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Phil: Use the Integral to Solve Problems Phil: Use the Integral to Solve Problems. That Is the Question Others who have no intention of ever studying the subject have this notion that calculus is impos- sibly difficult unless you happen to be a direct descendant of Einstein.
Much of calculus is really just very advanced algebra, geometry, and trig. It builds upon and is a logical exten- sion of those subjects. If you can do algebra, geometry, and trig, you can do calculus. But why should you bother — apart from being required to take a course? Why climb Mt. Why visit the Louvre to see the Mona Lisa? Why watch South Park? Like these endeavors, doing calculus can be its own reward. There are many who say that calculus is one of the crowning achievements in all of intellectual his- tory.
Oh, sure, I know calculus. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series.
Calculus For Dummies, 2nd Edition takes calculus out of the ivory tower and brings it down to earth. Whenever possible, I explain the calculus concepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry.
I then show you how the calculus concepts work in concrete examples. Only later do I give you the fancy calculus formu- las. All explanations are in plain English, not math-speak. It can be a great aid to true understanding of calculus — or any math topic for that matter — to focus on the why in addition to the how-to.
If you want to give your study of calculus a solid foun- dation, you should read these explanations. You can then read the remaining exposition only if you feel the need. I find the sidebars interesting and entertaining. What do you expect? I wrote them! But you can skip them without missing any essential calculus.
Minor note: Within this book, you may note that some web addresses break across two lines of text. Foolish Assumptions Call me crazy, but I assume. In short, without the pre-calculus stuff, you can see the calculus forest, but not the trees. No, not the dreaded w-word! Is that too much to ask? Icons Used in This Book Keep your eyes on the icons: Next to this icon are the essential calculus rules, definitions, and formulas you should definitely know. These are things you need to know from algebra, geometry, or trig, or things you should recall from earlier in the book.
Take note. This icon highlights common calculus mistakes. Take heed. This is a great place to go if you forget a formula. Where to Go from Here Why, Chapter 1, of course, if you want to start at the beginning. If you already have some background in calculus or just need a refresher course in one area or another, then feel free to skip around.
Visit www. In this part. Chapter 1 What Is Calculus? After reading this and the following two short chapters, you will understand what calculus is all about. Ready for your first calculus test? Circle True or False. True or False: Studying calculus is hazardous to your health. True or False: Calculus is totally irrelevant. False, false, false!
You made it through algebra, geometry, and trigonometry. And calculus is not a dead language like Latin, spoken only by academics. But the work of those engineers, scientists, and economists has a huge impact on your day-to-day life — from your microwave oven, cell phone, TV, and car to the medicines you take, the workings of the economy, and our national defense. At this very moment, something within your reach or within your view has been impacted by calculus.
So What Is Calculus Already? Calculus is basically just very advanced algebra and geometry. The rub, of course, is that darn other sense in which it is a new and more difficult subject.
Look at Figure On the left is a man pushing a crate up a straight incline. On the right, the man is pushing the same crate up a curving incline. The problem, in both cases, is to determine the amount of energy required to push the crate to the top. You can do the problem on the left with regular math. Chapter 1: What Is Calculus? For the straight incline, the man pushes with an unchanging force, and the crate goes up the incline at an unchanging speed. With some simple physics formulas and regular math including algebra and trig , you can compute how many calories of energy are required to push the crate up the incline.
Note that the amount of energy expended each second remains the same. For the curving incline, on the other hand, things are constantly changing. And the man pushes with a constantly changing force — the steeper the incline, the harder the push. As a result, the amount of energy expended is also changing, not every second or every thousandth of a second, but constantly changing from one moment to the next.
For the curving incline problem, the physics formulas remain the same, and the algebra and trig you use stay the same. Figure shows a small portion of the curving incline blown up to several times its size.
Each small chunk can be solved the same way, and then you just add up all the chunks. What makes the invention of calculus such a fantastic achievement is that it does what seems impossible: it zooms in infinitely.
Real-World Examples of Calculus So, with regular math you can do the straight incline problem; with calculus you can do the curving incline problem. Here are some more examples. With regular math you can determine the length of a buried cable that runs diagonally from one corner of a park to the other remember the Pythagorean theorem?
With calculus you can determine the length of a cable hung between two towers that has the shape of a catenary which is different, by the way, from a simple circular arc or a parabola. Knowing the exact length is of obvious importance to a power company planning hundreds of miles of new electric cable. See Figure Figure Without and with calculus.
With calculus you can compute the area of a complicated, nonspherical shape like the dome of the Minneapolis Metrodome. The weight, of course, is needed for planning the strength of the supporting struc- ture. Check out Figure Figure Sans and avec calculus. With regular math and some simple physics, you can calculate how much a quarterback must lead his receiver to complete a pass. You see many real-world applications of calculus throughout this book.
The differentiation problems in Part IV all involve the steepness of a curve — like the steepness of the curving incline in Figure In Part V, you do integra- tion problems like the cable-length problem shown back in Figure These problems involve breaking up something into little sections, calculating each section, and then adding up the sections to get the total.
More about that in Chapter 2. Before the Calculus Era and C. Defining Differentiation Differentiation is the process of finding the derivative of a curve. And because the slope of a curve is equivalent to a simple rate like miles per hour or profit per item , the derivative is a rate as well as a slope. Let me guess: run A sudden rush of algebra nostalgia is flooding over you. On a curve, the slope is constantly changing, so you need calculus to determine its slope.
And the slope of this line is the same at every point between A and B. But you can see that, unlike the line, the steepness of the curve is changing between A and B. At A, the curve is less steep than the line, and at B, the curve is steeper than the line. What do you do if you want the exact slope at, say, point C? Can you guess? Answer: You zoom in. Figure Zooming in on the curve. When you zoom in far enough — really far, actually infinitely far — the little piece of the curve becomes straight, and you can figure the slope the old- fashioned way.
The two graphs in Figure show a relation- ship between distance and time — they could represent a trip in your car. Figure Average rate and instanta- neous rate.
A regular algebra problem is shown on the left in Figure For the problem on the right, on the other hand, you need calculus. Using the derivative of the curve, you can determine the exact slope or steepness at point C. Just to the left of C on the curve, the slope is slightly lower, and just to the right of C on the curve, the slope is slightly higher.
The slope for this single infinitesimal point on the curve gives you the instantaneous rate in miles per hour at point C. Integration is the process of cutting up an area into tiny sections, figuring the areas of the small sections, and then adding up the little bits of area to get the whole area. Figure shows two area problems — one that you can do with geometry and one where you need calculus.
The shaded area on the left is a simple rectangle, so its area, of course, equals length times width. So what do you do? Why, zoom in, of course. Figure shows the top portion of a narrow strip of the weird shape blown up to several times its size. Figure For the umpteenth time, when you zoom in, the curve becomes straight. When you zoom in as shown in Figure , the curve becomes practically straight, and the further you zoom in, the straighter it gets.
After zooming in, you get the shape on the right in Figure , which is practically an ordinary trapezoid its top is still slightly curved.
Because you can compute the areas of rectangles, triangles, and trapezoids with ordi- nary geometry, you can get the area of this and all the other thin strips and then add up all these areas to get the total area. The horizontal axes show the number of hours after mid- night, and the vertical axes show the amount of power in kilowatts used by the city at different times during the day. Figure Total kilowatt- hours of energy used by a city during a single day.
The crooked line on the left and the curve on the right show how the number of kilowatts of power depends on the time of day. In both cases, the shaded area gives the number of kilowatt-hours of energy consumed during a typical hour period. The shaded area in the oversimplified and unrealistic problem on the left can be calculated with regular geometry. But the true relationship between the amount of power used and the time of day is more complicated than a crooked straight line.
Because of its weird curve, you need calculus to determine the shaded area. In the real world, the relationship between different variables is rarely as simple as a straight-line graph. Sorting Out Infinite Series Infinite series deal with the adding up of an infinite number of numbers. The following sequence of numbers is gener- ated by a simple doubling process — each term is twice the one before it: 1, 2, 4, 8, 16, 32, 64, ,.
Divergent series The preceding series of doubling numbers is divergent because if you continue the addition indefinitely, the sum will grow bigger and bigger without limit.
Divergent usually means — there are exceptions — that the series adds up to infinity. Divergent series are rather uninteresting because they do what you expect. You keep adding more numbers, so the sum keeps growing, and if you con- tinue this forever, the sum grows to infinity. Big surprise. Convergent series Convergent series are much more interesting. With a convergent series, you also keep adding more numbers, the sum keeps growing, but even though you add numbers forever and the sum grows forever, the sum of all the infinitely many terms is a finite number.
Achilles is racing a tortoise — some gutsy warrior, eh? Our generous hero gives the tortoise a yard head start. Figure shows the situ- ation at the start of the race and your first two photos.
You take your first photo the instant Achilles reaches the point where the tortoise started. The tortoise moves a tenth as fast as Achilles, so in the time it takes Achilles to travel yards, the tortoise covers a tenth as much ground, or 10 yards. If you do the math, you find that it took Achilles about 10 seconds to run the yards. Then you take your second photo when Achilles reaches point A, which takes him about one more second.
In that second, the tortoise has moved ahead 1 yard to point B. You take your third photo not shown when Achilles reaches point B and the tortoise has moved ahead to point C. Every time Achilles reaches the point where the tortoise was, you take another photo. There is no end to this series of photographs. And every single time Achilles reaches the point where the tortoise was, the tortoise has covered more ground — even if only a millimeter or a millionth of a millimeter.
This process never ends, right? Thus, the argument goes, because you can never get to the end of your infinite series of photos, Achilles can never catch or pass the tortoise. Well, as everyone knows, Achilles does in fact reach and pass the tortoise — thus the paradox. The mathematics of infinite series explains how this infinite series of time intervals sums to a finite amount of time — the precise time when Achilles passes the tortoise.
You see more of them in Part V. Calculus works because after you zoom in and curves look straight, you can use regular algebra and geometry with them. The zooming-in process is achieved through the mathematics of limits. With the slope formula from algebra, you can figure the slope of the line between 1, 1 and 2, 4. If you do the math, the slopes between 1, 1 and your moving point would look something like 2.
And with the almost magical mathematics of limits, you can conclude that the slope at 1, 1 is precisely 2, even though the sliding point never reaches 1, 1. The mathematics of limits is all based on this zooming-in process, and it works, again, because the further you zoom in, the straighter the curve gets. What Happens When You Zoom In Figure shows three diagrams of one curve and three things you might like to know about the curve: 1 the exact slope or steepness at point C, 2 the area under the curve between A and B, and 3 the exact length of the curve from A to B.
Chapter 3: Why Calculus Works 23 The first row of Figure shows a magnified detail from the three diagrams of the curve in Figure The second row shows further magnification, and the third row yet another magnification. This process is continued indefinitely. Figure One curve — three questions. Figure Zooming in to the microscopic level. After zooming in forever, an infinitely small piece of the original curve and the straight diagonal line are now one and the same.
You can think of the lengths 3 and 4 in Figure no pun intended as 3 and 4 millionths of an inch, no, make that 3 and 4 billionths of an inch, no, trillionths, no, gazillionths,. Figure Your final destina- tion — the sub, sub, sub. For the diagram on the left in Figure , you can now use the regular slope formula from algebra to find the slope at point C.
This is how differentiation works. For the diagram in the middle of Figure , the regular triangle formula from geometry gives you an area of 6.
Then you can get the shaded area inside the strip shown in Figure by adding this 6 to the area of the thin rectangle under the triangle the dark-shaded rectangle in Figure Then you repeat this process for all the other narrow strips not shown , and finally just add up all the little areas. This is how integration works.
And for the diagram on the right of Figure , the Pythagorean theorem from geometry gives you a length of 5. Then to find the total length of the curve from A to B in Figure , you do the same thing for the other minute sections of the curve and then add up all the little lengths. This is how you calculate arc length another integration problem. Well, there you have it. Two Caveats, or Precision, Preschmidgen Not everything in this chapter or this book for that matter will satisfy the high standards of the Grand Poobah of Precision in Mathematical Writing.
What would it mean to zoom in forever or an infinite number of times? So, if your pre-algebra and algebra are a bit rusty — you know, all those rules for algebraic expressions, equations, fractions, powers, roots, logs, factoring, quadratics, and so on — make sure you review the following basics.
Dealing with them requires that you know a few rules. The denominator of a fraction can never equal zero. Well, how many zeros would you need to add up to make 5? Definition of reciprocal: The reciprocal of a number or expression is its mul- tiplicative inverse — which is a fancy way of saying that the product of some- thing and its reciprocal is 1.
To get the reciprocal of a fraction, flip it upside down. Because 5 goes into 5 one time, and 5 goes into 10 two times, you can cancel a 5: 3 10 Also note that the original problem could have been written as. This illustrates a powerful principle: Variables always behave exactly like numbers. Find the least common denominator actually, any common denomi- nator will work when adding fractions , and convert the fractions.
In this problem, you have b d an a instead of a 2, a b instead of a 5, a c instead of a 3, and a d instead of an 8. You can 5 8 think of each of the numbers in the above solution as stamped on one side of a coin with the corresponding variable stamped on the other side.
Insights like this are the reason they pay me the big bucks. Chapter 4: Pre-Algebra and Algebra Review 33 x5y2 In the fraction, , three xs can be canceled from the numerator and denom- x3z x2y2 inator, resulting in the simplified fraction,.
Canceling works the same way with expressions as it does for single variables. Expressions always behave exactly like variables. You also need to know when you can cancel. The multiplication rule: You can cancel in a fraction only when it has an unbroken chain of multiplication through the entire numerator and the entire denominator.
Because the denomi- nator also has an unbroken chain of multiplication, canceling is allowed. Absolute Value — Absolutely Easy Absolute value just turns a negative number into a positive and does nothing to a positive number or zero. This is huge! Note that the 4 16 power is negative, but the answer of 1 is not negative. Do not distribute the power in this case.
Rooting for Roots Roots, especially square roots, come up all the time in calculus. So knowing how they work and understanding the fundamental connection between roots and powers is essential.
So, if you get a problem with roots in it, you can just convert each root into a power and use the power rules instead to solve the problem this is a very useful technique. No negatives under even roots.
The answer is thus 10 3. Break down into a product of all of its prime factors. Circle each pair of numbers. For each circled pair, take one number out. These two equations say precisely the same thing. On many newer-model log 3 calculators, you can compute log3 20 directly.
When are you ever going to need it? Algebraic factoring always involves rewriting a sum of terms as a product. What follows is a quick refresher course. Make sure you always look for a GCF to pull out before trying other factoring techniques.
Looking for a pattern After pulling out the GCF if there is one, the next thing to do is to look for one of the following three patterns. The first pattern is huge; the next two are much less important. Several definitions: A trinomial is a polynomial with three terms. The polynomial at the beginning of this para- graph, for instance, has a degree of 5.
A few standard techniques for factoring a trinomial like this are floating around the mathematical ether — you probably learned one or more of them in your algebra class. You can solve quadratic equations by one of three basic methods. Bring all terms to one side of the equation, leaving a zero on the other side. Set each factor equal to zero and solve using the zero product property.
Method 1 will work only if the quadratic is factorable. The quick test for that is a snap. The discriminant is the stuff under the square root symbol in the quadratic formula — see Method 2 below.
Since is a perfect square , the quadratic is factorable. But whether or not the quadratic is factorable, you can always solve it with the quadratic formula discussed in the next section. Plug the coefficients into the formula. Method 3: Completing the square The third method of solving quadratic equations is called completing the square because it involves creating a perfect square trinomial that you can solve by taking its square root.
Put the x2 and the x terms on one side and the constant on the other. Take half of the coefficient of x, square it, then add that to both sides. Factor the left side into a binomial squared. Notice that the factor always contains the same number you found in Step 3 —4 in this example. Differential calculus involves finding the slope or steepness of various functions, and integral calculus involves computing the area underneath functions.
What Is a Function? Basically, a function is a relationship between two things in which the numerical value of one thing in some way depends on the value of the other. Examples are all around us: The average daily temperature for your city depends on, and is a function of, the time of year; the distance an object has fallen is a function of how much time has elapsed since you dropped it; the area of a circle is a function of its radius; and the pressure of an enclosed gas is a function of its temperature.
The defining characteristic of a function A function has only one output for each input. Consider Figure The slot machine is not. The Coke machine is a function because after plugging in the inputs your choice and your money , you know exactly what the output is.
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