Higher Derivatives and the Mean Value Theorems: Quick Revision

A brief note on Higher Derivatives and the Mean Value Theorems. Read it before a test. Try each question at the end before you open its answer.

Higher derivatives

• Notation: $f''(x)$, $\dfrac{d^2y}{dx^2}$, $y_2$. The $n$th derivative is $f^{(n)}(x)$.
• Meaning: $f''$ is the rate of change of the slope. If $f$ is position, $f''$ is acceleration.
• Bend: $f'' > 0$ bends up. $f'' < 0$ bends down.
• Parametric: $\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)\Big/\dfrac{dx}{dt}$.

Rolle's theorem

• Statement: $f$ continuous on $[a, b]$, differentiable on $(a, b)$ and $f(a) = f(b)$ imply $f'(c) = 0$ for some $c$ in $(a, b)$.
• Meaning: a horizontal tangent between two points of equal height.
• Use: between two roots of $f$ there is a root of $f'$. This bounds the number of real roots.

The Mean Value Theorem

• Statement: $f$ continuous on $[a, b]$ and differentiable on $(a, b)$ imply $f'(c) = \dfrac{f(b) - f(a)}{b - a}$ for some $c$ in $(a, b)$.
• Meaning: a tangent parallel to the chord. The average rate equals an instantaneous rate.
• Uses: speed, constants ($f' = 0$ means $f$ is constant) and inequalities.

A worked example

Find the point $c$ of the Mean Value Theorem for $f(x) = x^2 + 1$ on $[1, 4]$.

1. The chord slope. $f(4) = 17$ and $f(1) = 2$. The slope is $\dfrac{15}{3} = 5$.
2. Solve $f'(c) = 5$. $2c = 5$, so $c = 2.5$.
3. Check. $2.5$ lies in $(1, 4)$.

Practice with new numbers

A program to try

The program checks Rolle's theorem and the Mean Value Theorem on $x^3 - 3x$, including the second derivative at $c$.

Mistakes to avoid

• Skipping a condition of the theorem.
• A point outside the open interval. Keep only the solutions inside $(a, b)$.
• Using the midpoint as $c$. That works only for a quadratic.
• A wrong parametric formula. Differentiate $\dfrac{dy}{dx}$ with respect to $t$, then divide by $\dfrac{dx}{dt}$.
• Reading $\dfrac{d^2y}{dx^2}$ as a power. The $2$ is the order of the derivative.

Check yourself

1. Find $y''$ for $y = \sin 4x$.

2. Does the graph of $y = x^3$ bend up or down at $x = -1$?

3. Find $c$ for Rolle's theorem with $f(x) = x^2 - 2x$ on $[0, 2]$.

4. Find $c$ for the Mean Value Theorem with $f(x) = x^2$ on $[0, 6]$.

5. A car travels $90$ km in $1.5$ hours. What does the Mean Value Theorem say?

Sources

• OpenStax, Calculus Volume 1, Chapters 3 and 4.
• NCERT, Mathematics Part I for Class XII, Chapter 5.