DOC PREVIEW
UMD MATH 220 - Describing Graphs of Functions

This preview shows page 1 out of 3 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 3 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Calculus 220, section 2.1 Describing Graphs of Functions notes by Tim Pilachowski Reminder: You will not be able to use a graphing calculator on tests! First, a quick scan of what we know so far. The slope of a curve at a point = slope of line tangent to the curve at that point = (instantaneous) rate of change of the curve at that point = first derivative evaluated at that point. Notations are ( )dxdyxf and ′. To find the first derivative of a given function we have the power rule (both general and specific), constant multiple rule, and sum rule. We also have that the derivative of the first derivative is the second derivative, with notations ()xf′′ and ( )22dxydfdxddxd=. Chapter 2 applies all this in various circumstances, and explores the meaning of it all. Example A. Consider the graph of 2xy = pictured to the left. Reading from left to right— From “forever left” ()∞− to x = 0, the curve is going down = graph is falling = the slope of the curve is negative. The graph “bottoms out” at the vertex (0, 0) where the slope of the curve = 0. From x = 0 onward to “forever right” ()∞ the curve is going up = the graph is rising = the slope of the curve is positive. In technical terms, the function ()2xxf = is decreasing on the interval 0<<∞−x , has a minimum at (0, 0), and is increasing on the interval ∞<<x0 . We can say that the minimum value of f is equal to 0 because there are no lower values in the range of f. In other words, the minimum here is an absolute minimum. Let’s go back to ()283+−= xxxf and take a closer look at the curve, pictured to the left below. increasing: decreasing: relative maximum: relative minimum: The function ()283+−= xxxf has no absolute maximum or minimum; the range is ∞<<∞−x. Vocabulary to know: relative extrema (plural) and relative extremum (singular).The function ( )225 xxf −= has a limited domain, –5 ≤ x ≤ 5, and range, 0 ≤ y ≤ 5. increasing: absolute maximum: decreasing: The minimum value of the function is 0. Because the minimum occurs at the endpoints of the domain it is called an endpoint extreme value or endpoint extremum. The maximum value of the function is 5. Compare the graphs of y = x2 and xy = on the interval 0 < x < ∞. Neither has an absolute maximum. Also both are increasing on this domain, but in different ways. 2xy =: xy =: Going back to ()283+−= xxxf — concave down: concave up: point of inflection: Now consider the graph of 3xy = — concave down: concave up: point of inflection: Note also that the curve “levels out” at this spot, and that, at x = 0, the slope of the curve is 0. We’ll say that this function is increasing for all values of x in the domain, ∞<<∞−x. (0, 2) (0, 0)Not all functions have unbounded domains. For example, ( )225xxf −= is not defined for values of x less than –5 or greater than 5. The domain of xy = is ∞<≤x0 . Rational functions will usually have limited domains: they are not defined for values of x that would put a 0 in the denominator. For example, as x approaches 2 from the left, the function ( )213−+=xxxf approaches ∞−. As x approaches 2 from the right, the function approaches ∞. vertical asymptote: As x approaches both ∞− and ∞, the value of f approaches 3. horizontal asymptote: Both x-intercepts and y-intercepts will also be of great interest and use to us in determining an accurate shape for the graph of a function. Now its time to put all of this terminology to work in describing the graph of a new function: ( )xexxf3=, whose graph is to the right. Numbers used correspond to the attributes listed in your text. 1) increasing/decreasing 2) maximum/minimum 3) points of inflection 4) x and y-intercepts 5) domain/undefined values 6) vertical/horizontal


View Full Document

UMD MATH 220 - Describing Graphs of Functions

Download Describing Graphs of Functions
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Describing Graphs of Functions and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Describing Graphs of Functions 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?