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For steady, incompressible, and inviscid flow along a streamline, the energy per unit volume remains constant:

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$

Breakdown of terms

• $P$ — Static pressure of the fluid (units: Pa or $N/m^2$)
• $\frac{1}{2}\rho v^2$ — Dynamic pressure (kinetic energy component)
• $\rho g h$ — Hydrostatic pressure (potential energy component)

Variables defined

• $\rho$ (rho) — Fluid density ($kg/m^3$)
• $v$ — Fluid velocity ($m/s$)
• $g$ — Acceleration due to gravity (approx $9.81\ m/s^2$)
• $h$ — Height above a reference level (m)

For any equation in the form $ax^2 + bx + c = 0$, the solutions are:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Worked example: solve $2x^2 + 3x - 5 = 0$

• $a = 2$, $b = 3$, $c = -5$
• Discriminant: $b^2-4ac = 9 - (4)(2)(-5) = 49$
• $x = \frac{-3 \pm 7}{4}$, so $x = 1$ or $x = -2.5$

When to use it: any quadratic that doesn't factor cleanly. The discriminant also tells you how many real solutions exist before you finish solving.

Eight-step cycle in the mitochondria that oxidizes acetyl-CoA, the entry point for the energy released by cellular respiration overall:

$C_6H_{12}O_6 + 6O_2 \rightarrow 6CO_2 + 6H_2O$

Net yield per turn

• NADH — 3 produced, feeds the electron transport chain
• FADH₂ — 1 produced, same pathway
• ATP — 1 produced directly via substrate-level phosphorylation

Unbalanced: $C_3H_8 + O_2 \rightarrow CO_2 + H_2O$

Balance one element at a time

• Carbon: 3 C on the left needs 3 $CO_2$ on the right
• Hydrogen: 8 H on the left needs 4 $H_2O$ on the right
• Oxygen: count on the right first ($6 + 4 = 10$), then fix the left

Balanced equation

$C_3H_8 + 5O_2 \rightarrow 3CO_2 + 4H_2O$

Rule of thumb: balance carbon and hydrogen first, oxygen last. It's usually the one that needs a fraction before you clear it.

A price ceiling is a legal maximum price set below the market equilibrium price.

What happens at that price

• Quantity demanded rises (it's cheaper, more people want it)
• Quantity supplied falls (producers earn less, so they supply less)
• The gap between the two is the shortage

Key idea: the shortage isn't a flaw in the market. It's the direct, predictable result of not letting price rise to clear it: the further below equilibrium the ceiling is set, the bigger the shortage.

For a composite function $f(g(x))$:

$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Worked example: differentiate $y = (3x^2 + 1)^5$

• Outer function: $u^5$, derivative $5u^4$
• Inner function: $u = 3x^2+1$, derivative $6x$
• Combine: $\frac{dy}{dx} = 5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4$

How to spot it: if you can describe the function as "something happening to something else," you need the chain rule.

The assassination of Archduke Franz Ferdinand triggered the war, but four longer-term forces made Europe ready to ignite:

• Militarism — an arms race and glorification of military strength across major powers
• Alliances — a web of defense treaties meant Austria-Hungary vs. Serbia became all of Europe
• Imperialism — competition over colonies bred resentment between rival empires
• Nationalism — ethnic pride and independence movements destabilized multi-ethnic empires

Why it matters: any one of these alone probably doesn't cause a world war. It's the combination that turned a regional assassination into a continental war within weeks.

A recursive function calls itself on a smaller version of the same problem, until it hits a base case that stops the recursion.

function factorial(n):
    if n == 0: return 1        # base case
    return n * factorial(n-1)  # recursive case

Tracing factorial(4)

• factorial(4) = 4 × factorial(3)
• factorial(3) = 3 × factorial(2)
• factorial(2) = 2 × factorial(1)
• factorial(1) = 1 × factorial(0)
• factorial(0) = 1 ← base case hit, now it unwinds: 1×1×2×3×4 = 24

The two things every recursive function needs: a base case that stops it, and a recursive case that makes real progress toward that base case.

For a data set: 2, 4, 4, 4, 5, 5, 7, 9

$\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$

Steps

• Mean: $\bar{x} = 5$
• Squared differences: 9, 1, 1, 1, 0, 0, 4, 16
• Average of those: $32/8 = 4$
• $\sigma = \sqrt{4} = 2$

What it means: most values in this set sit within about 2 points of the mean (5). A small $\sigma$ means tightly clustered data, a large one means spread out.

Weak: "There are many themes in The Great Gatsby." (states a fact, takes no position)

Strong: "Fitzgerald uses Gatsby's parties to expose the emptiness beneath the wealth of the 1920s American elite." (arguable, specific, previews the evidence)

Three things every strong thesis needs

• Takes a position — something a reasonable person could disagree with
• Is specific — names the actual text/evidence, not just a vague topic
• Previews the argument — hints at how you'll prove it, not just what you believe

Quick test: if your thesis could be the answer to "so what?", it's probably still too vague.