Calculus ii for dummies free pdf

The following formulas show how to format solution sets in interval notation. Never fear, the following formulas show you how to deal with absolute values in pre-calculus. Of course you use trigonometry, commonly called trig, in pre-calculus. And you use trig identities as constants throughout an equation to help you solve problems.

With this comprehensive study guide, you'll gain the skills and confidence that make all the difference. Calculus For Dummies, 2nd Edition provides a roadmap for success, and the backup you need to get there. Plus, an online component provides you with a collection of calculus problems presented in multiple-choice format to further help you test your skills as you go. Gives you a chance to practice and reinforce the skills you learn in your calculus course Helps you refine your understanding of calculus Practice problems with answer explanations that detail every step of every problem The practice problems in Calculus Practice Problems For Dummies range in areas of difficulty and style, providing you with the practice help you need to score high at exam time.

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The always-true, never-changing trig identities are grouped by subject in the following lists:. Get ahead in pre-calculus Pre-calculus courses have become increasingly popular with 35 percent of students in the U.

Often, completion of such a course is a prerequisite for calculus and other upper level mathematics courses. Pre-Calculus For Dummies is an invaluable resource for students enrolled in pre-calculus courses. By presenting the essential topics in a clear and concise manner, the book helps students improve their understanding of pre-calculus and become prepared for upper level math courses.

Grab the coordinate system on the left and right and stretch it by a factor of 4, pulling everything away from the y-axis, but keeping the y-axis in the center. Check these transformations out on your graphing calculator. The last horizontal transformation is a reflection over the y-axis.

Vertical transformations To transform a function vertically, you add a number to or subtract a number from the entire function or multiply the whole function by a number. Now, all vertical transforma- tions are made by placing a number somewhere on the right side of the equation outside the parentheses. Unlike horizontal transformations, vertical transformations work the way you expect: Adding makes the function go up, subtracting makes it go down, multiplying by a number greater than 1 stretches the function, and multiplying by a number less than 1 shrinks the function.

Look at these transformations on your graphing calculator. As you saw in the previous section, horizontal transformations change only the x-coordinates of points, leaving the y-coordinates unchanged. Conversely, vertical transformations change only the y-coordinates of points, leaving the x-coordinates unchanged.

The three main trig functions sine, cosine, and tangent and their reciprocals cosecant, secant, and cotan- gent all tell you something about the lengths of the sides of a right triangle that contains a given acute angle — like angle x in Figure The longest side of this right triangle or any right triangle , the diagonal side, is called the hypotenuse.

Figure Sitting around the campfire, studying a right triangle. SohCahToa uses the initial letters of sine, cosine, and tangent, and the initial letters of hypotenuse, opposite, and adjacent to help you remember the following definitions. To remember how to spell SohCahToa, note its pronunciation and the fact that it contains three groups of three letters each.

Look at Figure again. With the unit circle, however, you can find trig values for any size angle. The unit circle has a radius of one unit and is set in an x-y coordi- nate system with its center at the origin. Figure The so-called unit circle. Angles in the unit circle Measuring angles: To measure an angle in the unit circle, start at the positive x-axis and go counterclockwise to the terminal side of the angle.

Measuring angles with radians You know all about degrees. You can also use radians. Degrees and radians are just two different ways to measure angles, like inches and centimeters are two ways to measure length.

Definition of radian: The radian measure of an angle is the length of the arc along the circumference of the unit circle cut off by the angle. This is true not only of unit circles, but of circles of any size. Radians are preferred over degrees. In this or any other calculus book, some problems use degrees and others use radians, but radians are the preferred unit. Honey, I shrunk the hypotenuse Look at the unit circle in Figure again. Each of its sides is half as long.

This shows you, by the way, that shrinking a right triangle down or blowing it up has no effect on the trigonometric values for the angles in the triangle. Putting it all together Look at Figure Figure Quadrant I of the unit circle with three angles and their coordinates. So, the point must be , not the other way around.

Now for the whole enchilada. Because of the symmetry in the four quadrants, the three points in quadrant I in Figure have counterparts in the other three quadrants, giving you 12 known points. Figure The unit circle with 16 angles and their coordinates.

Chapter 6: The Trig Tango 71 These 16 pairs of coordinates automatically give you the cosine and sine of the 16 angles. Finally, you can find the cosecant, secant, and cotangent of the 16 angles because these trig functions are just the recip- rocals of sine, cosine, and tangent. Same caution: whenever sine, cosine, or tangent equals zero, the reciprocal function will be undefined.

Learn the unit circle. Knowing the trig values from the unit circle is quite useful in calculus. So quiz yourself. Then picture how these triangles fit into the four quadrants of the unit circle. Use the symmetry of the quadrants as an aid. With some practice, you can get pretty quick at figuring out the values for the six trig functions of all 16 angles. Try to do this without looking at something like Figure And quiz yourself with radians as well as with degrees. That would bring your total to trig facts!

These letters now tell you whether the various trig functions have positive or negative values in the different quadrants. The A in quadrant I tells you that All six trig functions have positive values in quadrant I. The S in quadrant II tells you that Sine and its reciprocal, cosecant are positive in quadrant II and that all other trig functions are negative there.

The T in quad- rant III tells you that Tangent and its reciprocal, cotangent are positive in quadrant III and that the other functions are negative there. Finally, the C in quadrant IV tells you that Cosine and its reciprocal, secant are positive there and that the other functions are negative. Graphing Sine, Cosine, and Tangent Figure shows the graphs of sine, cosine, and tangent, which you can, of course, produce on a graphing calculator.

Definitions of periodic and period: Sine, cosine, and tangent — and their reciprocals, cosecant, secant, and cotangent — are periodic functions, which means that their graphs contain a basic shape that repeats over and over indefinitely to the left and the right. The period of such a function is the length of one of its cycles.

If you know the unit circle, you can easily reproduce these three graphs by hand. If you start with these five points, you can sketch one cycle. The cycle then repeats to the left and right. You can use the unit circle in the same manner to sketch the cosine function. Definition of vertical asymptote: A vertical asymptote is an imaginary line that a curve gets closer and closer to but never touches as the curve goes up toward infinity or down toward negative infinity.

In Chapters 7 and 8, you see more vertical asymptotes and also some horizontal asymptotes. Inverse Trig Functions An inverse trig function, like any inverse function, reverses what the original function does. It works the 2 same for the other trig functions. The negative 1 superscript in the sine inverse function is not a negative 1 power, despite the fact that it looks just like it. Pretty weird that the same symbol is used to mean two different things.

Go figure. Consider sine inverse, for example. See the definition of the vertical line test in Chapter 5. To make sine inverse a function, you have to take a small piece of the vertical wave that does pass the vertical line test. The same thing goes for the other inverse trig functions. Tell the truth now — most people remember trig identities about as well as they remember nineteenth century vice-presidents.

They come in handy in calculus though, so a list of other useful ones is in the online Cheat Sheet at www. Plus the plain English meaning: Not lifting your pencil off the paper. The formal definition of a derivative involves a limit as does the definition of a definite integral. But understanding the mathematics of limits is nonetheless important because it forms the foundation upon which the vast architecture of calculus is built Okay, so I got a bit carried away.

In this chapter, I lay the groundwork for differentiation and integration by exploring limits and the closely related topic, continuity. Informal definition of limit the formal definition is in a few pages : The limit of a function if it exists for some x-value c, is the height the function gets closer and closer to as x gets closer and closer to c from the left and the right.

Note: This definition does not apply to limits where x approaches infinity or negative infinity. More about those limits later in the chapter and in Chapter 8. Got it? Let me say it another way. A function has a limit for a given x-value c if the function zeros in on some height as x gets closer and closer to the given value c from the left and the right.

Did that help? The arrow- number gives you a horizontal location in the x direction. Now, look at Table Figure The graphs of the func- tions of f, g, and h. We need to use limits in calculus because of discontinuous functions like g and h that have holes. Function g in the middle of Figure is identical to f except for the hole at 2, 7 and the point at 2, 5.

Actually, this function, g x , would never come up in an ordinary calculus problem — I only use it to illustrate how limits work. Keep reading. I have a bit more groundwork to lay before you see why I include it.

The important functions for calculus are the functions like h on the right in Figure , which come up frequently in the study of derivatives. This third function is identical to f x except that the point 2, 7 has been plucked out, leaving a hole at 2, 7 and no other point where x equals 2.

Imagine what the table of input and output values would look like for g x and h x. Can you see that the values would be identical to the values in Table for f x? For both g and h, as x gets closer and closer to 2 from the left and the right, y gets closer and closer to a height of 7. For all three functions, the limit as x approaches 2 is 7.

This brings us to a critical point: When determining the limit of a function as x approaches, say, 2, the value of f 2 — or even whether f 2 exists at all — is totally irrelevant. In a limit problem, x gets closer and closer to the arrow-number c, but technically never gets there, and what happens to the function when x equals the arrow-number c has no effect on the answer to the limit problem though for continuous functions like f x the function value equals the limit answer and it can thus be used to compute the limit answer.

Sidling up to one-sided limits One-sided limits work like regular, two-sided limits except that x approaches the arrow-number c from just the left or just the right. Figure p x : An illustration of two one- sided limits. However, both one-sided limits do exist. As x approaches 3 from the left, p x zeros in on a height of 6, and when x approaches 3 from the right, p x zeros in on a height of 2. As with regular limits, the value of p 3 has no effect on the answer to either of these one-sided limit problems.

And sometimes, like with p x , a piece does not connect with the adjacent piece — this results in a discontinuity. Here goes: Formal definition of limit: Let f be a function and let c be a real number. I think this is why calc texts use the 3-part definition. When we say a limit exists, it means that the limit equals a finite number.

Some limits equal infinity or negative infinity, but you nevertheless say that they do not exist. That may seem strange, but take my word for it. More about infinite limits in the next section. Remember asymptotes? Consider the limit of the function in Figure as x approaches 3. As x approaches 3 from the left, f x goes up to infinity, and as x approaches 3 from the right, f x goes down to negative infinity.

But x can also approach infinity or negative infinity. Limits at infinity exist when a function has a horizontal asymptote. Going right, the function stays below the asymptote and gradually rises up toward it. The following problem, which eventually turns out to be a limit problem, brings you to the threshold of real calculus. Say you and your calculus-loving cat are hanging out one day and you decide to drop a ball out of your second-story window.

After 1. Table Average Speeds from 1 Second to t Seconds As t gets closer and closer to 1 second, the average speeds appear to get closer and closer to 32 feet per second. It gives you the average speed between 1 second and t seconds: In the line immediately above, recall that t cannot equal 1 because that would result in a zero in the denominator of the original equation.

Figure shows the graph of this function. Chapter 7: Limits and Continuity 85 Figure f t is the average speed between 1 second and t seconds. And why did you get 0? Definition of instantaneous speed: Instantaneous speed is defined as the limit of the average speed as the elapsed time approaches zero. Linking Limits and Continuity Before I expand on the material on limits from the earlier sections of this chapter, I want to introduce a related idea — continuity.

This is such a simple concept. A continuous function is simply a function with no gaps — a function that you can draw without taking your pencil off the paper. Consider the four functions in Figure Whether or not a function is continuous is almost always obvious. Well, not quite. The two functions with gaps are not continuous everywhere, but because you can draw sections of them without taking your pencil off the paper, you can say that parts of those functions are continuous. Often, the important issue is whether a function is continuous at a particular x-value.

Continuity of polynomial functions: All polynomial functions are continuous everywhere. Continuity of rational functions: All rational functions — a rational function is the quotient of two polynomial functions — are continuous over their entire domains.

They are discontinuous at x-values not in their domains — that is, x-values where the denominator is zero. Consider whether each function is continuous there and whether a limit exists at that x-value.

So there you have it. If a function is continuous at an x-value, there must be a regular, two-sided limit for that x-value. Keep reading for the exception. When you come right down to it, the exception is more important than the rule. Consider the two functions in Figure In each case, the limit equals the height of the hole. The hole exception: The only way a function can have a regular, two-sided limit where it is not continuous is where the discontinuity is an infinitesimal hole in the function.

This bears repeating, even an icon: The limit at a hole: The limit at a hole is the height of the hole. This gave me zero distance. Function holes 0 often come about from the impossibility of dividing zero by zero. The derivative-hole connection: A derivative always involves the unde- fined fraction 0 and always involves the limit of a function with a hole. Chapter 7: Limits and Continuity 89 Sorting out the mathematical mumbo jumbo of continuity All you need to know to fully understand the idea of continuity is that a func- tion is continuous at some particular x-value if there is no gap there.

You must remember, however, that condition 3 is not satisfied when the left and right sides of the equation are both undefined or nonexistent. It may seem contrived or silly, but with mnemonic devices, con- trived and silly work. First, note that the word limit has five letters and that there are five 3s in this mnemonic.

Remembering that it has three parts helps you remember the parts — trust me. Note that the three types of discontinuity hole, infinite, and jump begin with three consecutive letters of the alphabet. Hey, was this book worth the price or what?

Did you notice that another way this mnemonic works is that it gives you 3 cases where a limit fails to exist, 3 cases where continu- ity fails to exist, and 3 cases where a derivative fails to exist? This chapter gets down to the nitty-gritty and presents several techniques for calculating the answers to limits problems.

Easy Limits A few limit problems are very easy. Okay, so are you ready? Limits to memorize You should memorize the following limits. If you fail to memorize the limits in the last three bullets, you could waste a lot of time trying to figure them out. The limit is simply the func- tion value. Beware of discontinuities. In that case, if you get a number after plugging in, that number is not the limit; the limit might equal some other number or it might not exist.

See Chapter 7 for a description of piecewise functions. What happens when plugging in gives you a non-zero number over zero? This is the main focus of this section. These are the interesting limit problems, the ones that likely have infinitesimal holes, and the ones that are important for differ- ential calculus — you see more of them in Chapter 9. Note on calculators and other technology.

With every passing year, there are more and more powerful calculators and more and more resources on the Internet that can do calculus for you. A calculator like the TI-Nspire or any other calculator with CAS — Computer Algebra System can actually do that limit problem and all sorts of much more difficult calculus problems and give you the exact answer.

The same is true of websites like Wolfram Alpha www. Many do not allow the use of CAS calculators and com- parable technologies because they basically do all the calculus work for you. Method one The first calculator method is to test the limit function with two numbers: one slightly less than the arrow-number and one slightly more than it.

The result, 9. Since the result, The answer is 10 almost certainly. Method two 2 The second calculator method is to produce a table of values. Hit the Table button to produce the table. Now scroll up until you can see a couple numbers less than 5, and you should see a table of values something like the one in Table These calculator techniques are useful for a number of reasons. Your calculator can give you the answers to limit problems that are impossible to do alge- braically.

Also, for problems that you do solve on paper, you can use your calculator to check your answers. This gives you a numerical grasp on the problem, which enhances your algebraic understand- ing of it. If you then look at the graph of the function on your calculator, you have a third, graphical or visual way of thinking about the problem. Many calculus problems can be done algebraically, graphically, and numerically. When possible, use two or three of the approaches.

Each approach gives you a different perspective on a problem and enhances your grasp of the relevant concepts. Gnarly functions may stump your calculator. By the way, even when the non-CAS-calculator methods work, these calcu- lators can do some quirky things from time to time.

This can result in answers that get further from the limit answer, even as you input numbers closer and closer to the arrow-number. Try plugging 5 into x — you should always try substitution first. You get 0 — no good, on to plan B. Now substitution will work. And note that the limit as x approaches 5 is 10, which is the height of the hole at 5, Conjugate multiplication — no, this has nothing to do with procreation Try this method for fraction functions that contain square roots.

Try this one: Evaluate lim. Try substitution. Plug in 4: that gives you 0 — time for plan B. Definition of conjugate: The conjugate of a two-term expression is just the same expression with subtraction switched to addition or vice versa.

The product of conjugates always equals the first term squared minus the second term squared. Now do the rationalizing. Now substitution works. Plug in 0: That gives you 0 — no good. The best way to understand the sandwich or squeeze method is by looking at a graph. Look at functions f, g, and h in Figure g is sandwiched between f and h.

The limit of both f and h as x approaches 2 is 3. So, 3 has to be the limit of g as well. Figure The sandwich method for solv- ing a limit.

Functions f and h are the bread, and g is the salami. Plug 0 into x. Try the algebraic methods or any other tricks you have up your sleeve. Knock yourself out. Plan C. Try your calculator.

It definitely looks like the limit of g is zero as x approaches zero from the left and the right. Table gives some of the values from the calculator table.

Table Table of Values for x g x 0 Error 0. Get it? To do this, make a limit sand- wich. Fooled you — bet you thought Step 3 was the last step.

Because the range of the sine function is from negative 1 to positive 1, whenever you multiply a number by the sine of anything, the result either stays the same distance from zero or gets closer to zero. Thus, will never get above x or below. Figure shows that they do. The function is thus continuous everywhere. Figures and and discussed in the sec- tion about making a limit sandwich. If we now alter it or connect to 0, 0 from the left or the right? The function is now drive through the origin, are you on one of the continuous everywhere; in other words, it has up legs of the road or one of the down legs?

But at 0, 0 , it seems to contradict Neither seems possible because no matter how the basic idea of continuity that says you can close you are to the origin, you have an infinite trace the function without taking your pencil number of legs and an infinite number of turns off the paper.

There is no last turn before you reach 0, 0. Now, keeping the line vertical, slowly between you and 0, 0 is infinitely long! It winds up and down with such increas- you pass over 0, 0. There are no gaps in the ing frequency as you get closer and closer to function, so at every instance, the vertical line 0, 0 that the length of your drive is actually infi- crosses the function somewhere.

On this long and function. How can winding road. And because you reconcile all this? I wish I knew. This is confirmed by con- sidering what happens when you plug bigger and bigger numbers into 1 : x the outputs get smaller and smaller and approach zero.

Determining the limit of a function as x approaches infinity or negative infinity is the same as finding the height of the horizontal asymptote.

That quo- tient gives you the answer to the limit problem and the height of the 3 asymptote. A horizontal asymptote occurs at this same value. If you have any doubts that the limit equals 0. All you see is a column of 0. Try substitution — always a good idea. No good. On to plan B. Now do the conjugate multiplication.

Remember those problems from algebra? One train leaves the station at 3 p.