At the end of this lesson, you, as a successful student, will be able to understand and to use the following functions
At the end of the session, you will be able to
ONE-TO-ONE FUNCTION
Make sure you know the definition of a one-to-one function and how to recognize from a graph if you a one-to-one function as illustrated on pages 466 and 467.
INVERSE FUNCTION
As you review the concept of inverse function at the bottom of page 467 and top of page 468, see if you can describe in your own words how you obtain an inverse function from a give function. Notice the use of the superscript "-1" to symbolize an inverse function.
RELATIONSHIPS BETWEEN DOMAINS, RANGES, AND GRAPHS
As you read Figure 7.2 at the top of page 469, particularly notice the switching between the domains and ranges of the functions f and the inverse functions.
The bottom of page 469 and on page 470 present an excellent description of the relationship between the graphs of functions f and the inverse functions. Notice the line y = x in Figures 7.2, 7.3, 7.4 and 7.5 is the line of symmetry between the graphs. Also the graphs on page 470 further illustrate the relationship between the graphs of the function f and the inverse function.
FIND THE RULE OF THE INVERSE FUNCTION
Examples 2 and 3 on pages 469 and 470 illustrate how to find the rule for the inverse function of a given function. Without looking at the solutions in the textbook, try working the problems on paper yourself. Before moving on, make sure you know the steps to finding the inverse.
You can now go to the Session 3 assignments to continue your assignment.
At the end of this session you will be able to
Define a logarithmic functions.
State the domain and range of a logarithmic function.
Describe the graph of a logarithmic function.
Describe the relationship between exponential and logarithmic functions.
DEFINITION OF LOGARITHMIC FUNCTION
At the bottom of page 31 you need to the know the definition of a logarithmic function so well that you can write it without looking at the definition . Pay particular attention to the restrictions on the base "a".
Immediately under the definition you will see the definition and symbols for natural and common logarithms.
DOMAIN AND RANGE OF LOGARITHMIC FUNCTION
You need to know the domain and range of the logarithmic function given on page 32 at the end of the definition.
GRAPH OF LOGARITHMIC FUNCTION
You should be able to describe the general nature of a the graph of a logarithmic function. In Figure 1.44 on page 33, your textbook author graphed base 2, base 3, base 5, and base 10 logarithmic functions.
RELATIONSHIP BETWEEN EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Exponential and logarithmic functions are inverse functions of each other. Thus, their domains and ranges are interchanged and their graphs are reflected through the line y = x.
You can now go to the Session 3 assignments to continue your assignment.
DEFINITION OF EXPONENTIAL FUNCTION
As you read the bottom of page 31 of Section 1.4 and the bottom of page 486 of Section 7.3 in your textbook, concentrate on learning the definition of the exponential functions and learning the domain and range of the exponential function given at the bottom of page 31..
RULES, LAWS OF EXPONENTS
You should know the "Rules (Laws) of Exponents" on page 488 well enough that you could write them without looking at the textbook.
GRAPH OF EXPONENTIAL FUNCTION
Also, at the bottom of page 32, learn the difference between the exponents for exponential growth, the first graph in Figure 1.43, and exponential decay, the right graph of Figure 1.43. Correspondingly, you want to know generally the appearance of the graphs for exponential growth and for exponential decay as you see in the graph in Figure 1.43 on page 32.
You can now go to the Session 3 assignments to continue your assignment.
At the end of this session you will be able to
DEFINITIONS USING ANGLE IN STANDARD POSITION
In the first paragraph below the box on page 50 of your textbook, the authors defined trigonometric functions with the independent variable being the measure of an angle in standard position on the coordinate plane. See Figure 1.68 on page 50.
You must know this six definitions on page 50 where x and y are coordinates of a point on the terminal ray of the angle and r is the radius between the origin and that point on the terminal ray.
DEFINITIONS USING A UNIT CIRCLE
Frequently, we define the trigonometric functions where the independent variable is the length on the arc of a unit circle. As you study this lesson with the unit circle, look at Figure 1.69 in the lower left corner of page 50.
Just as the angle is measured from the positive horizontal axis, the length of the arc on the unit circle is measured on the arc of the circle going away from the positive horizontal axis. Let us let t by the length on the arc. If the angle at the origin is measured in radians, then the length of the arc t is exactly the same numerical value as the angle measurement in radians. Radian measure is summarized on page 48 of your textbook.
Then the following are the definitions of the trigonometric functions using a unit circle:
- The sin (t) is defined to be the value of the second coordinate of P(x,y).
- The cos (t) is defined to be the value of the first coordinate of P(x,y).
- The tan (t) is defined to be the value of sin (t) / cos (t).
- The cot (t) is defined as the reciprocal of tan (t), that is cot (t) = 1 / tan (t).
- The sec (t) is defined as the reciprocal of cos (t), that is sec (t) = 1 / cos (t).
- The csc (t) is defined as the reciprocal of sin (t), that is csc (t) = 1 / sin (t).
Usually, in calculus and higher levels of mathematics, we use these definitions or, if we do use the definitions on page 50 of an angle in standard position on the coordinate plane, we use the angle measurement in radians, not degrees. Therefore you must know these six definitions of trigonometric functions using a unit circle.
DOMAINS AND RANGES OF THE SIX TRIGONOMETRIC FUNCTIONS
At the bottom of page 52, the domains and ranges of the six trigonometric functions are written underneath the graphs. They are also summarized in the lower right These definitions are also given in the upper left corner of the inside back cover of your textbook.
You need to know the domains, ranges, and periods of the sine, cosine and tangent functions.
SELECTED VALUES OF TRIGONOMETRIC FUNCTIONS
As you study and use calculus, you were experience many applications where you need to know the exact trigonometric values of selected values of the independent variable. By "exact values" we mean values found without the calculator and as illustrated in Table 14 on page 51 of your textbook.
Therefore, you must know these values without using your calculator and without looking at Table 14 on page 51. If you have forgotten them since you studied trigonometry or if you never learned these selected values, then take some time right now to do so.
TRIGONOMETRIC IDENTITIES
The Pythagorean identities following directly by using the Pythagorean Theorem in a unit circle as shown in Figure 1.74 on page 53. The resulting identity for sine and cosine is the identity (1) in the middle of page 53. You need to know that identity as well as you know how to spell your name. We use the Pythagorean identity to convert between the sine and cosine values. The companion Pythagorean identities for the other four trigonometric functions are in the next box on page 53.
The sum identities are the identities in the box (2) at the very bottom of page 539. Again, you need to know these identities for the sine and cosine functions so well that you use them without looking at the textbook.
The double-angle identities in the box (3) at the top of page 54, come directly for the sum identities. In our calculus, we will frequently use these double-angle identities. Therefore, additionally, you need to know these identities so well that you use them without looking at the textbook.
You can now go to the Session 3 assignments to continue your assignment.
INVERSE TRIGONOMETRIC FUNCTION
At the end of this session you will be able to
RELATIONSHIP BETWEEN TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS
Examples 7.18, 7.19, and 7.22 graphs on pages 520 and 521 illustrates well the relationship between trigonometric and inverse trigonometric functions. Careful observation of these graphs show the two graphs are reflected through the line y = x.
RESTRICTING THE DOMAINS AND RANGES
To better understand the inverse trigonometric function values, we will look at the "unit circle" definitions of trigonometric functions or you can look at the definitions of trigonometric functions of angles in standard position as on page 50. While values for trigonometric functions can be for any length of the arc or for any angle, the inverse trigonometric values are restricted to specific quadrants. Why is that? That is so we have a function and, thus, only one possible value when we find inverse trigonometric functions.
- Because the range of the inverse cosine is 0 ≤ y ≤ π (See page 519.), your inverse trigonometric function value must be in quadrant I or II.
- Because the range of the inverse sine is -π /2 ≤ y ≤ π /2 (See page 519.), your inverse trigonometric function value must be in quadrant I or IV.
- Because the range of the inverse tangent is -π /2 < y < π /2 (See page 519.), your inverse trigonometric function value must be in quadrant I or IV.
In conclusion, then we are finding inverse trigonometric values, you can only have one answer because you are evaluating functions. Mathematicians agree that the answers are in the ranges given on page 519 and in the quadrants I have specified above.
You need to know the domains and ranges for the inverse sine, cosine, and tangent functions as described above and as described on page 519. If you learned the domains and ranges of the sine, cosine, and tangent functions at the bottom of page 52 and apply the restrictions directly above here, then you will quickly learn the domains and ranges the inverse sine, cosine, and tangent functions as illustrated on page 519. Learn these domains and ranges before moving on in this lesson.
EVALUATING SELECTED VALUES OF INVERSE TRIGONOMETRIC FUNCTIONS
In Examples 1, 2, and 3 on pages 521 and 523, the textbook illustrates evaluating selected values of inverse trigonometric functions. You need to be able to do all six of these examples without looking at the textbook or using your calculator.
You can now go to the Session 3 assignments to continue your assignment.