7 Questions
What is the shape of the graph of the function f(x) = x^2, x ∈ R?
Parabola
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 increases
What can be said about the function f(x) = x^2, x ∈ R, for x < 0?
It is decreasing
What is the analytical definition of a function that is increasing on an interval I?
x1 < x2 in I ⇒ f(x1) < f(x2) for all x1, x2 ∈ I
What is the analytical definition of a function that is decreasing on an interval I?
x1 < x2 in I ⇒ f(x1) ≥ f(x2) for all x1, x2 ∈ 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?
It decreases
What is the analytical definition of a function that is constant on an interval I?
f(x) = c for all x ∈ I, where c is a constant
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.
Determining whether a function is increasing or decreasing using differentiation and graph analysis. Example: f(x) = x^2.
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