Lecture 14: The Second Derivative
Lecture 14: The Second Derivative
Learning Objectives
- I. Sketch the second derivative
- II. Interpret the second derivative as curvature.
- III. Take the second derivative of polynomials, exponentials, and logarithmic functions.
Due dates and Announcements
- Due Date: 1/18
Understanding the Second Derivative
- The second derivative is defined as the derivative of the derivative of a function. It quantifies the rate at which the first derivative (or the slope of the original function) is changing. In physical terms, if f(x) represents position, then f'(x) is velocity and f''(x) is acceleration.
- Notation: \frac{d^2}{dx^2}f(x) represents the second derivative of the function f(x).
- Mathematically expressed as:
\frac{d^2}{dx^2} f(x) = \frac{d}{dx}\left(\frac{d}{dx} f(x)\right)
Calculation Example: Second Derivative of a Polynomial
- To find the second derivative, let’s consider the function f(x) = x^4:
- First Derivative: f'(x) = 4x^3
- Second Derivative: f''(x) = 12x^2
Example Problem
Find the second derivative
- Task: Compute \frac{d^2}{dx^2} (2x + 3x^2).
- Solution Steps:
- Compute the first derivative: \frac{d}{dx}(2x + 3x^2).
2. After finding the first derivative, differentiate again to find the second derivative.
Review: Sketching Derivatives
Steps to Sketch the First and Second Derivatives:
- Identify Turnaround Points: Points where the function changes direction (potential maxima or minima).
- Determine Positive and Negative Regions: Indicate where the first derivative is positive (original function increasing) and where it is negative (original function decreasing). For the second derivative, positive means concave up, and negative means concave down.
- Estimate Magnitude: Consider the steepness of the slope for both derivatives.
- Identify Inflection Points: Points where the concavity of the original function changes (where f''(x)=0 or is undefined).
Notation Recap:
- Operator: The concept of differentiation itself.
- Fraction: Indicating the operation of taking a derivative.
- Prime: Used to denote first derivative (f'), second derivative (f''), etc.
Interpreting the Second Derivative as Curvature
Concavity of the Function f(x):
- Concave Up: Occurs when the second derivative f''(x) > 0. If a function is concave up at a critical point where f'(x)=0, it indicates a local minimum.
- Indicates that the function is curving upwards, resembling a U-shape.
- Concave Down: Occurs when f''(x) < 0. If a function is concave down at a critical point where f'(x)=0, it indicates a local maximum.
- Inflection Points: Occur when f''(x) = 0 and the sign of f''(x) changes around that point. These are points where the concavity of the function switches from up to down, or vice versa.
Example Curve Analysis:
- For the curve of a given function f(x), analyze the intervals where the second derivative indicates concavity:
- Determine intervals based on values where f''(x) = 0 or where f''(x) is undefined. These points mark potential changes in concavity.
Example Function Analysis
Let’s consider the function f(x) = x^3:
- First Derivative: f'(x) = 3x^2
- Second Derivative: f''(x) = 6x
- Concave Up when x > 0 (as f''(x) > 0).
- Concave Down when x < 0 (as f''(x) < 0).
Second Derivatives of Different Function Types
1. Polynomials
- For a polynomial of the form p(x) = a0 + a1x + a2x^2 + \dots + anx^n: The first derivative is a polynomial of degree n-1. The second derivative is a polynomial of degree n-2.
- The first derivative is also a polynomial of one degree lower.
- The second derivative decreases the degree by another 1. Example: Find the degree of \frac{d^2}{dx^2} (7 + 2x^3 + 2x).
- Solution: The original function is a polynomial of degree 3. The first derivative will be a polynomial of degree 2 (6x^2+2). The second derivative will be a polynomial of degree 1 (12x). Therefore, the second derivative of a degree 3 polynomial is a degree 1 polynomial.
2. Exponential Functions
- For f(x) = e^x:
- First Derivative: f'(x) = e^x
- Second Derivative: f''(x) = e^x
3. Logarithmic Functions
- For g(x) = \ln(x):
- First Derivative: g'(x) = \frac{1}{x}
- Second Derivative: g''(x) = -\frac{1}{x^2}
Biological Case Study: Scoliosis
Problem Statement:
- Scoliosis is a spinal condition where deviation from a straight line occurs.
- A healthy spine can be represented by the function h(x)=0.
- Scoliosis can be modeled mathematically by a curve like S(x) = -13.6x^4 + 36x^3 - 30x^2 + 7.9.
Tasks:
- A. Sketch the first and second derivatives.
- B. Find the second derivative.
- C. Identify intervals for concave up/down:
- Concave Up: When S''(x) > 0;
- Concave Down: When S''(x) < 0;
- Inflection points (where S''(x)=0) indicate points where the spine's curvature changes direction.
Procedure for Biological Function Analysis
Finding Local Extrema:
- A. To check for a local maximum at x_{\text{max}} = 0.191, confirm that:
- S'(0.191) = 0 (first derivative test, indicating a critical point).
- S''(0.191) < 0 (second derivative test, confirming it's a local maximum because the function is concave down at that point).
Finding Inflection Points:
- Solve S''(x) = 0 to find inflection points where the curvature changes.
Second Biological Example: Effects of Atmospheric CO2 on Ocean pH
- As atmospheric CO2 levels rise since the industrial revolution, there are implications for ocean productivity and ecosystems due to the formation of carbonic acid (H2CO_3). Model:
- p(t) = 8.12 - 0.00278t + 0.0000334t^2.
- Where t is the number of years since 1990.
Tasks:
- A. Use the first derivative p'(t) to analyze trends in ocean pH at t = 30 (2020):
- p'(t) = -0.00278 + 2(0.0000334)t
- Compute to find the trend.
- B. Use the second derivative to determine the rate of ocean acidification: