The Evolve website assets include practice exams, interactive activities and exercises, the Body Spectrum Online Coloring Book, and more! Ideal for one-semester courses and self-study, this clear, concise text covers disorders, diagnosis, and treatment with a strong focus on anatomy for an integrated discussion of each body system. Small bits of information are presented then immediately tested for comprehension using a combination of programmed and traditional review questions.
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Practical applications such as case reports, operative and diagnostic tests, and laboratory and x-ray reports demonstrate the use of medical terminology in the clinical environment. So, time is the independent variable, and distance fallen is the dependent variable; and you would say that distance is a function of time. Generally, we want to know what happens to the dependent or y-variable as the independent or x-variable goes to the right: Is the y-variable the height of the graph rising or falling and, if so, how steeply, or is the graph level, neither going up nor down?
These two equations are, in every respect, mathematically identi- cal. Students are often puzzled by function notation when they see it the first time.
It does not. If function notation bugs you, my advice is to think of f x as simply the way y is written in some foreign language. The area of a square depends on, and is a function of, the length of its side.
The two functions thus have different domains. When you input 3 for x, the output is 9. Composite functions A composite function is the combination of two functions. For example, the cost of the electrical energy needed to air condition your place depends on how much electricity you use, and usage depends on the outdoor temperature. Chapter 5: Funky Functions and Their Groovy Graphs 49 Because cost depends on usage and usage depends on temperature, cost will depend on temperature.
In function language, cost is a function of usage, usage is a function of temperature, and thus cost is a function of temperature. This last function, a combination of the first two, is a composite function. The machine meta- phor shows what I did here. The g machine turns the 3 into a 7, and then the f machine turns the 7 into a Figure Two function machines. Then you take the output, 7, and calculate f 7 , which equals The f function or f machine takes an input and squares it.
With composite functions, the order matters. Figure The Cartesian for Descartes or x—y coordinate system.
Think of the coordinate system or the screen on your graphing calculator as your window into the world of calculus. Virtually everything in your calculus textbook and in this book involves directly or indirectly the graphs of lines or curves — usually functions — in the x-y coordinate system.
Consider the four graphs in Figure These four curves are functions because they satisfy the vertical line test. No matter where you draw a vertical line on any of the four graphs in Figure , the line touches the curve at only one point. Try it. If, however, a vertical line can be drawn so that it touches a curve two or more times, then the curve is not a function.
The two curves in Figure , for example, are not functions. Figure These two curves are not functions because they fail the vertical line test. They are, however, relations. Definition of relation: A relation is any collection of points on the x-y coordi- nate system. You spend a little time studying some non—function relations in calculus — circles, for instance — but the vast majority of calculus problems involve functions.
Lines in the plane in plain English A line is the simplest function you can graph on the coordinate plane. Lines are important in calculus because you often study lines that are tangent to curves and because when you zoom in far enough on a curve, it looks and behaves like a line. Chapter 5: Funky Functions and Their Groovy Graphs 53 Hitting the slopes The most important thing about the line in Figure — at least for your study of calculus — is its slope or steepness. Notice that whenever x goes 1 to the right, y goes up by 3.
A good way to visualize slope is to draw a stairway under the line see Figure The ratio of rise to run, and thus the slope, always comes out the same, regardless of what size you make the steps. If you make the run equal to 1, however, the slope is the same as the rise because a number divided by 1 equals itself.
This is a good way to think about slope — the slope is the amount that a line goes up or down as it goes 1 to the right. Definitions of positive, negative, zero, and undefined slopes: Lines that go up to the right have a positive slope; lines that go down to the right have a negative slope.
Horizontal lines have a slope of zero, and vertical lines do not have a slope — you say that the slope of a vertical line is undefined. The answer of 3 agrees with the slope you can see in Figure Perpendicular lines have opposite reciprocal slopes. The old-fashioned way is to create a table of values by plugging numbers into x and calculating y.
If you plug 0 into x, y equals 5; plug 1 into x, and y equals 8; plug 2 into x, and y is 11, and so on. Table shows the results. This is a snap with a graphing calculator.
Slope-intercept and point-slope forms You can see that the line in Figure crosses the y-axis at 5 — this point is the y-intercept of the line. All lines, except for vertical lines, can be written in this form. The number tells you where the vertical line crosses the x—axis. Definition of a constant function: A line is the simplest type of function, and a horizontal line called a constant function is the simplest type of line. You can use any point on the line. Consider the line in Figure again.
Notice that both functions are symmetric with respect to the y-axis. In other words, the left and right sides of each graph are mirror images of each other. This makes them even functions. Such an even polynomial function can contain — but need not contain — a constant term like the 3 in the preceding function.
This makes sense because 3 is the same as 3x 0 and zero is an even number. These two functions illustrate odd symmetry. Figure shows the graphs of both these functions on the same x-y coordinate system.
When b is greater than 1, you have exponential growth. All such functions get higher and higher without limit as they go to the right toward positive infinity.
As they go to the left toward negative infinity, they crawl along the x-axis, always getting closer to the axis, but never touching it. You use these and related functions for figuring things like investments, inflation, and growing population. When b is between 0 and 1, you have an exponential decay function. The graphs of such functions are like exponential growth functions in reverse. Exponential decay functions also cross the y-axis at 0, 1 , but they go up to the left forever, and crawl along the x-axis to the right.
These functions model things that shrink over time, such as the radioactive decay of uranium. Logarithmic functions A logarithmic function is simply an exponential function with the x- and y-axes switched. If you want a refresher on logs, see Chapter 4. Both exponential and logarithmic functions are monotonic. A monotonic func- tion either goes up over its entire domain called an increasing function or goes down over its whole domain a decreasing function.
This makes them inverses of each other, which brings us to the next topic. It works the same way if you start with f x. It does not equal 1 , which is the reciprocal of f x f x. So why is the exact same symbol used for two different things? Beats me. One consequence of this symmetry is that if a point like 2, 4 is on one of the functions, the point 4, 2 will be on the other. I do the horizontal transformations first. Horizontal transformations Horizontal changes are made by adding a number to or subtracting a number from the input variable x or by multiplying x by some number.
The original function and both transformations are shown in Figure So every point on the new function is half of its original distance from the y-axis. The y-coordinate of every point stays the same; the x-coordinate is cut in half. Multiplying x by 16 16 a number less than 1 has the opposite effect. 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.
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