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"You can master mathematics if you practice enough final exam reviews of desired mathematical subjects. Being good at mathematics is a matter of practice". - O.J.B.
| Substitute given variables with custom <br> variables (e.g., AbcdEfG) | Solve the equation using your own variables, then mirror the <br> steps onto the original problem for proportional reasoning. |
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| Interpret the equal sign as <br> "converts to" | Thinking of "=" as "converts to" can facilitate substitutions & <br> manipulations in other mathematical expressions. |
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| Think in terms of ratios by <br> default | Viewing values as ratios can simplify problem-solving & <br> conceptual understanding. |
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| Understand the difference between analytical vs. numerical approaches| Exact solutions are often associated with analytical approaches while approximation or discretized solutions are often associated with numerical appoaches. |
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| Use software tools for conversion <br> to markdown or LaTeX | Convert equations for better inspection & rendering, ensuring <br> accuracy. |
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| Leverage Python & libraries <br> like SymPy | Write equations in Python for execution & manipulation, <br> aiding clarity & verification. |
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| Remember solutions on graphs <br> are line intersections | Graphical solutions typically correspond to intersection points <br> of lines or curves. |
|**Arbitrary**| Refers to chosen values or units that maintain internal consistency without relying on an external, standardized reference. Example: arbitrary units used in graphs and charts. |
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|**Arbitrary Units (a.u.)**| Used in graphs and charts, they represent a consistent measure but do not correspond to a standardized physical unit. They are meaningful within the given context but are not directly comparable to a universal scale. |
|**Definition**: A solution that, given the same initial conditions, **always produces the same result**. These solutions are fully **predictable** and **can be expressed in a closed mathematical form**. |**Definition**: A solution where the outcome is **probabilistic or dependent on unknown/uncontrollable factors**, even if the underlying equations are well-defined. |
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|**Characteristics**: <br> - No randomness or probability involved. <br> - Given a set of initial conditions and equations, the result is always the same. <br> - Typically derived using exact algebraic, calculus-based, or differential equation methods. |**Characteristics**: <br> - **Involves probabilities or randomness** in the results. <br> - Repeating the same conditions **does not always yield the same outcome**. <br> - Often appears in **quantum mechanics, chaotic systems, and stochastic processes**. |
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| **Examples**: <br> 1. **Newton’s Laws of Motion**: <br> - If you apply a known force to an object, its acceleration and trajectory can be determined exactly. <br> - Example: $( x(t) = x_0 + v_0t + \frac{1}{2}at^2 )$ (kinematics equation). <br> <br> 2. **Ohm’s Law in Circuits**: <br> - $( V = IR )$ gives the exact voltage given current $( I )$ and resistance $( R )$. <br> <br> 3. **Schrödinger's Equation for Simple Systems**: <br> - The time-independent Schrödinger equation for a hydrogen atom yields **exact energy eigenvalues** for electron states. | **Examples**: <br> 1. **Quantum Mechanics (Wavefunction Collapse)**: <br> - The Schrödinger equation **deterministically** evolves a wavefunction, but upon measurement, the outcome is probabilistic. <br> - Example: Measuring the spin of an electron in a superposition state gives a **random outcome** (e.g., 50% spin-up, 50% spin-down). <br> <br> 2. **Radioactive Decay**: <br> - The decay of a single nucleus follows a **probability distribution**, not a deterministic function. <br> - We can only predict **half-life**, but not when a specific atom will decay. <br> <br> 3. **Chaotic Systems (Butterfly Effect)**: <br> - Some classical systems, like **weather models**, follow deterministic equations but are **highly sensitive to initial conditions**, making long-term predictions effectively non-deterministic. <br> <br> 4. **Monte Carlo Simulations**: <br> - Used in optimization and physics, these rely on **random sampling** to approximate solutions to complex problems. |
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|**Summary**: <br> - **Deterministic solutions** provide exact answers **every time** for given conditions. |**Summary**: <br> - **Indeterministic (or non-deterministic) solutions** involve **probabilities** or **sensitive dependencies**, making exact results uncertain, even if the equations governing the system are known. |
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