MATH 5a Spring 2018 READING ASSIGNMENTS FOR CHAPTER 2

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1 MATH 5a Spring 2018 READING ASSIGNMENTS FOR CHAPTER 2 Note: Tere will be a very sort online reading quiz (WebWork) on eac reading assignment due one our before class on its due date. Due dates can be found on te calendar on LATTE. Reading Assignments As usual, ere we tell you wic pages of eac section you must read and wic you can skip. We also give comments to elp you wit te reading and ints to elp you wit te omework. I. Section 2.1. We will do everyting in Section 2.1. IMPORTANT: You ave probably studied functions before. However, it s possible tat you never got a very clear idea of wat a function actually is. You may ave tougt a function was simply a formula tat you could plug numbers into or tat you could grap. But te basic idea beind a function is muc deeper. Before you can study functions seriously you need to understand tis basic idea: a function is a relationsip between two quantities, were one quantity depends on te oter. For example, you know from experience tat te cost of of mailing a first-class letter depends on its weigt. Here we ave two quantities: cost and weigt. One quantity depends on te oter: te cost depends on te weigt. Terefore te cost is a function of te weigt. Here s anoter example. Suppose tat a population of bacteria is growing in a test tube and tat initially tere are 1000 bacteria in te tube. Suppose tat you know tat te population doubles every 5 ours. Again we ave two quantities: te size of te population and te time tat as passed since te experiment began, and te size of te population depends on ow muc time as passed. So te size of te population is a function of te time tat as passed since te experiment began. We ll see later tat te formula for tis function is f(t) = t. Here s one more example. Suppose tat you ave a circular oil spill. (In real life, it s unlikely tat an oil spill would be exactly circular, but often scientists studying suc penomena model it wit a simplified situation and ten progress to more complicated and realistic situations.) Clearly te area of te oil spill depends on te radius of te circle. So te area is a function of te radius. In tis case, it s easy to write down a formula for tis function. If we call te radius of te oil spill r and we call te area a, ten te formula is a = πr 2, wic we can also write as f(r) = πr 2 or a(r) = πr 2. As you read troug Capter 2, keep in mind tis basic idea of a function as a relationsip between two quantities. Here are some comments on te reading and ints for te omework: Make sure tat you know and understand te definition of a function tat s given in te middle of page 143. You will need to use it. We may also ask you to state it on quizzes or exams. Make sure tat you know all te vocabulary for functions given on page 143.

2 Since tis is a mat course, most of te functions we ll work wit can be expressed as matematical formulae. Examples 1 5 on pages in Section 2.1 are examples of suc functions. Notice tat in eac example you are asked to evaluate te function at a value of x; tis just means finding te f(x)-value tat corresponds to tat particular x. For example, consider te function f(x) = x If we evaluate te function at x = 4, we get f(4) = (4) 2 +1 = 17. So te function matces te x-value 4 wit te f(x)-value 17. Similarly, if we evaluate te same function at x = π, we get f(π) = (π) 2 +1 = π So te function matces te x-value π wit te f(x)-value π Finally, remember tat te f(x)-value is usually called te y value. It is also called te output of te function. Te x-value is called te input of te function. Example 3 on page 145 gives an example of a piecewise defined function. Suc functions will be important in tis course, so study tis example carefully. Make sure you understand ow to evaluate expressions like f(2 + ) or f(a + ). (See Example 4c on page 145). If you ave trouble doing tis, ten start by replacing te x in te formula for te function wit te word input. For example, f(x) = x can be rewritten as f(input) = (input) Now suppose tat you need to find f(2+). Te input ere is 2+. Since f( input) = (input) 2 + 1, we get f (2 + ) = ( 2 + ) = f(a + ) f(a) Example 4d on page 145 computes and simplifies te expression. We will empasize tis type of computation, since it is crucial in calculus. Note tat f(a + ) f(a) f(a) + f() f(a) is not te same as. Here s an example tat sows wy not: Solution: f(a + ) f(a) Example. Let f(x) = x 2. Compute = (a + )2 (a) 2 f(a + ) f(a) = a2 + 2a + 2 a 2 =. 2a + 2 = 2a +. Important: Tis is NOT te same as f(a) + f() f(a) = (a)2 + ( 2 ) (a) 2 = 2 = THIS IS WRONG! A function like f(x) = 7 is called a constant function. Tis means tat for any input x, te output is te constant 7. For example, f(3) = 7, f(0) = 7, f( 14) = 7, f(100) = 7, f( 1 ) = 7, f(a) = 7, f(a + 1) = 7, and so on. In oter words, f(anyting) = 7. 5 Hint for omework problems #21 and 23: In problem #21, you must find g(a 1). Remember tat te entire expression a 1 is te input. Similarly, in problem #23, you must find f(x + 1); in tis case te entire expression x + 1 is te input.

3 Hint for omework problem #31: You are asked to find two different expressions: f(x + 2) and f(x) + 2. Tey are not te same! Hint for omework problems #36 38: tese problems ask you (among oter tings) to f(a + ) f(a) find and simplify te expression. See Example 4d on page 145 for an f(a + ) f(a) f(a) + f() f(a) example. Remember tat. In problem #37, note tat f(x) = 5 is a constant function, so f(anyting) = 5. Hint for omework problem #56: you must solve a quadratic inequality. II. Section 2.2. We will do everyting in Section 2.2 except Example 2 (using a graping calculator), Example 3 (families of power functions) and Example 6 (te greatest integer function). Here are some comments on te reading and ints for te omework: If te definition of te grap of a function (top of page 153) and te picture next to it confuse you, ten just tink of a familiar example like f(x) = x 2. See Figure 3a on page 154, were te usual points are labeled, namely (0, 0), (1, 1), ( 1, 1), etc. We could also label a general point as (x, x 2 ). Since x 2 is just anoter name for te output f(x), so we could also label tis general point as (x, f(x)). Keeping tis in mind, reread te definition on page 153. You must know ow to accurately grap all te following functions: f(x) = x 2 ; f(x) = x 3 ; f(x) = x ; f(x) = x; f(x) = 1 x ; f(x) = 1 x 2. Tese functions are all sown in te text, as well as in a andout called called Te Library of Basic Functions, wic is posted on LATTE. Te best way to grap tese functions accurately is to remember te general sape of eac, along wit a few key reference points. For example, to grap f(x) = x 3 you need to remember te general S-like sape of te curve and to remember some reference points like (0, 0), (1, 1), (2, 8), ( 1, 1) and ( 2, 8). You also need to know ow to grap two functions wose graps we saw in Section 1.10: f(x) = c (constant function), and f(x) = mx + b (linear function). You do not need to know ow to grap g(x) = x 4, f(x) = x 5, f(x) = 4 x, or f(x) = 5 x. Te text discusses te vertical line test on page 157. Make sure you can explain wy tis test works.

4 III. Section 2.3: We will cover te material on page 163 troug te middle of page 165. We will not cover te material on te local maximum and minimum values of a function. Here are some comments on te reading and ints for te omework. Te section begins wit a discussion of ow to read function values from a grap. It goes on to discuss ow to find te domain and range of a function from its grap. Study Figure 2 at te bottom of page 163; it will elp you understand ow to find te domain and range of a function if you are just given te grap of te function. Te text ten defines wat it means for a function to be increasing on an interval or decreasing on an interval. Tis topic is essential in calculus. Here s an example: Te grap of te function f(x) wit domain (, + ) is sown below. Ten f(x) is increasing on te intervals (, 3) and (0, 5) f(x) is decreasing on te intervals ( 3, 0) and (5, + ) Notice two tings: 1. Te intervals are intervals on te x-axis, not te y-axis. We always use intervals on te x-axis to describe were a function is increasing or decreasing. 2. Te intervals are open intervals. Unfortunately, our textbook uses closed intervals. Calculus textbooks use open intervals, so it is a good idea for you to do te same. However, you may use eiter open or closed intervals, as long as you are consistent. IV. Section 2.4. We will cover everyting in Section 2.4. Here are some comments on te reading and ints for te omework. Te formula for te average rate of cange of a function f(x) between x = a and x = b is given in te blue box in te middle of page 173. Tis formula is quite straigtforward. Make sure tat you study te grap tat accompanies te definition in te blue box. It is of central importance in calculus. Note tat in Example 2b on page 174 you ave to compute. Remember tat we did tis kind of computation in Section 2.1. d(a + ) d(a) In Example 3c on pages te average rate of cange is negative. Tis is because te temperature is falling. In general, if a function measures a quantity tat is decreasing over te interval [a, b], ten te average rate of cange on tat interval will be negative.

5 f(5) f(1) Hint for omework problem #6: find by reading off te appropriate 5 1 coordinates from te grap tat s given. Te same idea works for problem #8. f(a + ) f(a) Hint for omework problem #19: you re finding for f(t) = 2 t. Tis is similar to work you did in Section 2.1. (See Example 2b on page 174.) V. Section 2.5. We will do everyting in Section 2.5 except orizontal srinking/stretcing of graps (middle of page 184 middle of page 185) and even and odd functions (middle of page 185 te end of te section). Here are some comments on te reading and ints for te omework: Tis section is quite straigtforward. Te main ting to understand about transformations is te following: 1. Vertical sifting. Wen you are graping, for example, a function of te form y = f(x) + 3, you are actually adding 3 to te y-coordinate of eac point on te grap of f(x). Te result is to sift te grap of f(x) up by 3 units. On te oter and, te function y = f(x) 3 sifts te grap of f(x) down by 3 units. 2. Horizontal sifting. Wen you grap a function of te form y = f(x + 4), you are actually subtracting 4 from te x-coordinate of eac point on te grap of f(x). (Tis is counter-intuitive.) Te result is to sift te grap of f(x) to te left by 4 units. On te oter and, te function y = f(x 4) sifts te grap of f(x) to te rigt by 4 units. 3. Vertical stretcing/srinking. Wen you grap a function of te form y = 2f(x), you are actually multiplying te y-coordinate of eac point on te grap of f(x) by 2. Te result is to stretc te grap of f(x) vertically by a factor of 2. On te oter and, te function y = 1 f(x) srinks te grap of f(x) vertically Reflecting graps. Wen you grap a function of te form y = f(x), you are actually multiplying te y-coordinate of eac point on te grap of f(x) by 1. Te result is to reflect te grap across te x-axis. Similarly, wen you grap y = f( x), you are multiplying te x-coordinate of eac point on te grap of f(x) by 1. Te result is to reflect te grap across te y-axis. Hint for omework problem #64: Here tere is no formula for te function g(x) so you can t grap te transformations by making use of a formula. However, you can easily find te x- and y-coordinates of a number of points on te grap of g(x). Ten you can apply te appropriate transformation to te x- or y-coordinate of eac of tese points. For example, in part (f), you would multiply te y-coordinate of eac point by 2.

6 IV. Section 2.6: We will do everyting in tis section except grapical addition (Example 2 on page 192) and finding te domain of compositions of functions (te domain parts of Example 3a on page 193 and Example 4 on page 194). Here are some comments on te reading and ints for te omework: In general, te material in tis section is quite straigtforward. Remember not to worry about domains of compositions of functions as you read te examples in te text. A comment on te notation for compositions of functions: you may find it easier to rewrite (f g)(x) as f(g(x)) and to rewrite (g f)(x) as g(f(x)). But you do need to know wat te symbol means. You will notice tat tere are a lot of omework problems involving decomposition of functions. Tis is someting tat te book doesn t pay sufficient attention to; it only does one example on te topic (Example 6). But we will empasize decomposition of functions in class, since it is essential to know for calculus. Hint for omework problem #27: Find g(2) from te grap. Ten use te result as te input for f(x). Te same kind of idea will work for # VII. Section 2.7: We will do everyting in tis section. Here are some comments on te reading and ints for te omework: Te basic idea of a one-to-one function, as sown in Figure 1 on page 199, sould make sense to you. Te algebraic definition (given in te blue box on page 199) may seem more difficult. Don t worry: in general we won t ask you to work wit tat definition. Instead, we ll use te orizontal line test (also given on page 199). So, for instance, in Example 3 on page 200, we would expect you to recognize tat te grap of f(x) is a non-orizontal line and terefore passes te orizontal line test, proving tat f(x) is one-to-one. You would not ave not use te algebraic definition of one-to-one, altoug of course you are welcome to. It may take a little time to absorb te definition of te inverse of a function given at te top of page 201. Read it several times, making sure you go troug te end of Example 4. It is extremely important to realize tat f 1 1 (x) does NOT mean f(x). (See te comment in te margin of page 227.) Instead, f 1 (x) means te inverse function of f(x). Te Inverse Function Property given in te blue box on page 201 is te key property of inverse functions. Wenever we ask you to ceck tat two functions are inverses (unless te functions are given by graps) you can use tis property. See Example 5 on page 202. Te algoritm for finding te formula for te inverse of a function is given in te middle of page 202. It is clear and straigtforward. Hint for omework problems #12 14, 20: you can do tese problems grapically, using te orizontal line test.

7 Hint for omework problem #43: te algebra needed in tis kind of problem is a little tricky. Start as follows: y = 1 x + 2 y(x + 2) = 1 yx + 2y = 1. Ten solve for x. VIII. Modeling wit Functions. Te last topic in Capter 3 is called Modeling wit Functions, and it covers applications of functions (word problems). We will cover tis topic using a a andout instead of te text. So your reading assignment for tis section is te andout called Introduction to Modeling wit Functions. Tis andout is posted on LATTE, in bot te reading assignments section and te andouts section.