Theorems of Calculus: Rolle’s & Mean Value Theorem

1. Introduction

Theorems of calculus bridge the gap between differentiation and function behavior over closed intervals. Two fundamental theorems—Rolle’s Theorem and the Mean Value Theorem (MVT)—provide conditions under which a function must have specific properties (e.g., horizontal tangents or average slopes). These theorems are essential for:

• Proving function properties (e.g., uniqueness of roots).
• Understanding rates of change.
• Solving optimization and approximation problems.

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2. Rolle’s Theorem

2.1 Definition

Rolle’s Theorem states that if a function $f$ satisfies the following conditions:

1. $f$ is continuous on the closed interval $[a,b]$.
2. $f$ is differentiable on the open interval $(a,b)$.
3. $f(a)=f(b)$ (i.e., the function has equal values at the endpoints).

Then, there exists at least one point $c\in (a,b)$ such that: $$f^{\prime }(c)=0$$ This means the function has a horizontal tangent (local maximum, minimum, or saddle point) somewhere in $(a,b)$.

2.2 Geometric Interpretation

• If a smooth curve starts and ends at the same height, it must have at least one point where its slope is zero (flat tangent).
• Example: A ball thrown upward and landing at the same height must momentarily stop (velocity = 0) before descending.

2.3 Proof Sketch

1. By the Extreme Value Theorem, $f$ attains a maximum or minimum on $[a,b]$.
2. If the extremum occurs at an interior point $c$, then $f^{\prime }(c)=0$ (by Fermat’s Theorem on critical points).
3. If the extremum is at the endpoints, $f$ is constant (trivial case where $f^{\prime }(x)=0$ everywhere).

2.4 Worked Example: Verification

Problem: Verify Rolle’s Theorem for $f(x)=x^2-3x+2$ on $[0,3]$. Solution:

1. Check continuity: $f(x)$ is a polynomial → continuous everywhere.
2. Check differentiability: Polynomial → differentiable everywhere.
3. Check $f(0)=f(3)$: $$f(0)=0-0+2=2$$ $$f(3)=9-9+2=2$$ Condition satisfied.
4. Find $f^{\prime }(x)$ and solve $f^{\prime }(c)=0$: $$f^{\prime }(x)=2x-3$$ $$2c-3=0⟹c=1.5$$ Since $1.5\in (0,3)$, Rolle’s Theorem is verified.

2.5 Limitations of Rolle’s Theorem

• Requires $f(a)=f(b)$: Fails if endpoints differ (e.g., $f(x)=x$ on $[0,1]$ has no $c$ with $f^{\prime }(c)=0$).
• Only guarantees existence: Does not specify how many such points $c$ exist.

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3. Mean Value Theorem (MVT)

3.1 Definition

The Mean Value Theorem generalizes Rolle’s Theorem. If a function $f$ satisfies:

1. $f$ is continuous on $[a,b]$.
2. $f$ is differentiable on $(a,b)$.

Then, there exists at least one point $c\in (a,b)$ such that: $$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$$ This represents the average rate of change of $f$ over $[a,b]$, achieved instantaneously at $c$.

3.2 Geometric Interpretation

• The tangent at $c$ is parallel to the secant line connecting $(a,f(a))$ and $(b,f(b))$.
• Example: If you drive from Kathmandu to Pokhara at an average speed of 50 km/h, at some instant your speedometer must read exactly 50 km/h.

3.3 Proof Sketch

1. Define a helper function: $$g(x)=f(x)-\left[f(a)+\frac{f(b)-f(a)}{b-a}(x-a)\right]$$ This adjusts $f(x)$ to satisfy $g(a)=g(b)$.
2. Apply Rolle’s Theorem to $g(x)$: There exists $c\in (a,b)$ such that $g^{\prime }(c)=0$.
3. Compute $g^{\prime }(x)=f^{\prime }(x)-\frac{f(b)-f(a)}{b-a}$, so: $$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$$

3.4 Worked Example: Verification

Problem: Verify MVT for $f(x)=x^3-3x+2$ on $[-1,2]$. Solution:

1. Check continuity/differentiability: Polynomial → satisfied.
2. Compute average slope: $$\frac{f(2)-f(-1)}{2-(-1)}=\frac{(8-6+2)-(-1+3+2)}{3}=\frac{4-4}{3}=0$$
3. Find $f^{\prime }(x)=3x^2-3$. Solve $f^{\prime }(c)=0$: $$3c^2-3=0⟹c^2=1⟹c=\pm 1$$ Only $c=1\in (-1,2)$ is valid. Thus, MVT is verified at $c=1$.

3.5 Comparison with Rolle’s Theorem

3.6 Applications of MVT

1. Proving inequalities: Show that $f^{\prime }(x)$ cannot always be positive/negative.
2. Uniqueness of solutions: If $f^{\prime }(x)\ne 0$ on $(a,b)$, then $f$ has at most one root in $[a,b]$.
3. Error estimation: Used in Taylor’s Theorem to bound approximation errors.
4. Physics: Relates average velocity to instantaneous velocity.

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4. Common Mistakes and Pitfalls

1. Ignoring differentiability: Forgetting to check if $f$ is differentiable on $(a,b)$ (e.g., $f(x)=|x|$ at $x=0$).
2. Misapplying conditions: Rolle’s Theorem requires $f(a)=f(b)$; MVT does not.
3. Assuming uniqueness: MVT guarantees at least one $c$, not necessarily all.
4. Arithmetic errors: Incorrectly computing $f(b)-f(a)$ or $b-a$.

4.1 Counterexample Analysis

Problem: Does MVT apply to $f(x)=\sqrt[3]{x}$ on $[-1,1]$? Solution:

• Continuity: Yes, $f$ is continuous everywhere.
• Differentiability: $f^{\prime }(x)=\frac{1}{3}{x}^{-2/3}$, which is undefined at $x=0$.
• Conclusion: MVT fails because $f$ is not differentiable on $(-1,1)$.

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5. Exam Tip: How to Score Full Marks

5.1 For Definitions (2–4 marks)

• State all conditions explicitly (continuity, differentiability, endpoint equality for Rolle’s).
• Write the conclusion correctly:
  • Rolle’s: $\exists c\in (a,b)$ such that $f^{\prime }(c)=0$.
  • MVT: $\exists c\in (a,b)$ such that $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$.

5.2 For Verification Problems (6–8 marks)

1. Check all conditions (1 mark each):
   • Continuity on $[a,b]$.
   • Differentiability on $(a,b)$.
   • For Rolle’s: $f(a)=f(b)$.
2. Compute derivatives correctly (2 marks).
3. Solve $f^{\prime }(c)=\text{target value}$ and verify $c\in (a,b)$ (3 marks).
4. Interpretation: Briefly explain why the theorem holds (e.g., "The function attains a critical point...").

5.3 Common Exam Questions

5.4 Sample Answer Structure

Question: Verify MVT for $f(x)=x^2-4x+3$ on $[1,3]$. Model Answer:

1. Conditions:
   • $f(x)$ is a polynomial → continuous on $[1,3]$ and differentiable on $(1,3)$.
2. Average slope: $$\frac{f(3)-f(1)}{3-1}=\frac{(9-12+3)-(1-4+3)}{2}=\frac{0-0}{2}=0$$
3. Derivative: $$f^{\prime }(x)=2x-4$$ Set $f^{\prime }(c)=0⟹2c-4=0⟹c=2$.
4. Conclusion: Since $2\in (1,3)$, MVT is verified at $c=2$.

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6. Practice Problems

1. Verify Rolle’s Theorem for $f(x)=\sin x$ on $[0,\pi ]$.
2. Verify MVT for $f(x)=x^3-2x^2+x$ on $[0,2]$.
3. Use Rolle’s Theorem to show that $f(x)=x^3-3x$ has exactly one root in $[0,2]$.
4. Find all $c$ in $(0,\pi )$ such that MVT holds for $f(x)=\cos x$ on $[0,\pi ]$.

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7. Summary Table

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8. References

• Stewart, J. (2015). Calculus: Early Transcendentals. Cengage.
• Thomas, G. B., Weir, M. D., & Hass, J. (2014). Thomas’ Calculus. Pearson.
• TU Syllabus for Mathematics I (MTH117), Semester 1.