{ "key_pair_value_system": true, "answer_rating_count": "", "question_feedback_html": { "html_star": "", "html_star_feedback": "" }, "answer_average_rating_value": "", "answer_date_js": "2024-06-09T15:00:10-04:00", "answer_date": "2024-06-09 15:00:10", "is_docs_available": "", "is_excel_available": "", "is_pdf_available": "", "count_file_available": 0, "main_page": "student_question_view", "question_id": "2520617", "url": "\/study-help\/questions\/find-equations-of-the-tangents-to-the-curve-x-2520617", "question_creation_date_js": "2024-06-09T15:00:10-04:00", "question_creation_date": "Jun 09, 2024 03:00 PM", "meta_title": "[Solved] Find equations of the tangents to the cur | SolutionInn", "meta_description": "Answer of - Find equations of the tangents to the curve x = 6t2 + 6, y = 4!.\\3 + 2 that pass through the point (12, 6). (Enter you | SolutionInn", "meta_keywords": "find,equations,tangents,curve,x,6t2,+,6,y,4,3,2", "question_title_h1": "Find equations of the tangents to the curve x = 6t2 + 6, y = 4!.3 + 2 that pass through the point (12, 6).", "question_title": "Find equations of the tangents to the curve x = 6t2 +", "question_title_for_js_snippet": "Find equations of the tangents to the curve x 6t2 6, y 4 3 2 that pass through the point (12, 6) (Enter your answers as a commaseparated list ) y 2x 30 X Need Help and Use a graph to estimate the coordinates of the rightmost point on the curve x Br 2t6, y er Then use calculus to find the exact coordinates m4 6 (e)(e) (am ) ,e X Need Help Find dy and dx dx 2 x e, y te t dy dx dry dx 2 For which values of t is the curve concave upward (Enter your answer using interval notation )The curve shown in the figure is the astroid X a cos3(9), y a sin3(9) ' A u , u u ' u (2) Find the area of the region enclosed by the astroid 31m2 8 Consider the following parametric curve x e' t, y 4et 2, osts 5 Set up an integral that can be used to exact length of the given curve L dt Find the exact length of the curve Need Help Read ItDate Pages solution jito find dy and dy with given that et and y tet let find the dy we have then da et 7 and we have y tet then td ( et , et d t ) so on solving this we get dy te tet 1 L SO dy dy da it portion get de at cancelled dy 7 te tet et Now let find day so we have dy tete from et above so again differente it one more time gives you the ecosuitfor d24Dates Pages ie f dy dy Apply the chain mille here 1 tettet et so, this equation is in the form of so we use quotient eville In our case let u ( tettet ) and V ed so U e ( tet tet ) tel ( derivative of tettet 02 V) et ( derivative of et ) Now apply Quotient emile d f textety ( tetjet ( tettet ) (et ) et (et ) 2 simplify the expression furthere we get atetet elet ate co 21 1 eat eat 0 24Date Payca So 024 1 pat Now let determine for which value of it the curve is concave upward So, A curve is concave upward i e 9 t 1 so from here itcan est be seen that for this equation inequality to be true confirmed when the denominator is not or never be zero, as it approach to indeterminant form pat to again for the function to be correct at 1 must be posilue ie 2t 170 Hence, At the point 1 it look to be upward point and for all the real number greater than ( is the given curve is looks upward Dates Pages page 1 solution 7 a find this area a a a let calculate the area enclosed by the astroid curve x aug320) and 4 asin 3 (0), we find both the or and if component seperately and then add them up to get the area, the know that formula for finding, the area of polar curve B Area 9 Ir (o) do So 10) for cordinates no) aces co) and we want area from o 10 to 27 ) 24 SoDates Pages page 2 2 ) for y cordinale r (o) asin 30 and we want to find the area from 0 0 too 21 So, Ay 1 asing ( 0) do Now calculate Is and 2) seperally and add them in last so 2 1 coco) 1co2(0)) ( ( ( ) ) 6052(0 ) Now proceed with integration D we use double angle identity for lo ( 20) which is 2 ( 20) 17 4 ( 40 ) Ax a It co ( 40 ) 602 ( 0) do 2 0 2 as we see 102(0) use fact the average over period of 0$6 25 is TOOS Let's compute it step by step L Jo Ve2t 2et 1 4ett Simplify L So Vest 2et 1dt This can be rewritten as L (et 1)2dt So, the integral simplifies to L So(et 1)dt Now, calculate the integral L et to (e5 5) (e 0) e' 5 1 e' 4 170 44Date Page pages 1 coo co)do 14 4 and for a w (40 ) this is an standard equation of integral which Become Zero own oto 2 71 so we left as a y 9 X y Add Haz similarly, for 2 0 ) Ay 20 ) a 2 sink co) do have same value as abillie as it ROus same parameters and also after expanding it the term become Zero So Ay Na 8 The total Area Aut Ay, Astroid Area Naz 4", "question_description": "\"image\"image\"image\"image\"image\"image\"image\"image\"image\"image\"image\"image
\"image\"image\"image\"image\"image\"image\"image\"image\"image\"image\"image\"image<\/div><\/div><\/div><\/figure> Find equations of the tangents to the curve x = 6t2 + 6, y = 4!.\\\"3 + 2 that pass through the point (12, 6). (Enter your answers as a commaseparated list.) y= 2x+30 X Need Help? and\" Use a graph to estimate the coordinates of the rightmost point on the curve x = Br 2t6, y = er. Then use calculus to find the exact coordinates. m4 6.(e)(e) (am ) ,e X Need Help? Find dy and- dx dx 2 x = e, y = te-t dy dx dry dx 2 For which values of t is the curve concave upward? (Enter your answer using interval notation.)The curve shown in the figure is the astroid X = a cos3(9), y = a sin3(9). _\\\\' A u , u u _\\\\' u (2) Find the area of the region enclosed by the astroid. _ 31m2 8 Consider the following parametric curve. x = e' - t, y = 4et\/2, osts 5 Set up an integral that can be used to exact length of the given curve. L = dt Find the exact length of the curve. Need Help? Read ItDate: Pages solution jito find dy and dy with given that = et and y = tet let find the dy we have then da = et .7 \" and we have y = tet then td ( et , + et d \/ t ) so on solving this we get [ dy = -te tet . 1 L SO dy - dy \/ da \/ it portion get de \/ at. cancelled dy 7 - te\"tet . et Now let find day . so we have dy = - tete from et above so again differente it one more time gives you the ecosuitfor d24Dates Pages ie f \/ dy ] - dy Apply the chain mille here? 1-tettet et so, this equation is in the form of so we use quotient eville In our case \" let u = (-tettet ) and V = ed. so U = - e . ( - tet tet ) = tel ( derivative of - tettet 02 V) = et ( derivative of et ) Now apply Quotient emile. d f-textety = ( tetjet - ( tettet ) (et ) et (et ) 2 simplify the expression furthere we get; atetet- elet = ate-co 21 - 1 eat eat 0 24Date: Payca So 024 - 1 pat Now let determine for which value of it the curve is concave upward . So, A curve is concave upward i.e 9 t - 1 so from here itcan est be seen that for this equation inequality to be true confirmed when the denominator is not or never be zero, as it approach to indeterminant form pat to again for the function to be correct at - 1. must be posilue ie 2t-170 Hence, At the point [ 1\/] it look to be upward point and for all the real number greater than ( \/] is the given curve is looks upward .Dates Pages page 1. solution :7 a find this area . -a a - a let calculate the area enclosed by the astroid curve x= aug320) and 4 = asin 3 (0), we find both the or and if component seperately and then add them up to get the area, the know that formula for finding, the area of polar curve B Area = 9 Ir (o) ] do So 10) for - cordinates no) = aces co) and we want area from o= 10 to 27 ) 24 SoDates Pages page 2 2.) for y - cordinale .. r (o) = asin 30. and we want to find the area from 0=0 too= 21. So, Ay = 1 [ [ asing ( 0)] do Now calculate Is and 2) seperally and . add them in last. so 2 1 coco) = 1co2(0)) - ( (?(@) ) 6052(0 ) Now proceed with integration D we use double angle identity for lo ( 20) which is 2 ( 20) = 17 4 ( 40 ) Ax = a It co ( 40 ) . 602 ( 0) do 2 0 2 as we see 102(0) use fact the average over period of 0$6 25 is TOOS]Let's compute it step by step: L = Jo Ve2t - 2et + 1 + 4ett Simplify: L = So Vest + 2et + 1dt This can be rewritten as: L = (et + 1)2dt So, the integral simplifies to: L = So(et + 1)dt Now, calculate the integral: L = [et + to = (e5 + 5) - (e+ 0) = e' +5-1 = e' +4 ~170.44Date: Page: pages 1 coo co)do = 14 4 and for a w (40 ) this is an standard equation of integral which Become Zero own oto 2 71 so we left as a y = 9 X y Add = Haz similarly, for 2 0 ) Ay = 20 ) - a 2 sink co) do have same value as abillie as it ROus same parameters and also after expanding it the term become Zero So Ay = Na 8 . The total Area = Aut Ay, Astroid Area = Naz 4", "transcribed_text": "", "related_book": { "title": "Modern Engineering Mathematics", "isbn": "1292253495, 9781292253497", "edition": "6th Edition", "authors": "Glyn James", "cover_image": "https:\/\/dsd5zvtm8ll6.cloudfront.net\/si.question.images\/book_images\/63a169bbd0ea7_20493.jpg", "uri": "\/textbooks\/modern-engineering-mathematics-6th-edition-9781292253497", "see_more_uri": "\/textbooks\/sciences-animal-anatomy-3416" }, "free_related_book": { "isbn": "0444875085", "uri": "\/textbooks\/the-lambda-calculus-its-syntax-and-semantics-1st-edition-978-0444875082-109757", "name": "The Lambda Calculus Its Syntax And Semantics", "edition": "1st Edition" }, "question_posted": "2024-06-09 15:00:10", "see_more_questions_link": "\/study-help\/questions\/sciences-medical-sciences-2023-February-20", "step_by_step_answer": "The Answer is in the image, click to view ...", "students_also_viewed": [ { "url": "\/why-is-it-possible-to-prepare-financial-statements", "description": "Why is it possible to prepare financial statements directly from an adjusted trial balance?", "stars": 3 }, { "url": "\/a-process-is-in-control-and-the-results-from-a-series-of", "description": "A process is in control and the results from a series of samples show the overall mean and standard deviation of all the units sampled to be x-double bar = 21.4500 inches and s = 0.0969 inches. 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