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Integration by Parts: “Undoing” the Product Rule for Derivatives Look at the derivative of a product of functions: d dv du uv = u +v ( ) dx dx dx Let’s use the differential form: d (uv ) =udv+ vdu And solve for udv udv =d (uv) −vdu Integrating both sides, we get: ∫udv = ∫d (uv) −∫vdu

Integration by Parts: “Undoing” the Product Rule for Derivatives Integrating both sides, we get: ∫udv = ∫d (uv) −∫vdu Or ∫udv =uv−∫vdu ∫vdu should be simpler that the original ∫udv The integral If two functions are not related by derivatives (substitution does not apply), choose one function to be the u (to differentiate) and the other function to be the dv (to integrate)

Replay QoE measurement Old way: QoE = Server + Network Modern way: QoE = Servers + Network + Browser Browsers are smart Parallelism on multiple connections JavaScript execution can trigger additional queries Rendering introduces delays in resource access Caching and pre-fetching HTTP replay cannot approximate real Web browser access to resources 0.25s 0.25s 0.06s 1.02s 0.67s 0.90s 1.19s 0.14s 0.97s 1.13s 0.70s 0.28s 0.27s 0.12s 3.86s 1.88s Total network time GET /wiki/page 1 Analyze page GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET GET combined.min.css jquery-ui.css main-ltr.css commonPrint.css shared.css flaggedrevs.css Common.css wikibits.js jquery.min.js ajax.js mwsuggest.js plugins...js Print.css Vector.css raw&gen=css ClickTracking.js Vector...js js&useskin WikiTable.css CommonsTicker.css flaggedrevs.js Infobox.css Messagebox.css Hoverbox.css Autocount.css toc.css Multilingual.css mediawiki_88x31.png 2 Rendering + JavaScript GET GET GET GET GET GET GET GET GET ExtraTools.js Navigation.js NavigationTabs.js Displaytitle.js RandomBook.js Edittools.js EditToolbar.js BookSearch.js MediaWikiCommon.css 3 Rendering + JavaScript GET GET GET GET GET GET GET GET GET GET GET 4 GET GET GET GET GET GET page-base.png page-fade.png border.png 1.png external-link.png bullet-icon.png user-icon.png tab-break.png tab-current.png tab-normal-fade.png search-fade.png Rendering search-ltr.png arrow-down.png wiki.png portal-break.png portal-break.png arrow-right.png generate page send files send files mBenchLab – [email protected] BROWSERS MATTER FOR QOE? send files send files + 2.21s total rendering time 6

3.8 Higher Derivatives The derivative of a function f(x) is a function itself f ´(x). It has a derivative, called the second derivative f ´´(x) If the function f(t) is a position function, the first derivative f ´(t) is a velocity function and the second derivative f ´´(t) is acceleration. d2y f ( x) 2 dx The second derivative has a derivative (the third derivative) and the third derivative has a derivative etc. d3y f ( x) 3 dx 4 d y (4) f ( x) 4 dx n d y (n) f ( x) n dx

9.8 public, protected and private Inheritance Base class member access specifier Type of inheritance public inheritance protected inheritance private inheritance public in derived class. Can be accessed directly by any non-static member functions, friend functions and nonmember functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. protected in derived class. Can be accessed directly by all Protected non-static member functions and friend functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Public Private Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. 2003 Prentice Hall, Inc. All rights reserved. 83

11 9.3 Inheritances Base class member access specifier Type of inheritance public inheritance protected inheritance private inheritance public in derived class. Can be accessed directly by any non-static member functions, friend functions and nonmember functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. protected in derived class. Can be accessed directly by all Protected non-static member functions and friend functions. protected in derived class. Can be accessed directly by all non-static member functions and friend functions. private in derived class. Can be accessed directly by all non-static member functions and friend functions. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. Public Private Hidden in derived class. Can be accessed by non-static member functions and friend functions through public or protected member functions of the base class. 2003 Prentice Hall, Inc. All rights reserved.

Integration by Parts Every differentiation rule has a corresponding integration rule. For instance, the Substitution Rule for integration corresponds to the Chain Rule for differentiation. The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts. The Product Rule states that if f and g are differentiable functions, then [f (x)g (x)] = f (x)g (x) + g (x)f (x) 3

//******************************************************************** // Geometry.java Java Foundations // // Demonstrates the use of an assignment statement to change the // value stored in a variable. //******************************************************************** public class Geometry { //----------------------------------------------------------------// Prints the number of sides of several geometric shapes. //----------------------------------------------------------------public static void main(String[] args) { int sides = 7; // declaration with initialization System.out.println("A heptagon has " + sides + " sides."); sides = 10; // assignment statement System.out.println("A decagon has " + sides + " sides."); sides = 12; System.out.println("A dodecagon has " + sides + " sides."); } } Code\Chap2\Geometry.java Java Foundations, 3rd Edition, Lewis/DePasquale/Chase 2 - 32

Integration by Parts d(uv) = udv + vdu udv = uv - vdu Show that xnexdx = xnex - nxn-1exdx + C let u = xn; dv = exdx then du = nxn-1dx; v = ex + C Thus, xnexdx = xnex - nxn-1ex dx + C 03/22/2019 rd 12

Consider instead this function. At x1, we have a maximum. The derivative f (x1) = 0. y f (x1) = 0 f (x) > 0 f (x) < 0 x1 x Here, as we move from left to right in the vicinity of x1, the slope is going from positive to zero to negative. The slope is decreasing. If a function is decreasing, its derivative is negative. Again here, the function is the slope or first derivative. So its derivative is the second derivative. Then, because the first derivative is decreasing, the second derivative must be negative: f (x1) < 0. To put all this together: At a maximum x1 , the first derivative f (x1) = 0 and the second derivative f (x1) < 0.

Integration by Parts: “Undoing” the Product Rule for Derivatives ∫x ln(x)dx Consider: We have no formula for this integral. Notice that x and ln(x) are not related by derivatives, so we cannot use the substitution method.

y f (x) < 0 f (x) > 0 f (x1) = 0 x1 x So as we move from left to right in the vicinity of x1, the slope is going from negative to zero to positive. It is increasing. Recall that if a function is increasing, its derivative is positive. In this case, the function itself is the slope or first derivative. So its derivative is the second derivative. Then, because the first derivative is increasing, the second derivative must be positive: f (x1) > 0. To put all this together: At a minimum x1 , the first derivative f (x1) = 0 and the second derivative f (x1) > 0.

KU Group Consumers What to Share with Each KU Group Instrumental Use: Must Instrumental Use: Must demonstrate value of product demonstrate value of to end users (clinician or their product to end users via clients) via informational product packaging and literature, conference marketing efforts. presentations and marketing efforts. How to Reach Each Group Use sales representatives, product training sessions, news media (print, broadcast and internet based forms); product Use news media placement in stores. Offer Use email, phone calls (targeted print, product demonstrations at and web links to news broadcast and internet conferences and tradeshows. stories regarding the based forms); product Offer assessment tools that product. Report in formal placement in stores; help determine if a product is annual performance demonstration packages appropriate for a client, or the report. and trial use of product. ideal configuration, accessories, etc. for product. Provide demonstration packages and trial use of product. Anticipated Knowledge Translation Outcomes Clinicians Policy Makers Researchers Brokers Manufacturers Strategic Use: If public funds were used for product development, demonstrate return on investment (ROI). Conceptual Use: Provide information on path to market, barriers encountered and carriers used to overcome barriers to provide product development insights to other researchers. Conceptual Use: Demonstrate ROI (financial, university publicity, student training, etc.). Strategic Use: Demonstrate ROI and identify partnering opportunities with companies producing complementary products. Innovation Outputs. Consumers can use products, leading to a better QoL. Clinicians may use products directly or recommend products, leading to improved QoL. Present findings at research oriented conferences Use formal reports, (RESNA, etc.). Use email, phone calls and research papers and power face to face meetings. point presentations. Researchers will gain a greater appreciation and understanding of the Policy makers can share Knowledge to Axtion (KTA) ROI information with process, leading to an oversight organizations increased liklihood that new (ex. Office of Management research will be conducted and Budget). using the Need to Knowledge and KTA models. Brokers can use ROI information to justify future investments in similar products. Face to face meetings may be most effective. Seek out manufacturers at offices, conferences and tradeshows. Manufacturers can increase ROI by teaming up with companies selling complementary products.