Presentation Transcript

1. Read the following slides…. Jot down important information into your notes. (You don’t need every little detail but have a good idea of what is being talked about.) Answer the questions at the end of the slides for tomorrow. Assignment: pg. 9 Problems 1-4 all, 6-18 even, 39-42 all, 44-50 even, 54, 55,56 (These will be due on Wednesday)
2. What is Calculus? Calculus is Latin for stone, and the ancient Romans used stones for counting and arithmetic. In its most basic sense, calculus is just that – a form of counting. After advanced algebra and geometry, it is the next step in higher mathematics, and is used for solving complex problems that regular mathematics cannot complete.
3. Calculus is concerned with comparing quantities which vary in a non-linear way. It is used extensively in science and engineering since many of the things we are studying (like velocity, acceleration, current in a circuit) do not behave in a simple, linear fashion. If quantities are continually changing, we need calculus to study what is going on.
4. Calculus was developed by two different men in the seventeenth century. Gottfried Wilhelm Leibniz (1646-1716), a self-taught German mathematician, and Isaac Newton (1642-1727), an English scientist, both developed calculus in the 1680s. While Leibniz invented it ten years later than Newton, he published his findings twenty years earlier, and that overlap led to decades of controversy about which man reached his conclusions first. Today it is generally agreed that both men developed calculus independently.
5. Calculus is the mathematics of change, of calculating problems that are continually evolving. This is possible by breaking such problems into infinitesimal steps, solving each of those steps, and adding all the results. Rather than doing each step individually, calculus allows these computations to be done simultaneously.
6. There are two primary branches of calculus: differential calculus and integral calculus. Differential calculus, or differentiation, is used primarily to determine the slope or steepness of a curve, also called a curve’s derivative. Slope is a rate of change in a curve – a very steep curve is changing very fast – and calculus is used when a curve is very complicated, such as calculating the slope of a mountain or the speed of a roller coaster.
7. Differential calculus involves any problem that may be graphed when the desired result is a single point on that graph. For example, if a rancher wants to construct a corral with a limited amount of fence, he can vary the lengths of the sides of the corral. Using calculus, he could determine which lengths would enclose the greatest area and make the largest corral. A graph could be drawn using every possible combination of lengths, and the highest point on that graph – the maximum – would signify the greatest area.
8. Integral calculus, or integration, deals with areas and volumes of complex figures, such as determining the greatest amount of space or volume beneath a dome in a stadium design in order to incorporate as many seats as possible. To find the area beneath a curve, integration breaks the area beneath the curve into minute pieces, determines the area of each piece, and adds them all together, or integrates them, into a final answer. Another example of integration would be determining the exact volume of water necessary to fill a wading pool in a zoo exhibit. The pool may have varying depths and irregular edges, and it would be extremely difficult to determine how much water is needed without being able to use the infinite sums that integration calculates.
9. What Is Calculus Used For? Both differential and integral calculus are useful to a wide variety of careers. The examples below are only a fraction of the ways professionals regularly use calculus.
10. Credit card companies use calculus to set the minimum payments due on credit card statements at the exact time the statement is processed by considering multiple variables such as changing interest rates and a fluctuating available balance. Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed. This research can help increase the rate of growth of necessary bacteria, or decrease the rate of growth for harmful and potentially threatening bacteria.
11. An electrical engineer uses integration to determine the exact length of power cable needed to connect two substations that are miles apart. Because the cable is hung from poles, it is constantly curving. Calculus allows a precise figure to be determined. An architect will use integration to determine the amount of materials necessary to construct a curved dome over a new sports arena, as well as calculate the weight of that dome and determine the type of support structure required.
12. A physicist uses calculus to find the center of mass of a sports utility vehicle to design appropriate safety features that must adhere to federal specifications on different road surfaces and at different speeds. An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits. A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions. This can create a realistic environment for movies or video games.
13. Obviously, a wide variety of careers regularly use calculus. Universities, the military, government agencies, airlines, entertainment studios, software companies, and construction companies are only a few employers who seek individuals with a solid knowledge of calculus. Even doctors and lawyers use calculus to help build the discipline necessary for solving complex problems, such as diagnosing patients or planning a prosecution case. Despite its mystique as a more complex branch of mathematics, calculus touches our lives each day, in ways too numerous to calculate.
14. Questions 1. Look up Newton’s contribution to Calculus. Write a short paragraph (3-4 sentences) describing what he did. 2. Look up Leibniz’s contribution to Calculus. Write a short paragraph (3-4 sentences) describing what he did. 3. What are the two branches of Calculus? Briefly explain what each branch does. 4. Find 2 things that Calculus allows us to do that Pre-Calculus does not. (Use a complete sentence for each thing.)