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L'Hôpital, Integration by Parts, and Partial Fractions Jeanne Tong & Marisa Borusiewicz Table of Contents Slide # Rapunzel’s Real World Applications Mickey Mouse’s Clubhouse of Fame Mulan’s teachings of Integration by Parts Integration of Partial Fractions with Nemo L'Hôpital’s Rule Story Time! Cinderella’s Analytical Example AP Multiple Choice with Pinocchio Simba’s Solution AP Conceptual Problem AP Conceptual Problem Solution AP Level Free Response with Solution Graphical Problem 3 4 5 6 7 8 9 10 11 12 13 14 15 Coloring Page Works Cited 16 17 *All pictures used are copyrighted by Disney. The Wonderful World of Real Life Application You may be wondering if integration by parts, partial fractions, or L’Hopital’s rule have any practical uses. Of course, the answer is yes! Integration by parts and partial fractions are variations of standard integration which has a multitude of purposes. One of the major purposes is integrating the rate of growth of Rapunzel’s hair. She can never seem to remember how long her hair is, but by using an equation for the rate of growth she can easily just integrate using by parts to solve for the length. Other minor uses can be found in the fields of physics, engineering, architecture, business, and chemistry. Modeling the change in mass, energy or momentum on both a micro- and macroscopic scale with equations allows a physicist to be able to study the interaction between objects in the universe. Integration is also useful when it comes to the motion of waves. Vibration, distortion under weight and fluid flow, such as heat flow, air flow, and water flow all involve integration. These may be helpful to engineers designing planes, ships, pipe systems, submarines, or magic carpets. Architects might need to consider these ideas when designing buildings, bridges, or structures with unequal forces acting upon it. Chemists utilize integration when finding the pH of titrations. During these experiments pH is often plotted and a curve is fitted to the data. Integrating these curves can be used to predict and analysis the pH trends. When modeling regression curves, analyzing population, or studying the kinematics of the cell process, integration can but used to help produce a model or curve and L’Hopital can be used to find the bounds and limits. Johann & Jacob Bernoulli Guillaume de L'Hôpital Born into a noble French family in Paris, L'Hôpital is associated with the L'Hôpital rule. After abandoning his military career, L'Hôpital continued to pursue his interests in the mathematical field. In 1691, L'Hôpital met Johann Bernoulli. Johann became L'Hôpital’s instructor, giving him private lectures. Later, in 1964, L'Hôpital made a deal with Bernoulli for an annual payment of 300 Francs in exchange for Bernoulli’s latest mathematical discoveries (essentially it was a bribe). Eventually, after the creation of L'Hôpital’s rule, Bernoulli was credited because he was unhappy with the unjust publicity of L'Hôpital’s work. The L'Hôpital rule is the epitome of limits in indeterminate forms. When a limit is indeterminate always remember this key phrase: “take it to the Hospital!” Use this formula to solve for integrals that resemble the method of integration by parts: udv uv vdu HINTS AND TIPS: to easily solve integrals using by parts, let u equal an easily differentiable function and let dv equal a function that can be easily integrated! If you see an integral with a denominator that looks relatively easy to factor, this means use the partial fractions method! Here are a few examples of when you should use the method of integration for partial fractions: x x 1 2 5x 6 dx (5 x 3) 2 2x 3 dx x 1 2 2x 1 4x 1 x x2 2 ( 4 x 5) x 2 x2 dx dx dx x 2x 3 2 3x 2 dx lim x c f(x) g(x) lim x c f'(x) g'(x) Indeterminate forms when taking the limit: 0 ( ) ( ) , , , , 0 ( ) ( ) If you encounter one of these indeterminate forms, this is a huge clue directing you to use L'Hôpital’s rule! An indeterminate form tells us that no specific limit is guaranteed to exist or the limit cannot be found. In order to use L'Hôpital’s rule, f and g must differentiable functions If it’s indeterminate, take it to the Hospital! STORY TIME! Once upon a time there was a lovely princess. But she had an enchantment upon her of a fearful sort which could only be broken by love's first kiss. She was locked away in a castle guarded by a terrible firebreathing dragon. Many brave knights had attempted to free her from this dreadful prison, but non prevailed. She waited in the dragon's keep in the highest room of the tallest tower for her true love and true love's first kiss. Analytical Example (with solution) Find the solution to the indefinite integral: Although this integral may look a little intimidating at first, partial fractions can make it a rather simple integral. Partial fractions can be useful when the denominator is easily factorable. (Here the denominator has been factored ) From this factored form , we can divide the fraction into the sum of two fractions, assigning the numerators separate variables, in this case A and B. Allowing this new from to be equal to the original, multiply both the new and the original by the denominator of the original. Distributing and simplifying , we find that the we are left with: In order to solve for the variables A and B, the terms that they are contained within, must simplify to zero. Substitute in the values that will make the terms equal to zero and solve for both A and B. B 4 3 We can substitute the values for A and B into the factored form of the integral. This allows us to divide the integral into 2 separate integrals From this form we can in integrate as normal and find the indefinite integral Don’t forget to use usub with this one! 8 1 4 ln( 2 x 1) ln( x 2 ) C 3 2 3 4 2x 1 ln C 3 x2 Don’t forget +C! AP Level Multiple Choice Find A) x cos x 2 x sin x 2 cos x C B) x cos x 2 x sin x 2 cos x C C) x sin x 2 x sin x 2 cos x C D) x 2 2 2 3 cos x C 3 E) x 2 cos udv x 2 sin x 2 x cos x C uv vdu Correct answer: B; integration by parts must be done twice A: positive and negative signs are incorrect Abracadabra! Reveal the answers! C: student may have mistakenly multiplied (u)(dv) instead of uv D: completely wrong answer; student didn’t use the by parts formula E: multiplied by the wrong variables when the student did integration by parts the second time, which led to incorrect integration using the by parts formula Simba’s Solution DON’T FORGET! udv uv vdu Remember that u and v must be functions of x and be continuous derivatives Find x cos x 2 x cos x 2 2 x cos u 2x du 2 xdx v sin x dv cos x x cos x 2 x sin x 2 sin xdx 2 Choice B x cos x 2 x sin x 2 cos x C 2 HINT: You must do integration by parts a second time in this problem! AP Conceptual Problem 0 3 6 9 f(x) 1 6 3 7 f’(x) 4 2 -5 -12 g(x) -2 -1 8 2 g’(x) 7 -3 1 10 Using the table above: 6 Solve 0 6 f ' ( x ) g ( x ) dx given f ( x ) g ' ( x ) dx 20 0 Take a moment to solve this AP conceptual problem and enjoy the soundtrack! AP Conceptual Problem Solution 6 6 Solve f ' ( x ) g ( x ) dx given f ( x ) g ' ( x ) dx 20 0 0 Remember: udv uv vdu Using the by parts formula above, set the proper functions equal to u and dv. In order to get du, take the derivative of u. To get v, you must take the integral of dv. u g ( x) v f (x) du g ' ( x ) dv f ' ( x ) 6 g ( x) f ( x) f ( x ) g ' ( x ) dx 0 6 You are given that f ( x ) g ' ( x ) dx 20 0 g ( x ) f ( x ) 60 20 Follow normal integration rules for definite integrals doing: b f ' ( x ) dx f ( x ) a f ( b ) f ( a ) b a Now just plug in the bounds and look on the table for values to solve! [ g ( 6 ) f ( 6 ) g ( 0 ) f ( 0 )] 20 ( 8 )( 3 ) ( 2 )( 1) 20 6 . Aladdin took the princess from the palace through the city The speed of Aladdin’s magic carpet is modeled by the function If the trip took one hour and a half what was his average speed? (measured in miles per minutes) b 1 ba For this problem we must start with the basic integral for an average value. f ( x ) dx a 90 1 ( 5000 90 0 5000 4 x e 2 x ) dx Plug in for the bounds and the equation into the integral 0 90 (x 90 0 4 e 2 x By the rules of integration we can remove the 5000 from within the integral and multiple it after we integrate. ) dx 0 udv uv vdu From here we must use by parts to integrate. You may use the traditional UV from of integration, but for this example we will complete this with table method. In order to use table method, it is most effective if you pick a u which is easily differentiable and a dv which you can easily integrate. From here, list the derivatives of u and the anti-derivatives of dv. Every other derivative of u must be negative. Integrating the original function can be done by combining the derivatives and anti-derivatives in a diagonal fashion as shown to the left. Each diagonal represents a term that will be added together to form in the integral. 5000 90 0 90 (x 4 e 2 x ) dx Finally, we can evaluate from 0 to 90 minutes and find the average speed! 0 This is really fast for a magic carpet! At this rate he can literally show Jasmine the whole world in just about 25 days! mi/min When the problem asks for average value over a time interval, you must have b 1 a in front of the integral Graphical Problem lim x 0 sin x x sin( 0 ) 0 0 0 Use L'Hôpital’s rule because when you take the limit as x goes to 0, you get an indeterminate form of 0/0. Take the derivative of the top and bottom separately…now the limit as x approaches 0 is 1. lim x 0 This is a graph of cos x cos( 0 ) 1 1 1 1 You can see that as the limit goes to 0, the graph approaches 1 sin x x y x y ' (0) 1 y sin( x ) y ' (0) 1 As you can see both of these graphs pass through zero, but the derivatives (slopes) equal 1, which allows you to find the limit of the function sin x as x goes to 0. x One of the bugs Timon pulls out of the log during 'Hakuna Matata' is wearing Mickey ears. Time for a coloring break! Use a lot of blue! Did you know the color blue has a calming effect because it stimulates the release of hormones that low blood pressure? In the Disney movie Hercules is the son of the two gods, Zeus and Hera, but, according to traditional Greek mythology, he is the son of Zeus and the mortal woman Alcmena . While this page may be skipped, it is advised not to. I know the calculus is very exciting, but it has been proven that taking a break from learning increases the amount of information you retain by forcing you to refocus your thoughts. So color on my friends! Works Cited • • • • • • All pictures are copyrighted by Disney. http://www.physicsforums.com (accessed: 5/12/12) http://www.intmath.com/applicationsintegration/applications-integrals-intro.php (accessed: 5/15/12) thickclouds.com (accessed: 5/17/12) http://wwwmath.ucdenver.edu/~wcherowi/courses/m4010/s08/csbernoul li.pdf (accessed: 5/13/12) Rob Larson: Analytical Calculus 8th edition textbook Analytical Exercises 1. Integrate 2. Integrate 3. Integrate x sec 2 xdx x e dx x ln xdx 2 2x 2 /2 4. Integrate e cos xdx x 0 5. Integrate 3 xe dx 3x 0 Analytical Examples 6. Integrate 7. Integrate 1 x 6x 8 2 x 6x 2 x2 dx dx x 2 2 8. Integrate 9. Integrate 10. Integrate x x 2 7 x 30 dx 6 x 20 3 4x 4x 2 dx 2 cos x sin x cos 2 x cos x dx Analytical Examples 2x cos tdt 11. Integrate 0 lim x 0 2x x 9 2 12. Integrate lim x 3 13. Integrate 14. Integrate x 6x 9 3 lim x 1/ 2 x x lim x 0 15. Integrate lim x 0 xe 4x x 1 cos( 2 x ) sin 2 x sin x x 3 AP Practice Multiple Choice Solve the integral: 1. x 4 2 2 2 x x e dx 2. lim x 2 0 x x5 2 a. 2e 2 b. 2 e 2 2 c. 4e 2 2 d. 2 e 2 8 2 e e. 3 2 a. 1 b. 2 3 c. 0 4 d. e. 5 1 3 3. Given that f ( x) x 2 a. 2 b. c. 2 e 2 e 1 d. 0 e. underfined 2 , find lim x 0 f ( x) 2 e 1 x 4. x 2 dx 2 5. 4 x4 x 1 a. 1 ln x 2 C x2 2 b. 1 ln x 2 C 2 x2 c. d. 1 ln x2 4 x2 1 x2 ln 4 e. 1 4 x2 ln a. b. 3 ln 3 7 ln 2 c. ln 3 x2 x2 C 2 3 3 C x 3 ln 3 5 ln 2 7 C 2 2 d. 7 ln 3 e. 3 ln 2 7 ln 3 AP Practice Free Response Let f ' ( x ) x 2 3 x 2 ln x , for which f(x) is continuous and differentiable with an initial condition that f(0)=0. a. Use antidifferentiation to find f(x) b. Find lim x 1 f ( x) x 1 2 c. Determine if any maximums or minimums exist on f(x) on the interval 0<x<2