CHAPTER 1 FIGURE 22 increasing decreasing FIGURE 23

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1 . EXERCISES. The graph o a unction is given. (a) State the value o. (b) Estimate the value o 2. (c) For what values o is 2? (d) Estimate the values o such that. (e) State the domain and range o. () On what interval is increasing?

2 2. The graphs o and t are given. (a) State the values o 4 and t 3. (b) For what values o is t? (c) Estimate the solution o the equation. (d) On what interval is decreasing? (e) State the domain and range o. () State the domain and range o t. varies over time. What do ou think happened when this person was 3 ears old? Weight (pounds) g Age (ears). The graph shown gives a salesman s distance rom his home as a unction o time on a certain da. Describe in words what the graph indicates about his travels on this da. 3. Figure was recorded b an instrument operated b the Caliornia Department o Mines and Geolog at the Universit Hospital o the Universit o Southern Caliornia in Los Angeles. Use it to estimate the range o the vertical ground acceleration unction at USC during the Northridge earthquake. 4. In this section we discussed eamples o ordinar, everda unctions: Population is a unction o time, postage cost is a unction o weight, water temperature is a unction o time. Give three other eamples o unctions rom everda lie that are described verball. What can ou sa about the domain and range o each o our unctions? I possible, sketch a rough graph o each unction. 5 8 Determine whether the curve is the graph o a unction o. I it is, state the domain and range o the unction The graph shown gives the weight o a certain person as a unction o age. Describe in words how this person s weight. You put some ice cubes in a glass, ill the glass with cold water, and then let the glass sit on a table. Describe how the temperature o the water changes as time passes. Then sketch a rough graph o the temperature o the water as a unction o the elapsed time. 2. Sketch a rough graph o the number o hours o dalight as a unction o the time o ear. 3. Distance rom home (miles) 8 AM NOON PM Time (hours) Sketch a rough graph o the outdoor temperature as a unction o time during a tpical spring da. 4. Sketch a rough graph o the market value o a new car as a unction o time or a period o 2 ears. Assume the car is well maintained. 5. Sketch the graph o the amount o a particular brand o coee sold b a store as a unction o the price o the coee. 6. You place a rozen pie in an oven and bake it or an hour. Then ou take it out and let it cool beore eating it. Describe how the temperature o the pie changes as time passes. Then sketch a rough graph o the temperature o the pie as a unction o time. 7. A homeowner mows the lawn ever Wednesda aternoon. Sketch a rough graph o the height o the grass as a unction o time over the course o a our-week period. 8. An airplane takes o rom an airport and lands an hour later at another airport, 4 miles awa. I t represents the time in minutes since the plane has let the terminal building, let t be

3 the horizontal distance traveled and t be the altitude o the plane. (a) Sketch a possible graph o t. (b) Sketch a possible graph o t. (c) Sketch a possible graph o the ground speed. (d) Sketch a possible graph o the vertical velocit. 9. The number N (in millions) o cellular phone subscribers worldwide is shown in the table. (Midear estimates are given.) t N (a) Use the data to sketch a rough graph o N as a unction o (b) Use our graph to estimate the number o cell-phone subscribers at midear in 995 and Temperature readings T (in F) were recorded ever two hours rom midnight to 2: PM in Dallas on June 2, 2. The time t was measured in hours rom midnight. t T (a) Use the readings to sketch a rough graph o T as a unction o t. (b) Use our graph to estimate the temperature at : AM. 2. I 3 2 2, ind 2, 2, a, a, a, 2 a, 2a, a 2, [ a ] 2, and a h. 22. A spherical balloon with radius r inches has volume V r 4 3 r 3. Find a unction that represents the amount o air required to inlate the balloon rom a radius o r inches to a radius o r inches Evaluate the dierence quotient or the given unction. Simpli our answer , 24. 3, 25., a h a h a a 3 h 3 h t. 3. h s Find the domain and range and sketch the graph o the unction h s Find the domain and sketch the graph o the unction F t t 2 6t t s G i i i 2 i 2 i i i 3 i 3 i Find an epression or the unction whose graph is the given curve The line segment joining the points, 3 and 5, The line segment joining the points 5, and 7, 47. The bottom hal o the parabola The top hal o the circle H t 4 t 2 2 t F 2 t , 27 3 Find the domain o the unction t st s 3 t 3. t u su s4 u 5 55 Find a ormula or the described unction and state its domain. 5. A rectangle has perimeter 2 m. Epress the area o the rectangle as a unction o the length o one o its sides.

4 2 52. A rectangle has area 6 m. Epress the perimeter o the rectangle as a unction o the length o one o its sides. 53. Epress the area o an equilateral triangle as a unction o the length o a side. 54. Epress the surace area o a cube as a unction o its volume An open rectangular bo with volume 2 m has a square base. Epress the surace area o the bo as a unction o the length o a side o the base. 56. A Norman window has the shape o a rectangle surmounted b a semicircle. I the perimeter o the window is 3 t, epress the area A o the window as a unction o the width o the window. (b) How much ta is assessed on an income o $4,? On $26,? (c) Sketch the graph o the total assessed ta T as a unction o the income I. 6. The unctions in Eample and Eercises 58 and 59(a) are called step unctions because their graphs look like stairs. Give two other eamples o step unctions that arise in everda lie Graphs o and t are shown. Decide whether each unction is even, odd, or neither. Eplain our reasoning g g Catherine Karnow 63. (a) I the point 5, 3 is on the graph o an even unction, what other point must also be on the graph? (b) I the point 5, 3 is on the graph o an odd unction, what other point must also be on the graph? 57. A bo with an open top is to be constructed rom a rectangular piece o cardboard with dimensions 2 in. b 2 in. b cutting out equal squares o side at each corner and then olding up the sides as in the igure. Epress the volume V o the bo as a unction o A unction has domain 5, 5 and a portion o its graph is shown. (a) Complete the graph o i it is known that is even. (b) Complete the graph o i it is known that is odd. 2 _ A tai compan charges two dollars or the irst mile (or part o a mile) and 2 cents or each succeeding tenth o a mile (or part). Epress the cost C (in dollars) o a ride as a unction o the distance traveled (in miles) or 2, and sketch the graph o this unction. 59. In a certain countr, income ta is assessed as ollows. There is no ta on income up to $,. An income over $, is taedatarateo%,uptoanincomeo$2,.anincome over $2, is taed at 5%. (a) Sketch the graph o the ta rate R as a unction o the income I Determine whether is even, odd, or neither. I ou have a graphing calculator, use it to check our answer visuall

5 .3 EXERCISES. Suppose the graph o is given. Write equations or the graphs that are obtained rom the graph o as ollows. (a) Shit 3 units upward. (b) Shit 3 units downward. (c) Shit 3 units to the right. (d) Shit 3 units to the let. (e) Relect about the -ais. () Relect about the -ais. (g) Stretch verticall b a actor o 3. (h) Shrink verticall b a actor o Eplain how each graph is obtained rom the graph o. (a) 5 (b) 5 (c) (d) 5 (e) 5 () The graph o is given. Match each equation with its graph and give reasons or our choices. (a) 4 (b) 3 (c) 3 (d) 4 (e) 2 6! (c) 2 (d) The graph o is given. Use it to graph the ollowing unctions. (a) 2 (b) ( 2 ) (c) (d) 6 7 The graph o s3 2 is given. Use transormations to create a unction whose graph is as shown. 3 #.5 =œ 3- $ 3 _6 _ % _3 3 _4 4. The graph o is given. Draw the graphs o the ollowing unctions. (a) 4 (b) _2.5

6 8. (a) How is the graph o 2 sin related to the graph o sin? Use our answer and Figure 6 to sketch the graph o 2sin. (b) How is the graph o s related to the graph o s? Use our answer and Figure 4(a) to sketch the graph o s Graph the unction b hand, not b plotting points, but b starting with the graph o one o the standard unctions given in Section.2, and then appling the appropriate transormations cos 4. 4 sin 3 5. sin The cit o New Orleans is located at latitude 3 N. Use Figure 9 to ind a unction that models the number o hours o dalight at New Orleans as a unction o the time o ear. To check the accurac o our model, use the act that on March 3 the sun rises at 5:5 AM and sets at 6:8 PM in New Orleans. 26. A variable star is one whose brightness alternatel increases and decreases. For the most visible variable star, Delta Cephei, the time between periods o maimum brightness is 5.4 das, the average brightness (or magnitude) o the star is 4., and its brightness varies b.35 magnitude. Find a unction that models the brightness o Delta Cephei as a unction o time. 28. Use the given graph o to sketch the graph o. Which eatures o are the most important in sketching? Eplain how the are used. 6. ( ) s 4 7. s s tan 23. sin (a) How is the graph o related to the graph o? (b) Sketch the graph o sin. (c) Sketch the graph o Find t, t, t, and t and state their domains , t s3, 3 36 Find the unctions (a) t, (b) t, (c), and (d) t t and their domains. 3. 2, 32. 2, 33. 3, 34. s, 35., t 2 36., t sin Find t h. 37., t 2, 38. 2, t 2, 39. s 3, t 2, h tan, t, h s Epress the unction in the orm t. 4. F F sin(s ) 43. F 44. s s3 u t scos t t s 2 t 2 t t cos t s 3 h h u t t h Epress the unction in the orm 47. H H sec 4 (s ) tan t tan t H s Use the table to evaluate each epression. (a) t (b) t (c) (d) t t (e) t 3 () t G t

7 5. Use the given graphs o and t to evaluate each epression, or eplain wh it is undeined. (a) t 2 (b) t (c) t (d) t 6 (e) t t 2 () 4 g Use the given graphs o and t to estimate the value o t or 5, 4, 3,..., 5. Use these estimates to sketch a rough graph o t. g 53. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed o 6 cm s. (a) Epress the radius r o this circle as a unction o the time t (in seconds). (b) I A is the area o this circle as a unction o the radius, ind A r and interpret it. 54. A spherical balloon is being inlated and the radius o the balloon is increasing at a rate o 2 cm s. (a) Epress the radius r o the balloon as a unction o the time t (in seconds). (b) I V is the volume o the balloon as a unction o the radius, ind V r and interpret it. 55. A ship is moving at a speed o 3 km h parallel to a straight shoreline. The ship is 6 km rom shore and it passes a lighthouse at noon. (a) Epress the distance s between the lighthouse and the ship as a unction o d, the distance the ship has traveled since noon; that is, ind so that s d. (b) Epress d as a unction o t, the time elapsed since noon; that is, ind t so that d t t. (c) Find t. What does this unction represent? 56. An airplane is ling at a speed o 35 mi h at an altitude o one mile and passes directl over a radar station at time t. (a) Epress the horizontal distance d (in miles) that the plane has lown as a unction o t. (b) Epress the distance s between the plane and the radar station as a unction o d. (c) Use composition to epress s as a unction o t. 57. The Heaviside unction H is deined b H t i t i t It is used in the stud o electric circuits to represent the sudden surge o electric current, or voltage, when a switch is instantaneousl turned on. (a) Sketch the graph o the Heaviside unction. (b) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t and 2 volts are applied instantaneousl to the circuit. Write a ormula or V t in terms o H t. (c) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t 5 seconds and 24 volts are applied instantaneousl to the circuit. Write a ormula or V t in terms o H t. (Note that starting at t 5 corresponds to a translation.) 58. The Heaviside unction deined in Eercise 57 can also be used to deine the ramp unction cth t, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph o the ramp unction th t. (b) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t and the voltage is graduall increased to 2 volts over a 6-second time interval. Write a ormula or V t in terms o H t or t 6. (c) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t 7 seconds and the voltage is graduall increased to volts over a period o 25 seconds. Write a ormula or V t in terms o H t or t Let and t be linear unctions with equations m b and t m 2 b 2. Is t also a linear unction? I so, what is the slope o its graph? 6. I ou invest dollars at 4% interest compounded annuall, then the amount A o the investment ater one ear is A.4. Find A A, A A A, and A A A A. What do these compositions represent? Find a ormula or the composition o n copies o A. 6. (a) I t 2 and h , ind a unction such that t h. (Think about what operations ou would have to perorm on the ormula or t to end up with the ormula or h.) (b) I 3 5 and h , ind a unction t such that t h. 62. I 4 and h 4, ind a unction t such that t h. 63. (a) Suppose and t are even unctions. What can ou sa about t and t? (b) What i and t are both odd? 64. Suppose is even and t is odd. What can ou sa about t? 65. Suppose t is an even unction and let h t. Is h alwas an even unction? 66. Suppose t is an odd unction and let h t. Is h alwas an odd unction? What i is odd? What i is even?

8 .6 EXERCISES. (a) What is a one-to-one unction? (b) How can ou tell rom the graph o a unction whether it is one-to-one? 2. (a) Suppose is a one-to-one unction with domain A and range B. How is the inverse unction deined? What is the domain o? What is the range o? (b) I ou are given a ormula or, how do ou ind a ormula or? (c) I ou are given the graph o, how do ou ind the graph o? 3 4 A unction is given b a table o values, a graph, a ormula, or a verbal description. Determine whether it is one-to-one t is the height o a ootball t seconds ater kicko. 4. t is our height at age t. 5. I is a one-to-one unction such that 2 9, what is 9? 6. Let 3 2 tan 2, where. (a) Find 3. (b) Find I t 3 e, ind t The graph o is given. (a) Wh is one-to-one? (b) What are the domain and range o? (c) What is the value o 2? (d) Estimate the value o The ormula C 5 9 F 32, where F , epresses the Celsius temperature C as a unction o the Fahrenheit temperature F. Find a ormula or the inverse unction and interpret it. What is the domain o the inverse unction? t 2. t cos 2. In the theor o relativit, the mass o a particle with speed v is m v s v 2 c 2 where m is the rest mass o the particle and c is the speed o light in a vacuum. Find the inverse unction o and eplain its meaning. m

9 2 26 Find a ormula or the inverse o the unction. 2. s e 3 ln 3 ; Find an eplicit ormula or and use it to graph,, and the line on the same screen. To check our work, see whether the graphs o and are relections about the line Use the given graph o to sketch the graph o , e (a) How is the logarithmic unction log a deined? (b) What is the domain o this unction? (c) What is the range o this unction? (d) Sketch the general shape o the graph o the unction log a i a. 32. (a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs o the natural logarithm unction and the natural eponential unction with a common set o aes Find the eact value o each epression. 33. (a) log 5 25 (b) 34. (a) ln e (b) log s 35. (a) log 2 6 log 2 5 log 2 2 (b) log 3 log 3 8 log (a) e 2 ln 5 (b) ln(ln e ) e Epress the given quantit as a single logarithm. 37. ln 5 5 ln ln a b ln a b 2 ln c 39. ln 2 2 ln ln sin log Use Formula to evaluate each logarithm correct to si decimal places. (a) log 2 (b) log e 2e 2 ; 4 42 Use Formula to graph the given unctions on a common screen. How are these graphs related? 4. log.5, ln, log, 42. ln, log, e, ; 44. Compare the unctions. and t ln b graphing both and t in several viewing rectangles. When does the graph o inall surpass the graph o t? CAS CAS 43. Suppose that the graph o log 2 is drawn on a coordinate grid where the unit o measurement is an inch. How man miles to the right o the origin do we have to move beore the height o the curve reaches 3 t? Make a rough sketch o the graph o each unction. Do not use a calculator. Just use the graphs given in Figures 2 and 3 and, i necessar, the transormations o Section (a) log 5 (b) ln 46. (a) ln (b) ln 47 5 Solve each equation or. 47. (a) 2 ln (b) e (a) e (b) ln (a) (b) ln ln 5. (a) ln ln (b) e a Ce b, where a b 5 52 Solve each inequalit or. 5. (a) e (b) ln 52. (a) 2 ln 9 (b) e log Find (a) the domain o and (b) and its domain. 53. s3 e ln 2 ln 55. Graph the unction s 3 2 and eplain wh it is one-to-one. Then use a computer algebra sstem to ind an eplicit epression or. (Your CAS will produce three possible epressions. Eplain wh two o them are irrelevant in this contet.) 56. (a) I t 6 4,, use a computer algebra sstem to ind an epression or t. (b) Use the epression in part (a) to graph t,, and t on the same screen. 57. I a bacteria population starts with bacteria and doubles ever three hours, then the number o bacteria ater t hours is n t 2 t 3. (See Eercise 25 in Section.5.) (a) Find the inverse o this unction and eplain its meaning. (b) When will the population reach 5,?

10 58. When a camera lash goes o, the batteries immediatel begin to recharge the lash s capacitor, which stores electric charge given b t e t a (The maimum charge capacit is and t is measured in seconds.) (a) Find the inverse o this unction and eplain its meaning. (b) How long does it take to recharge the capacitor to 9 o capacit i a 2? Find the eact value o each epression. 59. (a) sin (s3 2) (b) cos 6. (a) tan ( s3 ) (b) sec 2 6. (a) arctan (b) sin ( s2 ) 62. (a) cot ( s3 ) (b) arccos( 2) 63. (a) tan arctan (b) sin sin (a) tan sec 4 (b) 65. rove that cos sin s 2. sin(2 sin ( 3 5)) Simpli the epression. 66. tan sin 68. cos 2 tan ; 69 7 Graph the given unctions on the same screen. How are these graphs related? 69. sin, tan, 2 2 sin tan 7. Find the domain and range o the unction t sin 3 sin tan ; 72. (a) Graph the unction sin sin and eplain the appearance o the graph. (b) Graph the unction t sin sin. How do ou eplain the appearance o this graph? 73. (a) I we shit a curve to the let, what happens to its relection about the line? In view o this geometric principle, ind an epression or the inverse o t c, where is a one-to-one unction. (b) Find an epression or the inverse o c, where c. 67.