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 $c<em>1\approx0.88$ and $c</em>2\approx3.79.$
  • 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.