2019 ab frq

Apr 13, 2020 ... ... frq-calculus-bc.pdf?course=ap-calculus-bc This is a great question for both AB and BC students as we review for the AP Calculus exam. The ...

May 1, 2020 ... ... AB Practice Test (2019): https://apcentral.collegeboard.org/pdf/ap19-frq-calculus-ab.pdf College Board Announcement on Online Calculators ...2019 International Practice Exam Ab Mcq WebAP Calculus 2019 AB FRQ Solutions Louis A. Talman, Ph.D. Emeritus Professor of Mathematics Metropolitan State University of Denver May 18, 2019 1 Problem 1 1.1 Part a Because sh enter the lake at the rate E(t) = 20 + 15sin ˇt 6, with tin hours after midnight, the number, F, of sh that enter the lake ...CALCULUS AB/CALCULUS BC . 2019 SCORING GUIDELINES . Question 1 (a) 5 . Et d t 153.457690 . 0 . To the nearest whole number, 153 fish enter the lake from midnight to 5 . A. M. 1 2 : : integral 1 : answer (b) 1 . 5 . 50 Lt d t 6.059038 . 0 . The average number of fish that leave the lake per hour from midnight to 5 . A. M. is 6.059 fish per hour.2. Let f and g be the functions defined by f x 1 x ex 2 x and g x x 4 6.5 x 2 6 x 2. Let R and S be the two regions enclosed by the graphs of f and g shown in the figure above. Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid.Walkthrough of the 2019 AP Statistics FRQ #4 Join My Discord Study Server: https://discord.gg/8WGtt3rCorrections:Should be using pooled variance. The result...AP Calculus AB 2019 Free-Response Questions AB Calculus Diagnostic Test - Penguin Random House WebCALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes . Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. 1. People enter a line for an escalator at a rate modeled by the function r ...May 17, 2019 ... ... //www.bothellstemcoach.com/homework-help More Problem Solving Videos: https://www.bothellstemcoach.com/ap-calculus-ab-problem-solving.

Questions 1 and 2 are long free-response questions that require about 22 minutes each to answer and are worth 10 points each. Questions 3–8 are short free-response questions that require about 6 minutes each to answer. Questions 3–5 are worth 4 points each and questions 6–8 are worth 3 points each. Read each question carefully and completely.30, is a better model for the amount of grass. clippings remaining in the bin. Use. L t. to predict the time at which there will be 0.5 pound of grass. clippings remaining in the bin. Show the work that leads to your answer. 2014 AP. …2019 AP ® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B . Time—1 hour . Number of questions—4 . NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS. 3. The continuous function f is defined on the closed interval . −6 £ x £ 5. The figure above shows a portion of . the graph of . fAccording to the Bible, Abraham is the father of the Jewish people and Judaism. He made a covenant with God, who promised Abraham would be the father of a great nation. He is an an...Calculus 2019 AB FRQ Solutions Louis A. Talman, Ph.D. Emeritus Professor of Mathematics Metropolitan State University of Denver May 18, 2019 1 Problem 1 1.1 Part … Web2019 AB #5 (no calculator) (a) The top curve is h(x) and the bottom curve is g(x). ln. 3 Note: Did not need x + 3 since

The AP Calculus Course and Exam Description (Fall 2019) (.pdf/2.29MB), which has everything you need to know about the course and exam. An AP Calculus AB Exam from 2012, previously available only through your AP Course Audit account. Since this exam is now publicly available, you can use the questions without restriction.1: limits of integration. 2 : . 1 : answer with integral. 2019 The College Board. Visit the College Board on the web: collegeboard.org. AP® CALCULUS AB/CALCULUS BC. 2019 SCORING GUIDELINES. Question 3.2019 International FRQ. Chemistry Resource. Does anyone have access to the 2019 international FRQ grading or scoring rubric? My teacher had us take it but they never graded it so I can’t check to see where I lost or gained points. If you do have the answers/scoring rubric, that would be very appreciated! I have access to a lot of practice ... 2019 AP Calculus AB Exam Free Response Question #5Area of a region; integral by hand; Fundamental Theorem, FTC; volume with known cross sections; volume give... AP Calculus 2019 AB FRQ Solutions - MSU Denver AP® Calculus AB - College Board AP Calculus AB - AP Central AP Calculus AB - AP Central ... WEB2022 AP Calculus AB Free Response Questions Louis A. Talman, Ph.D. Emeritus Professor of Mathematics Metropolitan State University of ...

Lonely siren seattle reviews.

back to mr calculus. mr. calculus 2018 released free response solutions ab/bc 1 ab 2 bc 2 2019 RELEASED FREE RESPONSE SOLUTIONS – MR. CALCULUS 2019 AB #5 (no calculator) (a) The top curve is h(x) and the bottom curve is g(x). Free-Response Questions. Download free-response questions from past exams along with scoring guidelines, sample responses from exam takers, and scoring distributions. If you are using assistive technology and need help accessing these PDFs in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at ssd@info ...2019 AP ® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B . Time—1 hour . Number of questions—4 . NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS. 3. The continuous function f is defined on the closed interval . −6 £ x £ 5. The figure above shows a portion of . the graph of . fAP Calculus 2018 AB FRQ Solutions Louis A. Talman, Ph.D. Emeritus Professor of Mathematics Metropolitan State University of Denver May 22, 2018 1 Problem 1 1.1 Part a During the time interval 0 t 300, Z 300 0 r(t)dt= 44 Z 300 0 t 100 3 1 t 300 7 dt= 270 (1) people enter the line for the escalator. (Fortunately, the problem is calculator active ...May 17, 2019 ... 2019 AP Calculus BC Exam Free Response Question #5 It's a let's do calculus problem! Finding a value of k to make the derivative equal a ...

CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes . Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. 1. People enter a line for an escalator at a rate modeled by the function r given by ⎧ ⎪ t t. 3 7 ⎪⎪ 44 1 for 0 ££ t ⎪ ()(rt ()=⎨ 100 300− ) 300 ...2019 SCORING GUIDELINES. Question 2. 2 H 2 O 2 ( aq) → 2 H 2 O ( l) + O 2 ( g ) Hydrogen peroxide, H 2 O 2, decomposes according to the equation above. This reaction is thermodynamically favorable at room temperature. (a) A particulate representation of the reactants is shown below in the box on the left.2019 AP Calculus BC Exam Free Response Question #6Series question! Taylor polynomials from Taylor's formula; Taylor series you memorized (e^x); multiplying ...2021 AP Calculus AB and AP Calculus BC Exam Free Response #3The Spinning Toy problem!Full playlist: https://www.youtube.com/playlist?list=PL6iwkLfBjZixEcq-TH...1. The sizes, in square feet, of the 20 rooms in a student residence hall at a certain university are summarized in the following histogram. Based on the histogram, write a few sentences describing the distribution of room size in the residence hall. Summary statistics for the sizes are given in the following table. Mean.AP® CALCULUS AB 2019 SCORING COMMENTARY Question 5 Note: y contain spelling and grammatical errors. Overview In this problem R is identified as the region enclosed by the graphs of g x 23 c osx 2 and hx 62 x 1 2, the y-axis, and the vertical line x 2. In part (a) students R. A response should demonstrate the area interpretation ofAP Calculus 2017 AB FRQ Solutions - MSU Denver 2019 International Practice Exam: A P Microeconomics AP Calc AB 2019 Released FRQ Solutions - Murrieta Valley … AP Physics 1 2018 Free-Response Questions AP Calculus AB Exam from the 2017 Exam Administration AP Calculus AB 2018 Free-Response Questions AP Calculus 2018 AB FRQ Solutions - MSU ...Past Papers. Every single AP Calculus AB and BC past exam paper including full papers, mark schemes and written mark schemes/model solutions for the multiple choice and free response questions from 1969 onwards can be found below. AB and BC Options. There are two AP calculus options - AB and BC.In this video, we’ll unpack a sample free-response question—FRQ (No Calculator).Download questions here: https://tinyurl.com/3kmjcb23Stay motivated and keep ...AP Calculus AB Scoring Guidelines from the 2019 Exam Administration; scoring guidelines; 2019 AP exam administration; teacher resources; exam resources; exam practice; exam scoring information; exam preparation; course prepaÇX Û/HßoÂò ´ …2019 AP ® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B . Time—1 hour . Number of questions—4 . NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS. 3. The continuous function f is defined on the closed interval . −6 £ x £ 5. The figure above shows a portion of . the graph of . fAP Calculus AB 2019 Free-Response Questions AP Calculus BC - College Board AP Calculus 2018 AB FRQ Solutions - MSU Denver AP Calculus BC - AP Central AP Calculus AB - AP Central web2019 AP ® CHEMISTRY FREE-RESPONSE QUESTIONS . GO ON TO THE NEXT PAGE. -5-CHEMISTRY . Section II . Time—1 hour and 45

AP Calc AB 2019 Released FRQ Solutions - Murrieta Valley WebFREE-RESPONSE SOLUTIONS ~ 2019 AB Question AB-2 (a) vtP is differentiable and therefore continuous on 0.3,2.8 .

1. Let R be the region bounded by the graphs of y. sin p x. 3 and y x. 4 x , as shown in the figure above. Find the area of R. The horizontal line y 2 splits the region R into two parts. Write, but do not evaluate, an integral expression for the area of the part of R that is below this horizontal line.AP ® CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES © 2016 The College Board. Visit the College Board on the Web: www.collegeboard.org. Question 5We would like to show you a description here but the site won't allow us.We would like to show you a description here but the site won't allow us.8.7%. 63.5%. Jun 13. This is the 5th year of the AP Physics 2 exam, & each year, student learning & achievement has increased, from ~8% scores of 5 in 2014 to ~12.6% scores of 5 this year. In other words: this year’s AP Physics 2 students have achieved the highest % of scores of 3+ yet for this exam.20.051 pounds of bananas are removed from the display table during the first 2 hours the store is open. (b) f ′ ( 7 ) = − 8.120 (or − 8.119 ) After the store has been open 7 hours, the rate at which bananas are being removed from the display table is decreasing by 8.120 (or 8.119) pounds per hour per hour. (c) g ( 5 ) − f ( 5 ) = − 2. ...Abraham Lincoln’s Gettysburg Address is one of the most iconic speeches in American history. Delivered on November 19, 1863, at the dedication of the Soldiers’ National Cemetery in...When it comes to your vehicle’s safety, the ABS (Antilock Braking System) is a vital component that should never be compromised. If you’re in need of a replacement ABS control modu...

Kuta software solving proportions.

Drag racing games unblocked games.

2019 AP® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. 1. Fish enter a lake at a rate modeled by the function E given by . p t Et ()=20 +15 sin (6 ). Fish leave the lake at . a rate modeled by the function . L . given by . Lt ...The College Board. The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,600 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students ...2019 RELEASED FREE RESPONSE SOLUTIONS - MR. CALCULUS 2019 AB/BC #4 (no calculator) (a) h Since the radius of the barrel does not change, then dr dt =0 and I can substitute 1 for r right away. ... Microsoft Word - AP2019_AB_BC4.docx Created Date: 5/17/2019 5:47:52 AM ...AP Calculus AB/BC Exams 42 AP Calculus AB/BC Exams Regularly Scheduled Exam Date: Tuesday morning, May 14, 2019 Late-Testing Exam Date: Friday morning, May 24, 2019 Section I Total Time: 1 hour and 45 minutes Number of Questions: 45 (The number of questions may vary slightly depending on the form of the exam.) Percent of Total Score: …Show the work that leads to your answer. Indicate units of mea sure. (b)Use a left Riemann sum with the four subintervals indicated by the table to estimate the total amount of1 : answer. 2019 The College Board. Visit the College Board on the web: collegeboard.org.2019 AP Calculus AB Exam Free Response Question #5Area of a region; integral by hand; Fundamental Theorem, FTC; volume with known cross sections; volume give...CALCULUS AB: 2019 AP EXAM DETAILS AP Exam Date: Tuesday, May 14th 2019 (8 am) The exam is approximately three hours and 15 minutes long and has two parts — multiple choice and free response. Each section is worth 50% of the final exam grade. Section I: Multiple Choice — 45 questions; 1 hour and 45 minutes2019 AP ® CHEMISTRY FREE-RESPONSE QUESTIONS . GO ON TO THE NEXT PAGE. -5-CHEMISTRY . Section II . Time—1 hour and 45 minutes . 7 Questions . YOU MAY USE YOUR CALCULATOR FOR THIS SECTION. Directions: Questions 1–3 are long free-response questions that require about 23 minutes each to answer and are worth 10 points each.CALCULUS AB/CALCULUS BC . 2019 SCORING GUIDELINES . Question 1 (a) 5 . Et d t 153.457690 . 0 . To the nearest whole number, 153 fish enter the lake from midnight to 5 . A. M. 1 2 : : integral 1 : answer (b) 1 . 5 . 50 Lt d t 6.059038 . 0 . The average number of fish that leave the lake per hour from midnight to 5 . A. M. is 6.059 fish per hour. ….

1. The sizes, in square feet, of the 20 rooms in a student residence hall at a certain university are summarized in the following histogram. Based on the histogram, write a few sentences describing the distribution of room size in the residence hall. Summary statistics for the sizes are given in the following table. Mean.A review of the solutions to the 2019 AP Statistics FRQ. All six questions involve material that will be tested this year! All Subjects. AP Statistics. Exam Skills. 2019 FRQ Review. 2019 FRQ Review. april 20, 2020. Bob Amar. Resources. 📑 Summary ⏳ Timestamps 📚 Resources.The title captures it all. In this video, I narrate my thoughts through the AP Calculus AB/BC 2019 Free Response question 1. I highlight some key concepts fo...Abraham Lincoln had several professions over his lifetime, but most notably, he was the 16th president of the United States of America. Elected in 1860, Lincoln led the country thr... Sample: 3C. Score: 3. The response earned 3 points: no points in part (a), no points in part (b), 2 points in part (c), and 1 point in part. (d). In part (a) the property of definite integrals that is required is not stated, so the first point was not earned. • The exact value of ( ) 150 0 1 150 ∫ g t dt is 12 25sin . 125 16 • A response may present the average value formula in single or multiple steps. For example, the following response earns both points: ( )Walkthrough of the 2019 AP Calculus AB FRQ #2 Join My Discord Study Server: https://discord.gg/8WGtt3r If you'd like extra resources for AP Calculus, check...2019 International FRQ. Chemistry Resource. Does anyone have access to the 2019 international FRQ grading or scoring rubric? My teacher had us take it but they never graded it so I can’t check to see where I lost or gained points. If you do have the answers/scoring rubric, that would be very appreciated! I have access to a lot of practice ...Download free-response questions from past exams along with scoring guidelines, sample responses from exam takers, and scoring distributions. If you are using assistive technology and need help accessing these PDFs in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at [email protected]. Expand All.Are you an avid book lover? Do you have a passion for collecting rare and collectible books? If so, then you’re in luck. Abe Used Books is a treasure trove for book enthusiasts, of... 2019 ab frq, Download free-response questions from past exams along with scoring guidelines, sample responses from exam takers, and scoring distributions. If you are using assistive technology and need help accessing these PDFs in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at [email protected]., calculus 2019 released free response solutions ab/bc 1 ab 2 ab/bc 3 ab/bc 4 ab 5 ab 6 bc 2 bc 5 bc 6. back to mr calculus ..., 2. Let f and g be the functions defined by f x 1 x ex 2 x and g x x 4 6.5 x 2 6 x 2. Let R and S be the two regions enclosed by the graphs of f and g shown in the figure above. Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid., AP Calc AB 2019 Released FRQ Solutions - Murrieta Valley … AP Calculus Multiple-Choice Question Collection 1969-1998 - MathGuy.… Calculus AB Practice Exam - AP Central, 2019 AP® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. 1. Fish enter a lake at a rate modeled by the function E given by . p t Et ()=20 +15 sin (6 ). Fish leave the lake at . a rate modeled by the function . L . given by . Lt ... , AP Calculus 2019 AB FRQ Solutions - MSU Denver 2019 International Practice Exam: A P Microeconomics AP Calculus AB - AP Central ... Web Result2018 AP ® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes . Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE, AP Calculus 2019 BC FRQ Solutions Louis A. Talman, Ph.D. Emeritus Professor of Mathematics Metropolitan State University of Denver May 21, 2019 1 Problem 1 1.1 Part a Because sh enter the lake at the rate E(t) = 20+15sin ˇt 6, with tin hours after midnight, the number, F, of sh that enter the lake over the 5-hour period from t= 0 to t= 5, is ..., 2019 SCORING GUIDELINES. 5 points (1 + 1 + 3) Question 2. (a) 1 point: • One point is earned for calculating Dana's total benefit from purchasing 2 bottles of water and 1 unit of good X as $66 and for showing the work. $24 + $18 + $24 = $66 OR $42 + $24 = $66. (b) 1 point:, 2019 RELEASED FREE RESPONSE SOLUTIONS - MR. CALCULUS. So. This is the slope of the line tangent to. 2. '(2). are differentiable so they are both continuous. 4. 4., 2018 AP Calculus AB International Practice Exam from the 2018 administration ap calculus ab practice exam this exam may not be posted on school or personal. Skip to document. University; ... Section II: Free-Response Questions This is the free-response section of the 2018 AP Exam. It includes cover material and other administrative instructions ..., Part A (AB or BC): Graphing Calculator Required. 0 ≤ t ≤ 12, where R(t) is measured in vehicles per hour and t is the number of hours since 7:00 a.m. (t = 0). Values of R(t) for selected values of t are given in the table above. Use the data in the table to approximate Rʹ(5). Show the computations that lead to your answer., AP Calculus AB 2019 Free-Response Questions AB Calculus Diagnostic Test - Penguin Random House WebCALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes . Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. 1. People enter a line for an escalator at a rate modeled by the function r ..., Download free-response questions from past exams along with scoring guidelines, sample responses from exam takers, and scoring distributions. If you are using assistive technology and need help accessing these PDFs in another format, contact Services for Students with Disabilities at 212-713-8333 or by email at [email protected]. The ..., 19 videos 6,592 views Last updated on Apr 24, 2023. Step by Step worked out solutions for the 2019 AP Calculus AB Exam Free Response Questions. Play all., Part (a): The distribution of the sample of room sizes is bimodal and roughly symmetric with most room sizes falling into two clusters: 100 to 200 square feet and 250 to 350 square feet. The center of the distribution is between 200 and 300 square feet. The range of the distribution is between 150 and 250 square feet., The #1 social media platform for MCAT advice. The MCAT (Medical College Admission Test) is offered by the AAMC and is a required exam for admission to medical schools in the USA and Canada. /r/MCAT is a place for MCAT practice, questions, discussion, advice, social networking, news, study tips and more., AP Calculus AB Scoring Guidelines from the 2019 Exam Administration; scoring guidelines; 2019 AP exam administration; teacher resources; exam resources; exam practice; exam scoring information; exam preparation; course prepaÇX Û/HßoÂò ´ Created Date: 7/15/2019 9:20:31 PM, The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of over 6,000 of the world's leading educational institutions and is dedicated to ..., , In this video, we'll unpack a sample free-response question—FRQ (No Calculator).Download questions here: https://tinyurl.com/3kmjcb23Stay motivated and keep ..., May 18, 2019 ... Walkthrough of the 2019 AP Statistics FRQ #3 ✓ Join My Discord Study Server: https://discord.gg/8WGtt3r Website: ..., 2019 AP ® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B . Time—1 hour . Number of questions—4 . NO CALCULATOR IS ALLOWED FOR THESE QUESTIONS. 3. The continuous function f is defined on the closed interval . −6 £ x £ 5. The figure above shows a portion of . the graph of . f , No matter what course you are taking, we are a community that helps students earn college credit! AP Calculus BC Exam Discussion. This post intended for US test discussion. International discussion here . CollegeBoard has released a set of free response questions for Calculus BC: 2022 FRQ set. compared to physics c, it was a first grader exam., AP微积分AB/BC 简答题（Free Response Questions）精讲题目：2019 IPE 微积分AB知识点考察：velocity and speed授课教师简介：景楠老师，留学美国多年，金融数学&经济学双专业。从事留学考试多年，了解学生痛点以及考试易错知识点，能够快速帮学生诊断出问题所在，准确高效的解决学生学习中存在的问题 ..., 2019 AP Calculus AB Exam Free Response Question #5Area of a region; integral by hand; Fundamental Theorem, FTC; volume with known cross sections; volume give..., 2019 Ap Calculus Ab Frq are crucial milestones in one's educational and professional journey. They require a strategic approach, deep understanding, and effective preparation. 2019 Ap Calculus Ab Frq serves as your mentor, providing detailed insights into exam formats, study strategies, and invaluable tips, Step by Step worked out solutions for the 2019 AP Calculus AB Exam Free Response Questions, The continuous function f is defined on the closed interval −6 to 5. The figure above shows a portion of the graph of f, consisting of two line segments and ..., Directions: You have 1 hour and 40 minutes to answer all four of the following questions. It is suggested that you spend approximately 20 minutes each on Questions 1, 2, and 3 and 40 minutes on Question 4. Unless directions indicate otherwise, respond to all parts of all four questions. In your response, use substantive examples where appropriate., 2015 AP® CALCULUS AB FREE-RESPONSE QUESTIONS. 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org., We're excited to once again share with you our early solutions to this year's AP Calc AB Free Response Questions. We love seeing the new questions and working through them, always amazed at the original ways the College Board incorporates multiple concepts into a single question. Some of of our exam predictions turned out to be spot on, but we definitely missed the return to an area/volume ..., The pyruvate dehydrogenase complex (PDC) catalyzes the conversion of pyruvate to acetyl-CoA, a substrate for the Krebs (citric acid) cycle. The rate of pyruvate conversion is greatly reduced in individuals with PDC deficiency, a rare disorder. (a) Identify the cellular location where PDC is most active., 2019 AP® CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time—30 minutes Number of questions—2 . A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS. 1. Fish enter a lake at a rate modeled by the function E given by . p t Et ()=20 +15 sin (6 ). Fish leave the lake at . a rate modeled by the function . L . given by . Lt ...