It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, with introductions to multivariable calculus, differential equations, and numerical analysis.
Best of all, it includes practical exercises designed to simplify and enhance understanding of this complex subject. Introduction to integration Indefinite integrals Intermediate Integration topics Infinite series Advanced topics Practice exercises Confounded by curves?
Perplexed by polynomials? This plain-English guide to Calculus II will set you straight! The book covers techniques and applications of integration, infinite series, and differential equations, the whole time motivating the study of calculus using its applications. The authors include numerous solved problems, as well as extensive exercises at the end of each section. On top of all that, the rocket had to hit a moving target, the moon. All of these things were changing, and their rates of change were changing.
Much of modern economic theory, for example, relies on differentiation. In economics, everything is in constant flux. Prices go up and down, supply and demand fluctuate, and inflation is constantly changing. These things are con- stantly changing, and the ways they affect each other are constantly chang- ing. You need calculus for this. Differential calculus is one of the most practical and powerful inventions in the history of mathematics.
The derivative is just a fancy calculus term for a simple idea you know from algebra: slope. Slope, as you know, is the fancy algebra term for steepness. And steepness is the fancy word for. These are big words for a simple idea: Finding the steepness or slope of a line or curve. Throw some of these terms around to impress your friends.
By the way, the root of the words differ- ential and differentiation is difference — I explain the connection at the end of this chapter in the section on the difference quotient.
A steepness of 1 means that as the stickman walks one 2 foot to the right, he goes up 1 foot; where the steepness is 3, he goes up 3 2 feet as he walks 1 foot to the right. This is shown more precisely in Figure Negative slope: To remember that going down to the right or up to the left is a negative slope, picture an uppercase N, as shown in Figure Figure This N line has a Negative slope.
How steep is a flat, horizontal road? Not steep at all, of course. Zero steepness. So, a horizontal line has a slope of zero. Like where the stick man is at the top of the hill in Figure There are more. Variety is not the spice of the wall trying to figure out things like why some 2 author uses one symbol one time and a different mathematics. When mathematicians decide symbol another time, and what exactly does the on a way of expressing an idea, they stick d or f mean anyway, and so on and so on, or 2 to it — except, that is, with calculus.
Hold on to your hat. I all mean exactly the same thing: or df or strongly recommend the second option. I realize that no calculation is necessary here — you go up 2 as you go over 1, so the slope is automatically 2. But bear with me because you need to know what follows.
Now, take any two points on the line, say, 1, 5 and 6, 15 , and figure the rise and the run. You rise up 10 from 1, 5 to 6, 15 because 5 plus 10 is 15 or you could say that 15 minus 5 is And you run across 5 from 1, 5 to 6, 15 because 1 plus 5 is 6 or in other words, 6 minus 1 is 5. Table shows six points on the line and the unchanging slope of 2.
That was a joke. So why did I start the chapter with slope? Because slope is in some respects the easier of the two concepts, and slope is the idea you return to again and again in this book and any other calculus textbook as you look at the graphs of dozens and dozens of functions. A slope is, in a sense, a picture of a rate; the rate comes first, the picture of it comes second. Just like you can have a function before you see its graph, you can have a rate before you see it as a slope.
Calculus on the playground Imagine Laurel and Hardy on a teeter-totter — check out Figure Figure Laurel and Hardy — blithely unaware of the calculus implications.
Assuming Hardy weighs twice as much as Laurel, Hardy has to sit twice as close to the center as Laurel for them to balance.
And for every inch that Hardy goes down, Laurel goes up two inches. So Laurel moves twice as much as Hardy. A derivative is a rate. You can see that if Hardy goes down 10 inches then dH is 10, and because dL equals 2 times dH, dL is 20 — so Laurel goes up 20 inches. But a rate can be anything per anything. Again, a derivative just tells you how much one thing changes compared to another.
It tells you that for each mile you go the time changes 1 of an hour. We just saw miles per hour and hours per mile. Rates can be constant or changing. In either case, every rate is a derivative, and every derivative is a rate.
The rate-slope connection Rates and slopes have a simple connection. All of the previous rate examples can be graphed on an x-y coordinate system, where each rate appears as a slope.
Consider the Laurel and Hardy example again. Laurel moves twice as much as Hardy. The line goes up 2 inches for each inch it goes to the right, and its slope is thus 2 , or 2. One last comment. Well, you can think of dL as the run rise and dH as the run. That ties everything together quite nicely. The Derivative of a Curve The sections so far in this chapter have involved linear functions — straight lines with unchanging slopes.
Calculus is the mathematics of change, so now is a good time to move on to parabolas, curves with changing slopes.
You can see from the graph that at the point 2, 1 , the slope is 1; at 4, 4 , the slope is 2; at 6, 9 , the slope is 3, and so on. Unlike the unchanging slope of a line, the slope of a parabola depends on where you are; it depends on the x-coordinate of wherever you are on the parabola.
Table shows some points on the parabola 2 and the steepness at those points. Beginning with the original function, x , take the power and put it in 2 4 front of the coefficient. Reduce the power by 1. In this example, the 2 becomes a 1. So the derivative is 1 x 1 or just 1 x. The Difference Quotient Sound the trumpets! You come now to what is perhaps the cornerstone of dif- ferential calculus: the difference quotient, the bridge between limits and the derivative.
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. Point k is another stationary point because its derivative is zero.
After passing k, you go down to l, your final destination. Hope you enjoyed your trip. At j, the derivative is undefined. These points where the derivative is either zero or undefined are the critical points of the function. The x-values of these critical points are called the critical numbers of the function. However, not all critical points are necessarily local maxes or mins.
Point k, for instance, is a critical point but neither a max nor a min. Use a lot of these fancy plurals if you want to sound like a professor. A single local max or min is a local extremum. The absolute max is the highest point on the road from a to l. The absolute min is the lowest point. The function is also decreasing at point k, a hori- zontal inflection point, even though the slope and derivative are zero there. At all horizontal inflection points, a function is either increasing or decreasing.
At local extrema b, d, g, i, and j, the function is neither increasing nor decreasing. Inflection points c, e, h, and k are where the concavity switches from up to down or vice versa. Inflection points are also the steepest or least steep points in their immediate neighborhoods. Find the first derivative of f using the power rule. Set the derivative equal to zero and solve for x.
Because the derivative of f equals zero at these three critical numbers, the curve has horizontal tangents at these numbers. A curve has a horizontal tangent line wherever its derivative is zero, namely, at its stationary points.
A curve will have horizontal tangent lines at all of its local mins and maxes except for sharp corners like point j in Figure and at all of its horizontal inflection points. You can do this with either the first derivative test or the second derivative test.
I suppose you may be wondering why you have to test the critical numbers when you can see where the peaks and valleys are by just looking at the graph in Figure — which you can, of course, reproduce on your graph- ing calculator. Good point. So what else is new? The first derivative test The first derivative test is based on the Nobel-Prize-caliber ideas that as you go over the top of a hill, first you go up and then you go down, and that when you drive into and out of a valley, you go down and then up.
Take a number line and put down the critical numbers you found above:. Pick a value from each region, plug it into the first derivative, and note whether your result is positive or negative. Chapter 5 discusses even functions. These four results are, respectively, positive, negative, negative, and positive. Now, take your number line, mark each region with the appropriate positive or negative sign, and indicate where the function is increasing where the deriva- tive is positive and decreasing where the derivative is negative.
The result is a so-called sign graph. Conversely, because the function switches from decreasing to increasing at 2, you have the bottom of a valley there or a local minimum. To use the first deriv- ative test to test for a local extremum at a particular critical number, the function must be continuous at that x-value.
The second derivative test — no, no, anything but another test! The concavity of a function at a point is given by its second derivative: a posi- tive second derivative means the function is concave up, a negative second derivative means the function is concave down, and a second derivative of zero is inconclusive the function could be concave up, concave down, or there could be an inflection point there.
When this happens, you have to use the first derivative test. Now go through the first and second derivative tests one more time with another example. Find the first derivative of g. Set the derivative equal to zero and solve. Determine whether the first derivative is undefined for any x-values.
Plot the critical numbers on a number line, and then use the first derivative test to figure out the sign of each region. Figure shows the sign graph. Plug the critical numbers into g to obtain the function values the heights of these two local extrema. You could have used the second derivative test instead of the first deriva- tive test in Step 4.
An open interval like 2, 5 excludes the endpoints. Finding the absolute max and min is a snap. All you do is compute the critical numbers of the function in the given interval, determine the height of the func- tion at each critical number, and then figure the height of the function at the two endpoints of the interval.
The greatest of this set of heights is the absolute max; and the least, of course, is the absolute min. Chapter Differentiation and the Shape of Curves 2. Compute the function values the heights at each critical number. Determine the function values at the endpoints of the interval. The largest number in this list, 1. All you have to do is determine the heights at the critical numbers and at the endpoints and then pick the larg- est and smallest numbers from this list.
Second, the absolute max and min in the given interval tell you nothing about how the function behaves outside the interval. Function h, for instance, might rise higher than 1. Its absolute min is zero at 0, 0 of course. But now, say you alter the function slightly by plucking out the point at 0, 0 and leaving an infinitesimal hole there.
Now the function has no absolute minimum. Now consider g x in Figure Find the height of the function at each of its critical numbers. Includes index Getting ready for Calculus II?
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