Poetry for dummies pdf free download

Okay, so here goes. I keep repeating — have you noticed? You learned how to find the slope of a line in algebra. In Figure , I gave you the slope of the parabola at several points, and then I showed you the short-cut method for finding the derivative — but I left out the important math in the middle. That math involves limits, and it takes us to the threshold of calculus.

For a line, this is easy. You just pick any two points on the line and plug them in. You can see the line drawn tangent to the curve at 2, 4. Figure shows the tangent line again and a secant line intersecting the parabola at 2, 4 and at 10, Definition of secant line: A secant line is a line that intersects a curve at two points.

Now add one more point at 6, 36 and draw another secant using that point and 2, 4 again. Now, imagine what would happen if you grabbed the point at 6, 36 and slid it down the parabola toward 2, 4 , dragging the secant line along with it. Can you see that as the point gets closer and closer to 2, 4 , the secant line gets closer and closer to the tangent line, and that the slope of this secant thus gets closer and closer to the slope of the tangent?

So, you can get the slope of the tangent if you take the limit of the slopes of this moving secant. When the point slides to 2. Sure looks like the slope is headed toward 4. As with all limit problems, the variable in this problem, x2 , approaches but never actually gets to the arrow-number 2 in this case.

Herein lies the beauty of the limit process. A fraction is a quotient, right? You may run across other, equivalent ways. Figure is the ultimate figure for. Chapter 9: Differentiation Orientation Have I confused you with these two figures? They both show the same thing. Both figures are visual representations of. Figure shows this general definition graphically. Note that Figure is virtually identical to Figure except that xs replace the 2s in Figure and that the moving point in Figure slides down toward any old point x, f x instead of toward the specific point 2, f 2.

Plug any number into x, and you get the slope of the parabola at that x-value. Figure sort of sum- marizes in a simplified way all the difficult preceding ideas about the differ- ence quotient.

Like Figures , , and , Figure contains a basic slope stair-step, a secant line, and a tangent line. The slope of the tangent line is. If, for example, the y-coordinate tells you distance traveled in miles , and the x-coordinate tells you elapsed time in hours , you get the familiar rate of miles per hour. That slope is the average rate over the interval from x1 to x2.

When you take the limit and get the slope of the tangent line, you get the instan- taneous rate at the point x1 , y1. Again, if y is in miles and x is in hours, you get the instantaneous speed at the single point in time, x1.

Because the slope of the tangent line is the derivative, this gives us another definition of the derivative. To Be or Not to Be?

By now you certainly know that the derivative of a function at a given point is the slope of the tangent line at that point. These types of discontinuity are dis- cussed and illustrated in Chapter 7. Continuity is, therefore, a necessary condition for differentiability. Dig that logician-speak. See function f in Figure See function g in Figure Inflection points are explained in Chapter You also now know the mathematical foundation of the derivative and its technical definition involving the limit of the difference quotient.

Some of this material is unavoidably dry. If you have trouble staying awake while slogging through these rules, look back to the last chapter and take a peek at the next three chapters to see why you should care about mastering these differentiation rules. You may want to order up a latte with an extra shot.

Learning these first half dozen or so rules is a snap. If you get tired of this easy stuff, however, I promise plenty of challenges in the next section. The constant rule This is simple. End of story. To find its derivative, take the power, 5, bring it in front of the x, and then reduce the power by 1 in this example, the power becomes a 4.

To repeat, bring the power in front, then reduce the power by 1. Instead of all that, just use the power rule: Bring the 2 in front, reduce the power by 1, which leaves you with a power of 1 that you can drop because a power of 1 does nothing.

But the difference quotient is included in every calculus book and course because it gives you a fuller, richer understanding of calculus and its foun- dations — think of it as a mathematical character builder.

Or because math teachers are sadists. You be the judge. Make sure you remember how to do the derivative of the last function in the above list. The slope m of this line is 1, so the derivative equals 1. Or you can just memorize that the derivative of x is 1. But if you forget both of these ideas, you can always use the power rule. Rewrite functions so you can use the power rule. Makes no difference. A coefficient has no effect on the process of differentiation.

You just ignore it and differentiate according to the appropriate rule. The coef- ficient stays where it is until the final step when you simplify your answer by multiplying by the coefficient. Solution: You know by the power rule that the derivative of x 3 is 3x 2, so the derivative of 4 x 3 is 4 3x 2.

The 4 just sits there doing nothing. Then, as a final step, you simplify: 4 3x 2 equals 12x 2. By the way, most people just bring the 3 to the front, like this: , which gives you the same result.

Constants in problems, like c and k also behave like ordinary numbers. Be sure to treat them like regular numbers. Solution: Just use the power rule for each of the first four terms and the con- stant rule for the final term.

You still differentiate each term separately. The addition and subtraction signs are unaffected by the differentiation. A third option is to use the following mnemonic trick. Write these three down, and below them write their cofunctions: csc, csc, cot. Put a negative sign on the csc in the middle.

Look at the top row. The bottom row works the same way except that both derivatives are negative. Differentiating exponential and logarithmic functions Caution: Memorization ahead. This is a special function: ex and its multiples, like 5ex, are the only functions that are their own derivatives. Think about what this means. See Chapter 4 if you want to brush up on logs. Okay, ready for a challenge? The following rules, especially the chain rule, can be tough. The trick is knowing the order of the terms in the numerator.

And is it more natural to begin at the top or the bottom of a fraction? The top, of course. So the quotient rule begins with the derivative of the top. If you remember that, the rest of the numerator is almost automatic. Note that the product rule begins with the derivative of the first function you read as you read the product of two functions from left to right.

In the same way, the quo- tient rule begins with the derivative of the first function you read as you read the quotient of two functions from top to bottom. All four of these functions can be written in terms of sine and cosine, right? See Chapter 6. The other three functions are no harder. Give them a try. To make sure you ignore the inside, tempo- rarily replace the inside function with the word stuff.

All chain rule problems follow this basic idea. You do the derivative rule for the outside function, ignoring the inside stuff, then multiply that by the derivative of the stuff.

Differentiate the inside stuff. Now put the real stuff and its derivative back where they belong. The outside function is the sine function, so you start there, taking the derivative of sine and ignoring the inside stuff, x 2.

The derivative of sin x is cos x, so for this problem, you begin with cos stuff 2. Multiply the derivative of the outside function by the derivative of the stuff. Parentheses are your friend. For chain rule problems, rewrite a composite function with a set of parentheses around each inside function, and rewrite trig functions like sin2 x with the power outside a set of parentheses: sin x 2.

Okay, now that you know the order of the functions, you can differentiate from outside in: 1. The outermost function is stuff 3 and its derivative is given by the power rule.

Use the chain rule again. The derivative of sin x is cos x, so the derivative of sin stuff begins with cos stuff. Plug those things back in. This can be simplified a bit. It may have occurred to you that you can save some time by not switching to the word stuff and then switching back.

Make sure you. The argument of this natural logarithm function is x 3. This rule tells you to put the argument dx x of the function in the denominator under the number 1. You then finish the problem by x multiplying that by the derivative of x 3 which is 3x 2. Final answer after sim- plifying: 3. In the example above, ln x 3 , you first use the natural log rule, then, as a separate step, you use the power rule to differentiate x 3. At no point in any chain rule problem do you use both rules at the same time.

For example, with ln x 3 , you do not use the natural log rule and the power rule at the same time to come up with 1 2. The rate of movement of the noise- the chain rule is based on a very simple idea. So, how fast is riding a bike. If the biker goes four times as fast dL is the noisemaker moving compared to Hardy?

The noisemaker is the walker, then the biker goes 4 times 2, or 8 times moving 3 times as fast as Laurel, and Laurel is as fast as the walker, right? Here it is in symbols note that this is the on a teeter-totter? Recall that for every inch same as the formal definition of the chain rule Hardy goes down, Laurel goes up 2 inches. Differentiate 4x 2 sin x 3. This problem has a new twist — it involves the chain rule and the product rule.

How should you begin? Where do I begin? Your last computation tells you the first thing to do. Say you plug the number 5 into the xs in 4x 2 sin x 3. Because your last computation is multiplication, your first step in differentiating is to use the product rule.

Remember the product rule? In such cases, y is written explicitly as a function of x. This means that the equation is solved for y; in other words, y is by itself on one side of the equation.

For such a problem, you need implicit differentiation. Remember using the chain rule to differentiate something like sin x 3 with the stuff technique?

With implicit differentiation, a y works like the word stuff. But the concept is exactly the same, and you treat y just like the stuff. Here goes. Differentiate each term on both sides of the equation. For the second term, you use the regular power rule. And for the third term, you use the regular sine rule.

Divide for the final answer. So, if you want to evaluate the derivative to get the slope at a particular point, you need to have values for both x and y to plug into the derivative. Either way is fine. Take your pick. Now, you could multiply the whole thing out and then differentiate, but that would be a huge pain.

Or you could use the product rule a few times, but that would also be too tedious and time-consuming. The better way is to use loga- rithmic differentiation: 1. Take the natural log of both sides. Now use the property for the log of a product, which you remember of course if not, see Chapter 4.

Differentiate both sides. The right side of the first line gets multiplied by the second line. Figure The graphs of inverse functions, f x and g x. As with any pair of inverse functions, if the point 10, 4 is on one function, 4, 10 is on its inverse. And, because of the symmetry of the graphs, you can see that the slopes at those points are reciprocals: At 10, 4 the slope is 1 and at 4, 10 the slope is 3 3.

The algebraic explanation is a bit trickier, however. The point 4, 10 on g can be written as 4, g 4. Okay, so maybe it was a lot trickier. Scaling the Heights of Higher Order Derivatives Finding a second, third, fourth, or higher derivative is incredibly simple. The second derivative of a function is just the derivative of its first derivative. And the higher derivatives of sine and cosine are cyclical.

In Chapters 11 and 12, I show you several uses of higher derivatives — mainly second derivatives. But for now, let me give you just one of the main ideas in a nutshell. A second derivative tells you how fast the first derivative is changing — or, in other words, how fast the slope is changing. A third derivative tells you how fast the second derivative is changing, which tells you how fast the rate of change of the slope is changing. It gets increas- ingly difficult to get a handle on what higher derivatives tell you as you go past the second derivative, because you start getting into a rate of change of a rate of change of a rate of change, and so on.

Which is a good thing, because in this chapter you use deriva- tives to understand the shape of functions — where they rise and where they fall, where they max out and bottom out, how they curve, and so on. Then in Chapter 12, you use your knowledge about the shape of functions to solve real-world problems. Along your drive, there are several points of interest between a and l. All of them, except for the start and finish points, relate to the steepness of the road — in other words, its slope or derivative.

Thus the function is increasing and its slope and derivative are therefore positive. You climb the hill till you reach the top at b where the road levels out.

The road is level there, so the slope and derivative equal zero. Because the derivative is zero at b, point b is called a stationary point of the func- tion. To be a local max, b just has to be the highest point in its immediate neighborhood.

After reaching the crest of the hill at b, you start going down — duh. So, after b, the slope and derivative are negative and the function is decreasing. To the left of every local max, the slope is positive; to the right of a max, the slope is negative. Chapter Differentiation and the Shape of Curves The little down arrow between b and c in Figure indicates that this sec- tion of the road is curving down — the function is said to be concave down there.

As you can see, the road is also concave down between a and b. Concavity poetry: Down looks like a frown, up looks like a cup. Wherever a function is concave down, its derivative and slope are decreasing; wherever a function is concave up, its derivative and slope are increasing.

Okay, so the road is concave down until c where it switches to concave up. The point c is also the steepest point on this stretch of the road. Inflection points are always at the steepest — or least steep — points in their immediate neighborhoods. Be careful with function sections that have a negative slope. Point c is the steepest point in its neighborhood because it has a bigger negative slope than any other nearby point. But remember, a big negative number is actu- ally a small number, so the slope and derivative at c are actually the smallest of all the points in the neighborhood.

After point c, you keep going down till you reach d, the bottom of a valley. Point d is another stationary point because the road is level there and the derivative is zero. A scenic overlook: The absolute maximum After d, you travel up, passing e, which is another inflection point. You stop at the scenic overlook at g, another stationary point and another local max. No tangent line means no slope; and no slope means no derivative, or you can say that the derivative at j is undefined.

A sharp turning point like j is called a corner. Again, note that because the slope and the derivative are becoming smaller and smaller negative numbers on the way to k, they are actually increasing. It also seems to be one way of explaining the extraordinary longevity of the Japanese, especially on the island of Okinawa, where there are Some people have found their ikigai, while others are still looking, though they carry it within them.

Our ikigai is hidden deep inside each of us, and finding it requires a patient search. Please see the contents and read them clearly with the figures given in each chapter clearly and read them. Memorize them , the concepts clearly Showing results: of Published: Nov Downloads: 61 Pages: Published: Nov Downloads: 51 Pages: Big Data Analytics Strategies Configuration Management: Expert Guidance Face Processing: Advanced Modeling Formulas Of Acoustics springer Developing Information Systems: Practical Nine Algorithms That Changed This is a guide for teachers who want to present the natural world in their classroom by rearing the living things in the classroom.

It is based on standard 3 — life science: as a result of the children explorations and participation in simple investigation, children develop an understanding of This academic guide on Database Systems is meant for anybody interested in databases.

That is anyone can master the concepts of Databases, not only undergraduate students in Computer Science and Engineering but anybody, you name it. That is because majority of the chapters have been written with