4.2.1 Mean Value Theorem

Introduction & Context

• Previous lecture recap: focused on finding absolute extrema of functions.
• Today’s trajectory:
  • Use that knowledge to motivate and understand Rolle’s Theorem.
  • Show how Rolle’s Theorem forms the foundation of the Mean Value Theorem (MVT).
  • Demonstrate both theorems through an algebraic example (a cubic polynomial) and a real–world meteorology problem (atmospheric lapse rate).
• Key themes
  • Importance of continuity and differentiability as prerequisites.
  • Geometric intuition: connecting secant‐line slopes (average change) to tangent‐line slopes (instantaneous change).

Rolle’s Theorem (RT)

• Formal statement:
  • Let f be continuous on a closed interval [a,b] and differentiable on the open interval (a,b) with equal endpoint values f(a)=f(b).
  • Then \exists\,c\in(a,b) such that f'(c)=0.
• Intuition
  • If a smooth curve starts and ends at the same height, it must turn around somewhere → horizontal tangent.
  • Graphically, imagine a hill or valley between the endpoints.
• Conditions matter
  • Continuity on [a,b] ensures no breaks/holes.
  • Differentiability on (a,b) guarantees a well–defined slope everywhere inside.
• Role in calculus
  • Provides the critical “zero–slope” guarantee that underpins many later theorems (notably MVT).

Example 1 – Applying Rolle’s Theorem to f(x)=x^3-7x^2+10x

1. Verify hypotheses

• Polynomial ⇒ continuous and differentiable \forall x\in\mathbb R (no domain issues).
• Need two distinct inputs a,b with f(a)=f(b).

2. Find suitable endpoints via roots (easiest equal output = 0)

• Factor the function
  • f(x)=x(x^2-7x+10).
  • Quadratic factors: x^2-7x+10=(x-2)(x-5).
• Zeros: x=0,\,2,\,5.
• Choose (not uniquely) a=0 and b=5 ⇒ f(0)=f(5)=0.
  • Any pair of distinct roots would have sufficed; interval choice is flexible as long as f(a)=f(b).

3. Find all c\in(0,5) where f'(c)=0

• Compute derivative (transcript shows a small coefficient slip; we keep transcript’s form but note the issue):
  • Correct derivative should be f'(x)=3x^2-14x+10.
  • Transcript used 3x^2-7x+10 (and followed through). We record both to mirror discussion.
• Using the transcript’s quadratic f'(x)=3x^2-7x+10:
  • Apply quadratic formula x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} with a=3,\,b=-7,\,c=10.
  • Gives two roots c1\approx0.88 and c2\approx3.79.
  • Both lie in (0,5).
• Interpretation: within the chosen interval there are (at least) two interior points where the tangent is horizontal, satisfying Rolle’s Theorem.
• Visual confirmation
  • Graph shows a cubic entering at 0, exiting at 0, with a local max and min inside the interval ⇒ slopes 0 at those extrema.

From Rolle’s Theorem to the Mean Value Theorem

• The proof of MVT literally subtracts a secant‐line function from the original function to create a new function that meets RT’s endpoint‐equality condition.
• Therefore, RT provides the theoretical backbone for MVT.

Mean Value Theorem (MVT)

• Formal statement: If f is
  • continuous on [a,b] and
  • differentiable on (a,b),
  • then \exists\,c\in(a,b) such that
    \frac{f(b)-f(a)}{b-a}=f'(c).
• Geometric meaning
  • The slope of the secant line through (a,f(a)) and (b,f(b)) equals the slope of some tangent line inside the interval.
  • Guarantees at least one point where the instantaneous rate of change matches the average rate of change.
• Key distinctions from Rolle’s Theorem
  • RT requires f(a)=f(b)\implies secant slope $=0$, then finds f'(c)=0.
  • MVT allows any secant slope—RT is a special‐case MVT with slope 0.
• Practical significance
  • Basis for error bounds, numerical methods, proofs of inequalities, physics (average vs. instantaneous velocity), and applied sciences.

Example 2 – Meteorology & Lapse Rate

Background concepts

• Atmospheric lapse rate \gamma: rate at which temperature T decreases with altitude z.
  \gamma = \frac{dT}{dz}\,[\text{°C/km}]
• Typically \gamma<0 (temperature drops as altitude rises).
• Threshold: if |\gamma|>7\,\text{°C/km} in certain layers, conditions favor thunderstorms & tornadoes (assuming other factors cooperate).

Data provided

• T(2.9\,\text{km}) = 7.6\,\text{°C}
• T(5.6\,\text{km}) = -14.3\,\text{°C}
• Assume T(z) is continuous and differentiable for all relevant altitudes.

Applying the MVT

1. Check hypotheses: stated as continuous & differentiable ⇒ MVT applies on [2.9,5.6].
2. Compute secant (average) slope
   \frac{T(5.6)-T(2.9)}{5.6-2.9} = \frac{-14.3-7.6}{2.7} = \frac{-21.9}{2.7} \approx -8.1\,\text{°C/km}.
3. Conclusion from MVT
   • \exists\,c\in(2.9,5.6) such that T'(c) = -8.1\,\text{°C/km}.
   • Magnitude |\gamma|=8.1\,>7\,\text{°C/km} ⇒ at some altitude within the layer, the lapse‐rate criterion for severe weather is satisfied.
4. Meteorological interpretation
   • A forecaster can infer that potential exists for thunderstorms/tornadoes in that altitude band, subject to humidity, wind shear, and other atmospheric variables.
   • MVT does not specify where the critical altitude c lies or how long the condition persists—additional data required.

Diagram (verbal)

• Two plotted points (2.9, 7.6) and (5.6, –14.3).
• Secant line drawn through them.
• Somewhere between, a tangent line touches the (unknown) temperature curve with the same slope –8.1 °C/km.

Broader Connections & Caveats

• Continuity & differentiability are non‐negotiable: any break, cusp, or vertical tangent invalidates RT/MVT.
• Multiple interior points c often exist; the theorems guarantee at least one, not uniqueness.
• Direction for future study
  • Generalizations: Cauchy’s MVT, Darboux’s property, Taylor’s Theorem (built on MVT).
  • Applications: proving the Fundamental Theorem of Calculus, bounding numerical integration errors, analyzing motion (average vs. instantaneous velocity), proving inequalities.
• Transcript caution
  • Derivative coefficient slip (7 vs. 14) does not alter qualitative understanding but affects numeric roots; always verify algebra when applying RT/MVT in practice.

Key Takeaways & Study Tips

• Rolle’s Theorem: equal endpoints ⇒ at least one zero derivative inside.
• Mean Value Theorem: average rate equals some instantaneous rate within the interval.
• Both rely on the same smoothness assumptions and offer geometric insight into function behavior.
• Practice: compute secant slopes, then locate or approximate points where tangent slopes match.
• Real‐world relevance: weather forecasting, physics of motion, engineering stress analysis—any scenario comparing average vs. instantaneous change.
• When solving problems:
  1. Verify continuity & differentiability.
  2. Identify or construct suitable interval [a,b].
  3. Compute secant slope \frac{f(b)-f(a)}{b-a}.
  4. Apply RT/MVT to assert existence (and sometimes find) c.
  5. Interpret result in context.