ANSWERS: 7
  • Basically, calculus is the single most useful branch of mathematics developed. It has been proposed by some historians of science that the development of calculus will prove to have as great an important to humanity as the development of writing. Calculus is used in physics, engineering, chemistry, biology, economics, psychology and sociology. In other words, almost all areas of human scientific endeavour use calculus as an important tool.
  • Calculus is useful for solving non-linear equations. For example, say you were looking for the area inside a flat rectangle. A flat rectangle is linear, and finding the area would just be a simple matter of multiplying one side to one of the perpendicular sides, or 3 inch times 5 inch equals 15 square inches which is the area. What if it was a rectangular box? Same difference, still linear - The length times width times height may equal 3 times 5 times 4 equals 60 inches cubed. These are "linear equations". But what if it isn't linear? What if it's flat and curves around like a sine wave? Looping back and forth and never ending? And say you want to find the area of this flat sine wave from 0 to pi? This isn't linear at all. If you look at this curvy line from a distance, it's rounded. But what if you pin-pointed on one spot on that curvy line? And you keep getting closer and closer to that spot? Eventually, you would have a gnat's view of that one spot and it isn't curved at all. It would look like a straight line. Kind of like looking across the ocean. The earth is curving, but it looks dead straight for miles from here, cause you are so close to it. From this point, you can take a measurement that is pretty much linear, do the 3 inch times 5 inch routine and record it. Now move across by 3 inches and do it again, line still looks straight here also. Now add it to the prior one. Keep doing this until you reach your destination of pi, add them all together and you get your area. You've made your curve into very tiny little rectangles. It's not absolutely exact, cause the itty bitty curve part at each interval wasn't added in, but it's really close. If you took and just drew a straight line across the entire curve, the dome part you miss is huge, but since you made a hole bunch of little domes, it's getting closer to the true area. To make it closer to exact, keep making each small interval a little smaller. That makes the curve part more insignificant, making the answer more exact. The smallest interval would be zero inches, or whatever. But we can't do that. Can't divide by zero. But we can "approach" zero and get extremely close, say 1 zillionth of an inch maybe, then add up all zillion of these extremely thin rectangles. If you try to add up a zillion different rectangle areas, it may take a couple of hours, or so... Yep... Maybe even 3 hours considering you had to find the areas of a zillion rectangles in the first place..... (Bad joke). By the way, doesn't have to be an area. Look at the squiggly line for a road on a map. The smaller you make the intervals, the more accurate you come to the distance of that interval, and you just add all of them up to come to the answer- How far is it to Denver? Draw a straight line from your house to Denver, and you would be wayyyy off.. Unless you live in Denver. That, by the way, is why Calculus' integral sign looks like a wavy "S". It's a summation of little intervals as their distance approach zero. This is where calculus comes in. It allows you a quick mathematical procedure to find these irregular areas as their length "approaches" zero. It's impossible to master. There are so many different formulas to keep up with, so many different scenarios, it becomes endless. Nobody masters calculus, they just get used to the parts they need to know, and forget all the rest they learned in Calculus getting "up" to what they needed to know. In Calculus, you have a single integral, giving you length, a double integral giving you area, and a triple integral giving you depth area. Position is the derivative of velocity, and velocity is the derivative of acceleration. An integral goes the opposite direction where acceleration is the integral of velocity which is the integral of position. Everything we do on earth can be broken into areas, positions, accelerations, and velocities. Everything. So if you have a hankering to know the exact force a 2,127ton meteorite with a 26degree angular velocity of 63,000Kph, spinning at 3,800kph will cause when it strikes a planet moving at -64 degree angular velocity of 40,300Kph, striking it at an angle of 42 degrees on a surface spinning at 4,600kph at a 12 degree angle, I suggest you break out a calculus book. Or simplified... What is the charge in a 47uF capacitor in a particular circuit at 3.5uS? This is NOT linear. Calculus is the quickest and most accurate way to measure this and makes all these HDTV's, PDA's, and the occasionally successful Mars landings possible. Once you learn up to the part you need, it takes a couple of minutes to pull off the answers, instead of the rest of your life.
  • Differential calculus can be used for finding the local slope and curvature of a function and important points on a curve: maximum, minimum, inflection points, etc. These points and slopes can be the solution to a wide variety of problems related to physics, mechanical and electrical design, perhaps chemistry and other fields where a problem can be represented by an equation with variables. One example, if you can derive total fuel consumption as a function of acceleration there may be a minimum point of the function that allows you to reach your target speed with the least fuel - this is especially important with airplanes and rockets. Integral calculus can be used to find the area under curves which can represent the amount of work done, distance gone, energy exerted, stress, angle and deflection of beams and a host of other scientific and mechanical properties that can be characterized by equations. One overly simple example: if I push on a 1 Kg rock in space with a constant force of "A" newtons in the same direction where A represents a constant. The acceleration (a) as a function of time: a = A meters per second squared. The graph will be a horizontal line with respect to time (t). The speed (v for velocity) is the integral of the acceleration: v = A * t + B where t is time (how many seconds) and B is the speed it was going when we started. At+B is the integral of A with respect to time. The graph will be a line sloping upward with slope of A units with respect to t. The distance traveled is the integral of the speed. One half A * t squared + B * t + C where C is how far away it was when we started. ( .5At^2 + Bt + C) The graph will be a parabola with increasing slope with respect to t. Now, if a rocket starts with a mass of 60,000 Kg of which 50,000 is fuel and uses 500 Kg of fuel per second and the force generated by the fuel burn increases with altitude up to 15,000 m and decreases thereafter and the air resistance goes up with speed and down with altitude, you will need some calculus to determine how fast it will be going and how far it has gone when it runs out of fuel.
  • Calculus is the controversial concept provided by Newton way back in 17th century.It has unlimited number of uses.If you once look at physics you will find many uses of calculus in deriving expressions and formulas. It is also used in finding the areas of curves and volumes and areas of surfaces.
  • Besides all the calculations that calculus is useful for, calculus is the basis for our whole physical model. Formulas that we hear in high school are always simplified for linear problems. In general you have to use calculus. Without calculus Newton could not have calculated that the elliptical orbits of the planets resulted from a central force. That means we would not know about gravity. All the mechanical equations can be derived generally only through calculus. (e.g. You know that work = force x distance, but when the force changes as you move along you need integration) We could not prove that vibrations follow a sin curve, and we could not calculate waves without calculus. In electrodynamics the Maxwell equations are typically in an integral or differential form. Only through that we get to the wave properties of light. And when it comes to atoms and electron orbitals you definitely need calculus to solve (approximate) the differential Schrödinger equation. They say that the Schrödinger equation is the equation most often used in science, It allows us to calculate the microscopic properties of matter.
  • Calculus is useful for defining solutions to problems that best are posed in infinitely small steps (in 'limit'). What I mean by this: Suppose, for instance, you leave a window open to the outside and the outside air is at a temperature of 50 derees and the inside air is 70. The volume of air in the room is 1000 cubic feet, and wind blows into the room (and out the open door on the other side) at a rate of one cubic foot per second. Now, you might naively say that .1% of the air is replaced per second, so in 1000 seconds, it will all be replaced by outside air, so after 250 seconds the temperature will be 65, after 500, 60, after 750, 55 and after 1000, 50. Of course, there is a problem with this, and that is that the outside air mixes with the inside air. Before long, the air blowing out the doorway at the back is a mixture of indoor and outdoor air as well. So you might say well, after 250 seconds, the air in the room is 75% indoor air and 25% outdoor air and thus at 65 degrees. But after 500 seconds, this 75% indoor air has been diluted not to 50% but to 75% of 75%, or 9-16ths, 56.25%. In 750 seconds, it will be 75% of 75% of 75%, and after 1000, it will be 75% of that, about 31.64% total. And so the air in the room is in fact at a temperature of 56.33 degrees, not all the way down to 50. BUT WAIT, you say. Why stop there? Every tenth of a second, one ten-thousandth of the indoor air is replaced by outdoor-air, so it in fact should not be 3 quarters to the 4th power, but .9999 to the ten-thousandth power. How about .999999 to the millionth power? Well, this is where calculus takes over. Newton contemplated this sort of problem and defined the conditions for... the continuous case, where the time increment goes to ZERO and the fraction of the room's air replaced by outdoor air goes to ZERO... the LIMIT of this case. He in fact was not concerned with this particular classic elementary problem in differential equations (which I was assigned to do in 5th grade as a punishment for leaving a window open and did successfully I might add, without prior knowledge of calculus), but with falling bodies at the time. The point is, calculus is used when you don't want to define a continuous valued function with a bunch of little discrete steps, but you want to evaluate it as if the little steps were in fact zero in length. That's what it was invented for. Of course, if you WANT to solve a difference equation instead of differential equation, they have their places in nature too.
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