MTH 241: Business and Social Sciences Calculus

Transcription

1 MTH 241: Business and Social Sciences Calculus F. Patricia Medina Department of Mathematics. Oregon State University January 28, 2015 Section 2.1

2 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.

3 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.

4 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.

5 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)).

6 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.

7 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.

8 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.

9 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.

10 Completely describe the asymptotes and extremes of the following graph:

11 Consumer Price Index