MTH 241: Business and Social Sciences Calculus F. Patricia Medina Department of Mathematics. Oregon State University January 28, 2015 Section 2.1
Increasing and decreasing Definition 1 A function is increasing on an interval (a,b) if nor any pair of points x 1 and x 2 with x 1 < x 2 we have f (x 1 ) < f (x 2 ). Note: A function is increasing at a point x = x 0 if f is increasing in some open interval containing x 0. Definition 2 A function is decreasing on an interval (a,b) if nor any pair of points x 1 and x 2 with x 1 < x 2 we have f (x 1 ) > f (x 2 ). Note: A function is decreasing at a point x = x 0 if f is decreasing in some open interval containing x 0.
Extreme points. Relative Maximum and Minimum. Definition 3 A relative maximum point is a point at which the graph changes from increasing to decreasing; a point that is the highest in its neighborhood. Definition 4 A relative minimum point is a point at which the graph changes from decreasing to increasing; a point that is the lowest in its neighborhood. Note: They happen at the x value of the point.
Definition 5 The maximum value or absolute maximum value of a function is the largest value that the function attains on its domain. Definition 6 The minimum value or absolute minimum value of a function is the smallest (least) value that the function attains on its domain. Note: These absolute maximum and absolute minimum points would also be classified as relative extrema.
Concavity Definition 7 A function f (x) is concave up at a point x = c if there is an open interval on the x-axis containing c where the graph of f is above its tangent line. Observe that f is concave up is the slope of its graph increases as we move through the point (c,f (c)). Definition 8 A function f (x) is concave down at a point x = c if there is an open interval on the x-axis containing c where the graph of f is below its tangent line. Observe that f is concave down is the slope of its graph decreases as we move through the point (c,f (c)).
Inflection point Definition 9 An inflection point is a point on the graph of a function at which the function is continuous and at which the graph changes from being concave up to concave down or vice versa.
Intercepts, domain. Intercepts. These are the points at which the graph of the function crosses or touches the axes. Note that a function can have at most one y intercept and between no and infinitely many x intercepts. Domain. The domain of the function should be clearly represented in a graph of the function. Be aware of the points in the x-axis where the function is undefined. Clearly, the these points don t belong to the domain of the function.
Asymptotes If the graph of y = f (x) approaches a horizontal line y = L as x goes to or, then the line y = L is called horizontal asymptote. This occurs when OR lim f (x) = L, x lim f (x) = L, x If the graph of y = f (x) approaches the vertical line x = K as x goes to K from one side or the other, that is, if lim x K + f (x) = or, then the line x = K is called vertical asymptote. Inclined asymptotes: The graph of the function y = f (x) becomes closer and closer to a line as x tends to or.
Describing a Graph Completely describe the following graph: 1 Intervals in which the function is increasing or decreasing. 2 Relative minimum points and relative minimum points (if any). Absolute maximum or minimum? 3 Concavity. 4 x intercept(s) and y intercept(s). 5 Domain. 6 Asymptotes.
Completely describe the asymptotes and extremes of the following graph:
Consumer Price Index