Interpreting Second Derivatives in Motion and Curve Analysis

Introduction

Key Concepts

1. Understanding Second Derivatives

The second derivative of a function, denoted as \( f''(x) \) or \( \frac{d^2f}{dx^2} \), represents the derivative of the first derivative \( f'(x) \). It provides information about the curvature and concavity of the original function \( f(x) \).

2. Concavity and Convexity

Concavity describes the direction a curve bends. If \( f''(x) > 0 \) for all \( x \) in an interval, the function is concave upward on that interval, resembling a cup that can hold water. Conversely, if \( f''(x) < 0 \), the function is concave downward, akin to a frown.

**Example:** Consider the function \( f(x) = x^3 - 3x^2 + 2x \). First, find the second derivative:

$$ f'(x) = 3x^2 - 6x + 2 \\ f''(x) = 6x - 6 $$

Setting \( f''(x) = 0 \) gives \( x = 1 \). Testing intervals around \( x = 1 \) determines concavity changes.

3. Points of Inflection

A point of inflection occurs where the function changes its concavity, i.e., where \( f''(x) \) changes sign. This is identified by solving \( f''(x) = 0 \) and verifying a sign change in \( f''(x) \) around the critical point.

**Example:** Using the previous function \( f(x) = x^3 - 3x^2 + 2x \), \( f''(1) = 0 \). Checking \( x = 0 \) and \( x = 2 \):

• At \( x = 0 \), \( f''(0) = -6 \) (concave downward)
• At \( x = 2 \), \( f''(2) = 6 \) (concave upward)

Hence, \( x = 1 \) is a point of inflection.

4. Second Derivative Test for Extrema

The second derivative test helps determine the nature of critical points (where \( f'(x) = 0 \)). If \( f''(x) > 0 \) at a critical point, the function has a local minimum there. If \( f''(x) < 0 \), there's a local maximum. If \( f''(x) = 0 \), the test is inconclusive.

**Example:** For \( f(x) = x^3 - 3x^2 + 2x \), critical points are found by setting \( f'(x) = 0 \):

$$ 3x^2 - 6x + 2 = 0 \\ x = \frac{6 \pm \sqrt{36 - 24}}{6} = \frac{6 \pm \sqrt{12}}{6} = 1 \pm \frac{\sqrt{3}}{3} $$

Evaluating \( f''(x) \) at these points determines if they are maxima or minima.

5. Application in Motion: Acceleration

In the context of motion, the second derivative of the position function \( s(t) \) with respect to time \( t \) represents acceleration \( a(t) \), i.e., \( a(t) = s''(t) \). Analyzing \( a(t) \) provides insights into the changes in velocity and the behavior of the moving object.

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

$$ s'(t) = 3t^2 - 12t + 9 \quad (\text{Velocity}) \\ s''(t) = 6t - 12 \quad (\text{Acceleration}) $$

Setting \( s''(t) = 0 \) gives \( t = 2 \), indicating a change in acceleration.

6. Curve Sketching Using Second Derivatives

Second derivatives are instrumental in sketching the graph of a function by determining concavity, points of inflection, and the nature of critical points.

• Find the first and second derivatives.
• Determine critical points by setting \( f'(x) = 0 \).
• Use the second derivative to test concavity and classify critical points.
• Identify points of inflection by solving \( f''(x) = 0 \) and checking for sign changes.

**Example:** For \( f(x) = \ln(x) \), determine concavity and points of inflection.

$$ f'(x) = \frac{1}{x} \\ f''(x) = -\frac{1}{x^2} $$

Since \( f''(x) < 0 \) for all \( x > 0 \), the function is concave downward everywhere, and there are no points of inflection.

7. Second Derivatives of Parametric Equations

For parametric equations defined by \( x(t) \) and \( y(t) \), the second derivative \( \frac{d^2y}{dx^2} \) is computed using:

$$ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \\ \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} $$

This allows for analyzing the concavity and curvature of parametric curves.

**Example:** Given \( x(t) = t^2 \) and \( y(t) = t^3 \), find \( \frac{d^2y}{dx^2} \).

$$ \frac{dy}{dt} = 3t^2, \quad \frac{dx}{dt} = 2t \\ \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2} \\ \frac{d^2y}{dt^2} = 3t \\ \frac{d^2y}{dx^2} = \frac{3t}{2t} = \frac{3}{2} $$

The second derivative is a constant \( \frac{3}{2} \), indicating constant concavity.

8. Higher-Order Derivatives

While the second derivative provides significant information about concavity and acceleration, higher-order derivatives (third, fourth, etc.) can offer deeper insights into the behavior and properties of functions, although they are beyond the scope of standard Calculus BC curriculum.

Comparison Table

Aspect	First Derivative	Second Derivative
Definition	Rate of change or slope of the function	Rate of change of the first derivative; concavity
Interpretation	Determines increasing/decreasing intervals and local extrema	Determines concave up/down and points of inflection
Application in Motion	Velocity (first derivative of position)	Acceleration (second derivative of position)
Second Derivative Test	Not applicable	Classifies critical points as local minima or maxima
Graphical Implications	Slopes tangents to the curve	Curvature and bending of the curve

Summary and Key Takeaways