PPT-Sec 4.3 – Monotonic Functions and the First Derivative Test

Monotonicity defines where a function is increasing or decreasing A function is monotonic if it is increasing or decreasing on an interval d a b c   Monotonicity of Interval IncreasingDecreasing. 10 D51545 Waldbr57512ol Germany email alzerhorstfreenetde CHRISTIAN BERG Department of Mathematics University of Copenhagen Universitetsparken 5 DK2100 Denmark email bergmathkudk Abstract A function 0 is said to be completely monotonic if 1 0 for 10 D51545 Waldbrol Germany alzerwmax03mathematikuniwuerzburgde Department of Mathematics University of Copenhagen Universitetsparken 5 DK2100 Denmark bergmathkudk Abstract We prove i Let 1 and h 1 1 where 0 is a polynomial of degree 1 with r Notation dx dx y 00 f 00 Thus dx dx dy dx Example Find the second derivatives of the following functions a 2 x y 00 2 b y 00 c 5 4 5 y 00 The 64257rst derivative gives information about whether a funct ion increases or decreases In fact A d Data Using a Difference of Monotonic Representation. VLDB ‘11, Seattle. Guy . Sagy. , . Technion. , Israel. . Daniel Keren, Haifa University, Israel. . Assaf. . Schuster, . Technion. , Israel. Points of Inflection. Section 4.3a. Writing: True . or . False – A . critical point . of. a function always signifies . an . extreme. value . of the . function. Explain.. FALSE!!! – Counterexample???. Vocabulary Test #1. Solicitous. Induce. Loathsome. Disdain. Pedantry. Derivative. Esteem. Collaborate. . TotS. Vocab Test #2. w. anton. b. alm. d. ulcet. s. ubmissive. r. everence. h. usbanded. w. arrant . AKA “Shortcuts”. Review from 3.2. 4 places derivatives do not exist:. Corner. Cusp. Vertical tangent (where derivative is undefined). Discontinuity (jump, hole, vertical asymptote, infinite oscillation). Example. For. . find the derivative of . f. and state the domain of . f’. . The derivative can be regarded as a new function. Example. Given the graph of the function, . f. . can. not . be. . seen. Rainer Kaenders. University . of. Cologne. GeoGebra Conference Linz 2011. Functions. . can. not . be. . seen. … but . can. . be. . represented. . GeoGebra . can. . Convert 105 degrees to radians. Convert 5. π. /9 to radians. What is the range of the equation y = 2 + 4cos3x?. 7. π. /12. 100 degrees. [-2, 6]. Derivatives of Trigonometric Functions. Lesson 3.5. Objectives. We will learn about:. Concavity -Points of Inflection - The Second Derivative Test. Review. If a functions wants to switch from decreasing to increasing. or visa versa, what are its options of approach/attack! (There is only three options). Chapter 3.5. Proving that .  . In section 2.1 you used a table of values approaching 0 from the left and right that . ; but that was not a proof. Because you will need to know this limit (and a related one for cosine), we will begin this section by proving this through geometry. Slope of the Tangent Line. If . f. is defined on an open interval containing . c. and the limit exists, then . . and the line through (. c. , . f. (. c. )) with slope . m. is the line tangent to the graph of . The Second Derivative and the Function. The first derivative tells us where a function is increasing or decreasing. But how can we tell the manner in which a function is increasing or decreasing?. For example, if .