Second Derivative
Definition
The second derivative of a function represents the rate at which the slope of the original function is changing. It measures how the rate of change of the first derivative changes.
Analogy
Imagine you are driving a car on a curvy road. The first derivative tells you how fast your speed is changing, while the second derivative tells you how quickly your acceleration (change in speed) is changing. If the second derivative is positive, it means you're accelerating faster and taking sharper turns.
Related terms
Concavity: Concavity refers to whether a graph curves upward or downward. A graph is concave up when its second derivative is positive, meaning it's bending upwards like a smiley face.
Inflection Point: An inflection point occurs where the concavity of a graph changes. It happens when the second derivative changes sign, indicating that the curve transitions from being concave up to concave down or vice versa.
Critical Point: A critical point is where either the first or second derivative equals zero or does not exist. These points can be local maxima, minima, or points of inflection depending on their behavior around them.
"Second Derivative" appears in:
Study guides (5)
AP Calculus AB/BC - 5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points
AP Calculus AB/BC - 5.6 Determining Concavity
AP Calculus AB/BC - 5.7 Using the Second Derivative Test to Determine Extrema
AP Calculus AB/BC - 7.9 Logistic Models with Differential Equations
AP Calculus AB/BC - 9.2 Second Derivatives of Parametric Equations
Additional resources (1)
Practice Questions (20+)
What is the second derivative of the function f(x) = 3x^4 - 8x^2 + 2x - 5?
What is the second derivative of the function f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1?
The number of people in a livestream is decreasing at a variable rate, and the first derivative represents the rate at which people are leaving the room with respect to time. What are the units of the second derivative?
The temperature of a room is decreasing at a constant rate, and the first derivative represents the rate at which the temperature is decreasing. What are the units and what is the sign of the second derivative.
The profit of an influencer is given by P(x) = 500x - 100x^2, where x represents the number of likes they obtain on a post. What is the second derivative of the profit function with respect to the number of likes?
The total assets of a stock trader can be modeled by A(t)=1000-t^3. What is the second derivative of the assets with respect to time.
The cost of producing x units of octopus plushies is given by C(x) = 1000 + 10x + 0.01x^2. What is the second derivative of the cost function with respect to the number of units produced?
What is the first derivative of a function if its second derivative is equal to zero at a particular point?
At what points can the first and second derivatives of a function be equal to each other?
If the second derivative of a function is negative, what can be said about the concavity of the graph?
If the second derivative of a function is positive, what can be said about the concavity of the graph?
If the second derivative of a function is zero at a particular point, what can be said about the behavior of the function at that point?
If the second derivative of a function is zero for an interval, what can be said about the behavior of the function on that interval?
What can be inferred about the behavior of a function if its second derivative is always positive?
When the second derivative of a function changes sign from positive to negative, what can be inferred about the behavior of the function?
Which of the following is the correct notation for the second derivative of y with respect to x?
What does the second derivative of an implicit function represent?
If the second derivative of a function is negative, what does it indicate?
What does it mean if the second derivative of a function is zero at a specific point?
Which of the following is true about a function whose second derivative is positive?
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