

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6. Board Exam Summary Table