Lesson 6-3: Rates of Change & Advanced Optimization

Quadratic-Drag Model

Consider vertical motion with air resistance simplified as a quadratic term. Let position be $s(t)$ meters and use a polynomial surrogate for teaching:

Then

This model differs from any cubic used elsewhere in the course and captures non-constant acceleration.

Event Timing

1. Time to peak height: solve $v(t)=0$ for t.
2. Feasible interval: if $h_0>0$, ensure $s(t)ge 0$ while airborne.
3. Sensitivity: analyze how $alpha$ shifts the peak time and height.

Lab A: Tank Draining

1. V(t)=pi r^2 h(t), ; dh/dt proportional to -sqrt{h}
2. Relate dV/dt to dh/dt and parameters
3. Compute instantaneous dh/dt at given h with units

Lab B: Expanding Rectangle

1. Area A=xy with dx/dt, dy/dt given
2. Find dA/dt at an instant
3. Explain sign and magnitude

Lab C: Cooling Coffee

1. Newton's law: dT/dt=-k(T-T_a)
2. Solve and interpret instantaneous cooling rate
3. Discuss effect of k

Lab D: Traffic Flow

1. q=k v, with Greenshields v=v_max(1-k/k_j)
2. Find dq/dk and optimal density
3. Interpret physically

Definitions

Sign conventions matter: negative velocity indicates direction opposite to the positive axis; acceleration shows change in velocity.

Example

If $s(t)=t^3-6t^2+9t$, then:

At t=2, v=-3 (moving negative), a=0 (instantaneous constant speed).

Definitions

At optimum for smooth P, set MR = MC.

Example

If $C(x)=2x^2+5x+10$, $R(x)=20x-x^2$, then $P(x)=-3x^2+15x-10$, optimal at$x=2.5$ (thousand units), $P(2.5)=8.75$ (ten-thousand currency units).

Strategy

1. Identify variables and given rates (units!).
2. Write a constraint equation linking variables.
3. Implicitly differentiate with respect to time t.
4. Substitute known values at the instant; solve for unknown rate.

Classic Example

A ladder of fixed length L leans against a wall, sliding down: $x^2+y^2=L^2$. Differentiate to find dy/dt in terms of dx/dt and x,y.

Expanding Circle with Constraint

Circle area $A=pi r^2$ grows at a constant rate $dA/dt=k$. Find $dr/dt$ and discuss behavior as r increases.

Cone Filling with Leak

Right cone with radius-height ratio fixed. Inflow $q_{in}$, leak proportional to height q_{out}=eta h. Compute $dh/dt$.

Express volume $V= frac{1}{3}pi r^2 h$ with $r=kh$; then dV/dt=q_{in}-eta h and solve for $dh/dt$.

Set A: Velocity & Acceleration

1. Differentiate s(t) to v,a
2. Sign analysis and turning points
3. Units check

Set B: Marginal Analysis

1. Compute MC, MR
2. Find optimal output where MR=MC
3. Interpret economics units

Set C: Related Rates

1. Write constraint
2. Implicitly differentiate
3. Evaluate at instant with units

Set D: Optimization with Constraint

1. Reduce variables
2. Differentiate and solve
3. Check feasibility and bounds

Project A: Motion Profile

• Speed/acceleration limits
• Piecewise v(t)
• Feasibility & smoothness

Project B: Inventory Policy

• EOQ with constraint
• Sensitivity to D,S,H
• Units and feasibility

Project C: Signal Timing

• Queue model
• Optimize cycle time
• Practical constraints

Project D: Risk-Limited Plan

• EV with variance penalty
• Select policy
• Report sensitivity

Project E: Pump Scheduling

• Power cost curve
• Meet demand profile
• Optimize switching times

Project F: Warehouse Layout

• Travel-time model
• Decision variables
• Optimize and interpret

Problem A: Shadow Length

1. Person walks away from lamp
2. Find rate of shadow change
3. Units check

Problem B: Conical Tank

1. Inflow/leak rates
2. dh/dt at instant
3. Parameter sensitivity

Problem C: Expanding Sphere

1. dV/dt known
2. Find dr/dt
3. Interpret growth

Problem D: Sliding Ladder

1. x^2+y^2=L^2
2. Given dx/dt
3. Find dy/dt with sign

Problem E: Heating Coil

1. T(t) model
2. dT/dt at t0
3. Units and meaning

Problem F: Draining Tank

1. Torricelli form
2. Find dh/dt
3. Compare with measurement

Studio 1: Production Mix

• Maximize profit: P = 40x + 30y subject to 2x + y ≤ 100, x + 2y ≤ 80.
• Feasible region: vertices (0,0), (50,0), (40,20), (0,40).
• Corner point analysis: P(40,20) = 2200 maximum.
• Optimal: produce 40 units of X, 20 units of Y for $2200 profit.

Studio 2: Transportation Cost

• Minimize cost: C = 2x₁₁ + 3x₁₂ + 4x₂₁ + x₂₂ with supply/demand constraints.
• Supply: x₁₁ + x₁₂ = 50, x₂₁ + x₂₂ = 30. Demand: x₁₁ + x₂₁ = 40, x₁₂ + x₂₂ = 40.
• Solve using transportation simplex or corner point method.
• Optimal allocation minimizes total shipping cost across network.

Studio 3: Portfolio Optimization

• Maximize return: R = 0.08x + 0.12y - 0.001(x² + y²) with x + y ≤ 10000.
• Quadratic objective with linear constraint, interior maximum possible.
• ∇R = (0.08 - 0.002x, 0.12 - 0.002y), set equal to λ(1,1).
• Optimal: balanced portfolio based on risk-return trade-off.

Studio 4: Facility Location

• Minimize sum of distances: f(x,y) = Σᵢ wᵢ√[(x-aᵢ)² + (y-bᵢ)²].
• Weighted Fermat point problem with multiple customer locations.
• Use iterative methods or geometric median algorithms.
• Optimal location balances customer proximity with demand weights.

Studio 5: Resource Allocation

• Budget constraint: c₁x₁ + c₂x₂ + c₃x₃ = B, maximize utility U(x₁,x₂,x₃).
• Lagrange multipliers: ∇U = λ(c₁,c₂,c₃) at optimum.
• Marginal utility per dollar spent must be equal across all goods.
• Optimal spending pattern depends on individual preferences.

Studio 6: Network Flow

• Maximize flow through network with capacity constraints on edges.
• Conservation at nodes: flow in = flow out (except source/sink).
• Linear programming formulation with flow variables on edges.
• Max-flow min-cut theorem determines network bottlenecks.

Studio 7: Inventory Management

• Multi-item EOQ: minimize Σᵢ [(Dᵢ/Qᵢ)Sᵢ + (Qᵢ/2)Hᵢ] subject to warehouse space.
• Space constraint: Σᵢ vᵢQᵢ ≤ V where vᵢ is volume per unit.
• Lagrange method or heuristic allocation rules.
• Balance holding costs against setup costs and space limitations.

Studio 8: Geometric Optimization

• Minimize perimeter of rectangle inscribed in ellipse x²/a² + y²/b² = 1.
• Rectangle vertices: (±x,±y) on ellipse, perimeter P = 4(x+y).
• Constraint: x²/a² + y²/b² = 1, use substitution or Lagrange.
• Optimal rectangle depends on ellipse aspect ratio a/b.

Studio 9: Scheduling Optimization

• Minimize completion time: schedule n jobs on m machines.
• Precedence constraints and machine capacity limitations.
• Critical path method or integer programming formulation.
• Optimal schedule minimizes project duration and resource conflicts.

Studio 10: Energy Optimization

• Minimize energy: E = ½∫[u'(x)]² dx subject to boundary conditions.
• Calculus of variations: Euler-Lagrange equation u''(x) = 0.
• Solution: u(x) = ax + b, linear interpolation minimizes energy.
• Applications: heat distribution, membrane deformation, optimal control.

Case Study A: Pharmacokinetics

• Two-compartment model (conceptual)
• Elimination rate via derivative
• Interpret half-life

Case Study B: Wind Farm Output

• Power ~ v^3 model
• Marginal change with wind speed
• Capacity factor discussion

Case Study C: Traffic Ramp Metering

• Throughput vs density
• Optimal setpoint via derivative
• Constraints and safety

Case Study D: Cooling Systems

• Heat transfer rate ~ ΔT
• Optimize coolant flow subject to bounds
• Units sanity check

Block A: Units & Domains

1. Check units at each step
2. Mark undefined points
3. Confirm feasible intervals

Block B: Sensitivity

1. Small-Δ parameter change
2. Estimate impact on optimum
3. Report percent change

Block C: Verification

1. Test boundary cases
2. Use alternate method cross-check
3. Numerical sanity test

Block D: Presentation

1. Write assumptions explicitly
2. Include diagrams where helpful
3. State final answer with units