Week-09

Maths 1: Week 09 - The Master Encyclopedia of Optimization extremes

1. The Genesis: The Search for the Peak 📜

1.1 The Logic of Efficiency

Until now, we have learned to find the “Speed” of a function. But why does speed matter? Because in the real world, we want to know when speed is Zero.

Think about it: When a ball is thrown into the air, its velocity slows down until it hits the very peak. At that exact split-second, its speed is Zero. By finding where the derivative is zero, we find the “Peaks” and “Valleys” of the world.

This is Optimization. It is how Amazon minimizes its delivery costs, how engineers maximize bridge strength, and how investors find the peak of a market trend. In the second half of this week, we introduce the opposite of moving fast: the Accumulation of area, the birth of Integration.

1.2 The Philosophical Intuition

If the Derivative is about the Moment, then Optimization is about the Best Moment. And Integration is the story of how many “Moments” were spent to create a “Whole.”

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2. Axiomatic Foundations: The Critical Points 🏛️

2.1 Maxima & Minima (Extrema)

A function $f(x)$ has an “Extremum” at $x=c$ if it is higher (Max) or lower (Min) than all surrounding points.

• Critical Point Criterion: To find these points, we search for where $f^{\prime }(x)=0$ or $f^{\prime }(x)$ is undefined.
• The Global vs. Local Standard:
  • Local: The king of its neighborhood.
  • Global: The absolute champion across the entire domain.

2.2 Fermat’s Theorem

If $f$ has a local extremum at $c$, and $f^{\prime }(c)$ exists, then $f^{\prime }(c)=0$. Warning: The reverse is not always true! A derivative of zero doesn’t guarantee a peak—it could be a “Saddle Point” (like a plateau where you keep going).

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3. Validation Mechanics: Which Way is Up? 🛠️

3.1 The First Derivative Test (The Directional Check)

1. If $f^{\prime }$ changes from $(+)$ to $(-)$: You just climbed a peak. (Local MAX).
2. If $f^{\prime }$ changes from $(-)$ to $(+)$: You just hit the bottom. (Local MIN).

3.2 The Second Derivative Test (The Concavity Check)

• $f^{\prime }(c)=0$ AND $f^{\prime \prime }(c)<0$: The graph is “Opening Down” at a flat point. MUST be a MAX.
• $f^{\prime }(c)=0$ AND $f^{\prime \prime }(c)>0$: The graph is “Opening Up” at a flat point. MUST be a MIN.

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4. Mean Value Theorem: The Speeding Ticket 🖋️

4.1 The Theory

If a function is smooth and continuous on $[a,b]$, then there must be at least one point $c$ in between where your “Instantaneous Speed” matches your “Average Speed.”

• Physical Metaphor: If you drive 100 miles in 2 hours (average 50mph), at some point your speedometer MUST have read exactly 50mph. You can’t reach the average without hitting it at least once.

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5. Introduction to Integration: Riemann Sums 📈

5.1 The Area Problem

How do you find the area of a shape with a curvy roof?

• The Strategy: Chop the area into many thin rectangles. Add up the area of the rectangles.
• The Integration Axiom: As the number of rectangles ($n$) goes to infinity, the sum becomes the perfect area under the curve.
• The Symbol: ${\int }_a^{b}f(x)dx$.

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6. The Encyclopedia of Worked Examples (10 Case Studies) 📚

Case 1: Finding Critical Points

Problem: Find critical points of $f(x)=x^3-12x+5$.

• Step 1: $f^{\prime }(x)=3x^2-12$.
• Step 2: $3x^2-12=0⟹x^2=4⟹x=\pm 2$.
• Result: Critical points at $2$ and $-2$.

Case 2: Identifying Max vs Min

Problem: Is the point $x=2$ in Case 1 a Max or Min?

• Step 1: $f^{\prime \prime }(x)=6x$.
• Step 2: $f^{\prime \prime }(2)=12$. Since $12>0$, the graph is opening up.
• Result: $x=2$ is a Local Minimum.

Case 3: Maximizing Area (The Fence Problem)

Problem: You have 100m of fence to build a rectangle against a wall. Maximize the area.

• Logic: Perimeter $P=2x+y=100$. (Only 3 sides). $y=100-2x$.
• Area: $A=x\cdot y=x(100-2x)=100x-2x^2$.
• Optimize: $A^{\prime }=100-4x=0⟹x=25$.
• Dimensions: $25\text{m}\times 50\text{m}$.
• Result: Max area $=1,250{\text{m}}^2$.

Case 4: Calculating Increasing/Decreasing Intervals

Problem: On which interval is $f(x)=x^2$ increasing?

• Logic: Where is $f^{\prime }(x)>0$? $2x>0⟹x>0$.
• Result: $(0,\infty )$.

Case 5: Root of a Derivative (Horizontal Tangent)

Problem: Does $y=x^2-4x+7$ have a horizontal tangent?

• Calculation: $y^{\prime }=2x-4$. $2x-4=0⟹x=2$.
• Result: Yes, at $x=2$.

Case 6: Approximating Area with 2 Rectangles (Riemann)

Problem: Estimate area under $f(x)=x^2$ from 0 to 2 using 2 rectangles (Right Sum).

• Step 1: Width $\Delta x=(2-0)/2=1$.
• Step 2: Heights at $x=1$ and $x=2$. $f(1)=1,f(2)=4$.
• Sum: $(1\times 1)+(1\times 4)=5$.
• Result: 5 units.

Case 7: Finding the Points of Inflection

Problem: Where does $f(x)=x^3-3x^2$ change concavity?

• Step 1: $f^{\prime }=3x^2-6x$.
• Step 2: $f^{\prime \prime }=6x-6$.
• Step 3: $6x-6=0⟹x=1$.
• Result: Inflection point at $(1,-2)$.

Case 8: The Closed Interval Method

Problem: Find absolute max/min of $f(x)=x^2$ on $[-1,2]$.

• Test Critical Points: $f^{\prime }(x)=0$ at $x=0$. $f(0)=0$.
• Test Endpoints: $f(-1)=1$, $f(2)=4$.
• Result: Absolute Max is at $x=2$ (Value 4). Absolute Min is at $x=0$ (Value 0).

Case 9: Slope of an Integral

Problem: What is the derivative of ${\int }_0^x{t}^{2}dt$?

• The Fundamental Theorem: It is just the original function $x^2$.
• Result: $x^2$.

Case 10: Optimizing Profit

Problem: Profit $P(x)=-x^2+10x-5$. At what sales volume $x$ is profit max?

• Calculation: $P^{\prime }=-2x+10=0⟹x=5$.

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7. Fundamental “How-To” Recipes 🍳

Recipe: The Optimization Masterplan

1. Objective: Find the equation for the quantity you want to Max/Min.
2. Constraint: Use external info to substitute variables so you only have ONE variable ($x$).
3. Differentiate: Find the critical points.
4. Confirm: Use the 2nd derivative to make sure you found a peak and not a valley.

Recipe: Sketching using Calculus

1. Dots: Plot the X/Y intercepts.
2. Peaks/Valleys: Plot the Critical Points ($f^{\prime }=0$).
3. Arcs: Use $f^{\prime \prime }$ to decide if it bends up or down between the dots.
4. Connect: You now have a perfect topological map.

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8. Encyclopedia Mastery Challenge 🏆

1. The Mystery Point: Can a function have $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)=0$? Sketch $y=x^3$ at $x=0$. Is it a max or min?
2. The MVT Proof: If you started at $(0,0)$ and ended at $(1,1)$, why must there be a point where the slope is 1?
3. The Rectangle Convergence: If we use $n=1,000,000$ rectangles, is the area more or less accurate than using $100$ rectangles? Why?
4. The Rolle Theorem: If $f(a)=f(b)$ and the function is continuous, show that there must be a point between them where you are “not moving” ($f^{\prime }=0$).

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🚀 Master Status: You have completed the Encyclopedic Expansion of Optimization and Accumulation. You now hold the keys to finding the “Best” outcome in any system. 助