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?
- Parabola (correct)
- Circle
- Hyperbola
- Line
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?
- It first increases and then decreases
- It increases (correct)
- It remains constant
- It decreases
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?
- It is decreasing (correct)
- It is constant
- It is strictly decreasing
- It is increasing
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?
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?
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?
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?
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.
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Description
Determining whether a function is increasing or decreasing using differentiation and graph analysis. Example: f(x) = x^2.
