Stewart’s Calculus 8th Edition, Section 1.1, Question 3

Stewart’s Calculus 8th Edition, Section 1.1, Question 3 SAT / ACT Prep Online Guides and Tips This posts contains aTeaching Explanation. You can buyCalculus by Stewarthere. Why You Should Trust Me:I’m Dr. Fred Zhang, and I have a bachelor’s degree in math from Harvard. I’ve racked up hundreds and hundreds of hours of experienceworking withstudents from 5thgradethroughgraduate school, and I’m passionate about teaching. I’ve read the whole chapter of the text beforehand and spent a good amount of time thinking about what the best explanation is and what sort of solutions I would have wanted to see in the problem sets I assigned myself when I taught. Question:The graph of a function f is given.Page in 8th Edition: 19 Short Answers: f(1) = 3 f(-1) ~ -.3 f(x)=1 for x = 0 or 3 f(x)=0 for approximately x=-0.6 The domain of x are real numbers between -2 and 4 (or [-2,4], and the range are real numbers between -1 and 3, or [-1,3]. f is increasing on the interval [-2,1) Homework Answer:Same as Short Answers. Motivated Answers: The question is giving you the graph of the function f. This means that to figure out what f(x) is, we need to look at the y-value of the graph at x. To figure out f(1), we can take put a ruler vertically (up down) on the graph when x=1 and see how high the graph is, which is the same thing as the y-value of the graph. We can count boxes on the graph paper to see the y-value is 3. Just like a), we put a ruler vertically at x=-1, and the graph seems to show a y-value of about -.3 (it could be -0.2 or -0.5, but that’s our best guess by eyeballing it). This means f(-1)~-0.3 The question wants us to find all values of x where f(x)=1. Since 1 is the output of f, and the output means to y-values, we can take a ruler, put it horizontally at 1, and look at where the ruler hits the graph. We see the rule hits the graph two times, once when x is 0, and another time when x = 3. We do the same thing as c), but put the ruler horizontally at 0, which happens to be the x-axis. The graph hits the ruler at x=-.6 approximately. You have to find the domain and range of f. The domain of any function is all valid inputs, or stated the same way, all valid x-values. We can see from the graph that the graph spans the x-range of -2 though 4 (we can count boxes). To write this in interval notation, we write the range is [-2,4]. We use solid brackets here because the graph seems to include the endpoints.The range of f is all valid outputs of f. Stated the same way, these are all valid y-values of the graph. We can see the graph spans the y-range of -1 through 3, or [-1,3]. If you look at the graph you can see that f seems to be increasing throughout the first part of it, from x-values of -2 to 1. Writing this in interval notation, we get [-2,1). We use a parenthesis ) instead of bracket ] because at the point 1, the function is no longer increasing. Video Solution: Get full textbook solutions for just $5/month. PrepScholar Solutions has step-by-step solutions that teach you critical concepts and help you ace your tests. With 1000+ top texts for math, science, physics, engineering, economics, and more, we cover all popular courses in the country, including Stewart's Calculus. Try a 7-day free trial to check it out.

Henry Ford and the Auto Assembly Line

Henry Ford and the Auto Assembly Line Cars changed the way people lived, worked, and enjoyed leisure time; however, what most people don’t realize is that the process of manufacturing automobiles had an equally significant impact on the industry. The creation of the assembly line by Henry Ford at his Highland Park plant, introduced on December 1, 1913, revolutionized the automobile industry and the concept of manufacturing worldwide. The Ford Motor Company Henry Ford was not a newcomer to the business of automobile manufacturing. He built his first car, which he christened the â€Å"Quadricycle,† in 1896. In 1903, he officially opened the Ford Motor Company and five years later released the first Model T. Although the Model T was the ninth automobile model Ford created, it would be the first model which would achieve wide popularity. Even today, the Model T remains an icon for the still-existing Ford Motor Company. Making the Model T Cheaply Henry Ford had a goal of making automobiles for the multitudes. The Model T was his answer to that dream; he wanted them to be both sturdy and cheap. In an effort to make Model T’s cheaply at first, Ford cut out extravagances and options. Buyers couldn’t even choose a paint color; they were all black. By the end of production, however, the cars would be available in a wide variety of colors and with a wide variety of custom bodies. The cost of the first Model T was set at $850, which would be approximately $21,000 in today’s currency. That was cheap, but still not cheap enough for the masses. Ford needed to find a way to cut down the price even further. Highland Park Plant In 1910, with the aim of increasing manufacturing capacity for the Model T, Ford built a new plant in Highland Park, Michigan. He created a building that would be easily expanded as new methods of production were incorporated. Ford consulted with Frederick Taylor, creator of scientific management, to examine the most efficient modes of production. Ford had previously observed the assembly line concept in slaughterhouses in the Midwest and was also inspired by the conveyor belt system that was common in many grain warehouses in that region. He wished to incorporate these ideas into the information Taylor suggested to implement a new system in his own factory. One of the first innovations in production that Ford implemented was the installation of gravity slides that facilitated the movement of parts from one work area to the next. Within the next three years, additional innovative techniques were incorporated and, on December 1, 1913, the first large-scale assembly line was officially in working order. Assembly Line Function The moving assembly line appeared to the onlooker to be an endless contraption of chains and links that allowed Model T parts to swim through the sea of the assembly process. In total, the manufacturing of the car could be broken down into 84 steps. The key to the process, however, was having interchangeable parts. Unlike other cars of the time, every Model T produced on Fords line used the exact same valves, gas tanks, tires, etc. so that they could be assembled in a speedy and organized fashion. Parts were created in mass quantities and then brought directly to the workers who were trained to work at that specific assembly station. The chassis of the car was pulled down the 150-foot line by a chain conveyor and then 140 workers applied their assigned parts to the chassis. Other workers brought additional parts to the assemblers to keep them stocked; this reduced the amount of time workers spent away from their stations to retrieve parts. The assembly line significantly decreased the assembly time per vehicle and increased the profit margin. Assembly Line Customization As time passed, Ford used assembly lines more flexibly than he is generally given credit for. He used multiple parallel lines in a start-stop mode to adjust output to large demand fluctuations. He also used sub-systems which optimized extraction, transportation, production, assembly, distribution, and sales supply chain systems.   Perhaps his most useful and neglected innovation was the development of a way to mechanize production and yet customize the configuration of each Model T as it rolled off the block. Model T production had a core platform, a chassis consisting of engine, pedals, switches, suspensions, wheels, transmission, gas tank, steering wheel, lights, etc. This platform was continually being improved. But the body of the car could be any one of several types of vehicles: auto, truck, racer, woody wagon, snowmobile, milk wagon, police wagon, ambulance, etc.  At peak, there were eleven basic model bodies, with 5,000 custom gadgets that were manufactured by external companies that could be selected by the customers. Impact of the Assembly Line on Production The immediate impact of the assembly line was revolutionary. The use of interchangeable parts allowed for continuous workflow and more time on task by laborers. Worker specialization resulted in less waste and a higher quality of the end product. Sheer production of the Model T dramatically increased. The production time for a single car dropped from over 12 hours to just 93 minutes due to the introduction of the assembly line. Ford’s 1914 production rate of 308,162 eclipsed the number of cars produced by all other automobile manufacturers combined. These concepts allowed Ford to increase his profit margin and lower the cost of the vehicle to consumers. The cost of the Model T would eventually drop to $260 in 1924, the equivalent of approximately $3,500 today. Impact of the Assembly Line on Workers The assembly line also drastically altered the lives of those in Ford’s employ. The workday was cut from nine hours to eight hours so that the concept of the three-shift workday could be implemented with greater ease. Although hours were cut, workers did not suffer from lower wages; instead, Ford nearly doubled the existing industry-standard wage and began paying his workers $5 a day. Ford’s gamble paid off- his workers soon used some of their pay increases to purchase their own Model Ts. By the end of the decade, the Model T had truly become the automobile for the masses that Ford had envisioned. The Assembly Line Today The assembly line is the primary mode of manufacturing in the industry today. Automobiles, food, toys, furniture, and many more items pass down assembly lines worldwide before landing in our homes and on our tables. While the average consumer does not think of this fact often, this 100-year-old innovation by a car manufacturer in Michigan changed the way we live and work forever. Sources and Further Reading Alizon, Fabrice, Steven B. Shooter, and Timothy W. Simpson. Henry Ford and the Model T: Lessons for Product Platforming and Mass Customization. Design Studies 30.5 (2009): 588–605. Print.Upward, Geoffrey C. A Home for Our Heritage: The Building and Growth of Greenfield Village and Henry Ford Museum. Dearborn, Michigan: The Henry Ford Museum Press, 1979. Print.Wilson, James M. Henry Ford Vs. Assembly Line Balancing. International Journal of Production Research 52.3 (2014): 757–65. Print.