4. Given a series circuit with a source (V), source resistance (R.), and load resistance (R₁), show that maximum power transfer occurs when R. = R₁ . Start with the expression for current in the circuit, then with the expression for power dissipated in the load. (Hint: You will need to use calculus to solve).

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Thevenin Theorem

Any linear circuit, no matter how complex, maybe simplified to an equivalent circuit utilizing only a single voltage source and series resistance coupled to a load, according to Thevenin's Theorem. The superposition principle says that all foundational equations must be linear, and the notion of "linear" is still the same (i.e., there should not be any exponentials or roots). This is particularly the case when operating with passive elements like resistors, inductors, and capacitors. However, some components (particularly some gaseous and semiconductor devices like tunnel diode, Schottky diode, transistors like BJT, MOSFET) are nonlinear, meaning their current resistance fluctuates with voltage and current. As a result, circuits made up of these parts are known as nonlinear circuits.