Podcast
Questions and Answers
What is the shape of the graph of the function f(x) = x^2, x ∈ R?
What is the shape of the graph of the function f(x) = x^2, x ∈ R?
What happens to the height of the graph as we move from left to right along the graph to the right of the origin?
What happens to the height of the graph as we move from left to right along the graph to the right of the origin?
What can be said about the function f(x) = x^2, x ∈ R, for x < 0?
What can be said about the function f(x) = x^2, x ∈ R, for x < 0?
What is the analytical definition of a function that is increasing on an interval I?
What is the analytical definition of a function that is increasing on an interval I?
Signup and view all the answers
What is the analytical definition of a function that is decreasing on an interval I?
What is the analytical definition of a function that is decreasing on an interval I?
Signup and view all the answers
What happens to the height of the graph as we move from left to right along the graph to the left of the origin?
What happens to the height of the graph as we move from left to right along the graph to the left of the origin?
Signup and view all the answers
What is the analytical definition of a function that is constant on an interval I?
What is the analytical definition of a function that is constant on an interval I?
Signup and view all the answers
Study Notes
Analyzing Functions Using Differentiation
- The function f(x) = x^2, x ∈ R is a parabola.
Graph Analysis
- To the right of the origin, the graph continuously increases as we move from left to right.
- To the left of the origin, the graph continuously decreases as we move from left to right.
Definitions of Increasing, Decreasing, and Constant Functions
Increasing Function
- A function f is increasing on an interval I if x1 < x2 in I ⇒ f(x1) < f(x2) for all x1, x2 ∈ I.
Decreasing Function
- A function f is decreasing on an interval I if x1 < x2 in I ⇒ f(x1) ≥ f(x2) for all x1, x2 ∈ I.
- A function f is strictly decreasing on an interval I if x1 < x2 in I ⇒ f(x1) > f(x2) for all x1, x2 ∈ I.
Constant Function
- A function f is constant on an interval I if f(x) = c for all x ∈ I, where c is a constant.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Determining whether a function is increasing or decreasing using differentiation and graph analysis. Example: f(x) = x^2.
