All Flashcards
What are the differences between the average rate of change and instantaneous rate of change?
Average rate of change: over an interval | Instantaneous rate of change: at a single point.
What are the differences between linear and quadratic functions in terms of their rates of change?
Linear: constant rate of change | Quadratic: changing rate of change.
What are the differences between concave up and concave down?
Concave Up: Increasing rate of change | Concave Down: Decreasing rate of change
Compare the average rate of change of and over the interval [1, 3].
f(x): Constant rate of change = 2 | g(x): Changing rate of change, average rate of change = 4
What is the difference between a secant line and a tangent line?
Secant line: intersects the curve at two points | Tangent line: touches the curve at one point.
Compare the concavity of and .
f(x): Concave up | g(x): Concave down
Compare the average rate of change of a linear function with a positive slope and a linear function with a negative slope.
Positive slope: Average rate of change is positive | Negative slope: Average rate of change is negative
Compare the average rate of change of and as x approaches infinity.
f(x): Rate of change remains constant | g(x): Rate of change increases without bound
Compare the concavity of and around x = 0.
f(x): Changes concavity at x=0 (inflection point) | g(x): Always concave up
Compare the average rate of change of and .
f(x): Average rate of change is 1 | g(x): Average rate of change is 1
Explain the difference between average rate of change in linear and quadratic functions.
Linear functions have a constant average rate of change, while quadratic functions have a changing average rate of change.
How is the average rate of change related to the slope of a secant line?
The average rate of change is equal to the slope of the secant line connecting two points on the function's graph.
What does concavity tell you about the rate of change of a function?
Concavity indicates whether the rate of change is increasing (concave up) or decreasing (concave down).
How does the sign of the average rate of change relate to whether a function is increasing or decreasing?
A positive average rate of change indicates the function is increasing, while a negative average rate of change indicates the function is decreasing.
Explain how to determine if a quadratic function is accelerating or decelerating.
If the average rate of change is increasing, the function is accelerating. If it's decreasing, the function is decelerating.
What is the significance of the vertex of a quadratic function in terms of rate of change?
The vertex represents the point where the rate of change changes direction (from decreasing to increasing or vice versa).
What does a constant average rate of change imply about the function?
It implies that the function is linear.
How does concavity relate to the second derivative of a function?
If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.
How do you determine the concavity of a quadratic function?
If the coefficient of the term is positive, the function is concave up. If it is negative, the function is concave down.
What does the average rate of change approaching zero signify?
It signifies that the function's values are not changing significantly over the given interval, indicating a horizontal segment or a turning point.
What is a linear function?
A function with a constant rate of change (slope), resulting in a straight line.
What is a quadratic function?
A function with a changing rate of change, creating a curved graph.
Define average rate of change.
The change in output divided by the change in input over a specific interval.
What is a secant line?
A line that connects two points on a curve.
Define concavity.
The direction of the curve of a function (either concave up or concave down).
What does 'concave up' mean?
The function is increasing at an increasing rate.
What does 'concave down' mean?
The function is increasing at a decreasing rate.
What is the average rate of change for a linear function?
It is constant and equal to the slope of the line.
What is the average rate of change for a quadratic function?
It changes as you move along the curve.
What is the relationship between the rate of change of a quadratic function and a linear function?
The rate of change of a quadratic function is linear.