9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to find derivatives of functions To use derivatives to find slopes of tangents to curves verage and Instantaneous Rates of Cange: Te Derivative ] pplication Preview In Capter 1, Linear Equations and Functions, we studied linear revenue functions and defined te marginal revenue for a product as te rate of cange of te revenue function. For linear revenue functions, tis rate is also te slope of te line tat is te grap of te revenue function. In tis section, we will define marginal revenue as te rate of cange of te revenue function, even wen te revenue function is not linear. Tus, if an oil compan s revenue (in tousands of dollars) is given b R 100, 0 were is te number of tousands of barrels of oil sold per da, we can find and interpret te marginal revenue wen 0,000 barrels are sold (see Eample 4). We will discuss te relationsip between te marginal revenue at a given point and te slope of te line tangent to te revenue function at tat point. We will see ow te derivative of te revenue function can be used to find bot te slope of tis tangent line and te marginal revenue. verage Rates of Cange For linear functions, we ave seen tat te slope of te line measures te average rate of cange of te function and can be found from an two points on te line. However, for a function tat is not linear, te slope between different pairs of points no longer alwas gives te same number, but it can be interpreted as an average rate of cange. We use tis connection between average rates of cange and slopes for linear functions to define te average rate of cange for an function. verage Rate of Cange Te average rate of cange of a function f() from a to b is defined b f(b) f(a) verage rate of cange b a = f () (b, f (b)) Te figure sows tat tis average rate is te same as te slope of te segment joining te points (a, f(a)) and (b, f(b)). (a, f (a)) m = f (b) f (a) b a a b EXMPLE 1 Total Cost Suppose a compan s total cost in dollars to produce units of its product is given b C() 0.01 5 1500 Find te average rate of cange of total cost for (a) te first 100 units produced (from 0 to 100) and (b) te second 100 units produced.
610 Capter 9 Derivatives Solution (a) Te average rate of cange of total cost from 0 to 100 units is C(100) C(0) 100 0 (b) Te average rate of cange of total cost from 100 to 00 units is C(00) C(100) 00 100 (0.01(100) 5(100) 1500) (1500) 100 4100 1500 100 6900 4100 100 600 100 800 100 6 dollars per unit (0.01(00) 5(00) 1500) (4100) 100 8 dollars per unit EXMPLE Elderl in te Work Force Figure 9.18 sows te percents of elderl men and of elderl women in te work force in selected census ears from 1890 to 000. For te ears from 1950 to 000, find and interpret te average rate of cange of te percent of (a) elderl men in te work force and (b) elderl women in te work force. (c) Wat caused tese trends? Figure 9.18 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 0.0% 10.0% 0.0% 68.3 7.6 63.1 Elderl in te Labor Force, 1890-000 (labor force participation rate; figs. for 1910 not available) 8.3 55.6 54.0 7.3 7.3 41.8 6.1 41.4 1890 1900 190 1930 1940 1950 1960 1970 1980 1990 000 Source: Bureau of te Census, U.S. Department of Commerce 7.8 30.5 10.3 4.8 10.0 Men Women 19.3 17.6 18.6 8. 8.4 10.0 Solution (a) From 1950 to 000, te annual average rate of cange in te percent of elderl men in te work force is Cange in men s percent 18.6 41.4 Cange in ears 000 1950.8 0.456 percent per ear 50 Tis means tat from 1950 to 000, on average, te percent of elderl men in te work force dropped b 0.456% per ear. (b) Similarl, te average rate of cange for women is Cange in women s percent 10.0 7.8 Cange in ears 000 1950. 0.044 percent per ear 50 In like manner, tis means tat from 1950 to 000, on average, te percent of elderl women in te work force increased b 0.044% eac ear. (c) In general, from 1950 to 1990, people ave been retiring earlier, but te number of women in te work force as increased dramaticall.
9.3 verage and Instantaneous Rates of Cange: Te Derivative 611 Instantaneous Rates of Cange: Velocit noter common rate of cange is velocit. For instance, if we travel 00 miles in our car over a 4-our period, we know tat we averaged 50 mp. However, during tat trip tere ma ave been times wen we were traveling on an Interstate at faster tan 50 mp and times wen we were stopped at a traffic ligt. Tus, for te trip we ave not onl an average velocit but also instantaneous velocities (or instantaneous speeds as displaed on te speedometer). Let s see ow average velocit can lead us to instantaneous velocit. Suppose a ball is trown straigt upward at 64 feet per second from a spot 96 feet above ground level. Te equation tat describes te eigt of te ball after seconds is f() 96 64 16 Figure 9.19 sows te grap of tis function for 0 5. Te average velocit of te ball over a given time interval is te cange in te eigt divided b te lengt of time tat as passed. Table 9.4 sows some average velocities over time intervals beginning at 1. 160 = 96 + 64 16 18 96 64 3 Figure 9.19 1 3 4 5 6 TBLE 9.4 verage Velocities Time (seconds) Heigt (feet) verage Velocit (ft sec) Beginning Ending Cange ( ) Beginning Ending Cange ( ) ( ) 1 1 144 160 16 16 1 16 1 1.5 0.5 144 156 1 1 0.5 4 1 1.1 0.1 144 147.04 3.04 3.04 0.1 30.4 1 1.01 0.01 144 144.3184 0.3184 0.3184 0.01 31.84 In Table 9.4, te smaller te time interval, te more closel te average velocit approimates te instantaneous velocit at 1. Tus te instantaneous velocit at 1 is closer to 31.84 ft/s tan to 30.4 ft/s. If we represent te cange in time b, ten te average velocit from 1 to 1 approaces te instantaneous velocit at 1 as approaces 0. (Note tat can be positive or negative.) Tis is illustrated in te following eample. EXMPLE 3 Velocit Suppose a ball is trown straigt upward so tat its eigt f() (in feet) is given b te equation f() 96 64 16
61 Capter 9 Derivatives were is time (in seconds). (a) Find te average velocit from 1 to 1. (b) Find te instantaneous velocit at 1. Solution (a) Let represent te cange in (time) from 1 to 1. Ten te corresponding cange in f() (eigt) is f(1 ) f(1) 396 64(1 ) 16(1 ) 4 396 64 164 Te average velocit 96 64 64 16(1 ) 144 16 64 16 3 16 3 16 V av is te cange in eigt divided b te cange in time. f(1 ) f(1) V av 3 16 3 16 (b) Te instantaneous velocit V is te limit of te average velocit as approaces 0. V S0 V av S0 (3 16) 3 ft/s Note tat average velocit is found over a time interval. Instantaneous velocit is usuall called velocit, and it can be found at an time, as follows. Velocit Suppose tat an object moving in a straigt line as its position at time given b f(). Ten te velocit of te object at time is provided tat tis limit eists. V S0 f( ) f() Te instantaneous rate of cange of an function (commonl called rate of cange) can be found in te same wa we find velocit. Te function tat gives tis instantaneous rate of cange of a function f is called te derivative of f. Derivative If f is a function defined b f(), ten te derivative of f() at an value, denoted f (), is f () S0 f( ) f() if tis limit eists. If f (c) eists, we sa tat f is differentiable at c. Te following procedure illustrates ow to find te derivative of a function f() at an value.
9.3 verage and Instantaneous Rates of Cange: Te Derivative 613 Derivative Using te Definition Procedure To find te derivative of f() at an value : 1. Let represent te cange in from to.. Te corresponding cange in f() is f( ) f() f( ) f() 3. Form te difference quotient and simplif. f( ) f() 4. Find lim to determine f (), te S0 derivative of f(). Eample Find te derivative of 1. Te cange in from to is.. Te cange in f() is 3. 4. f( ) f() f() 4. f( ) f() 4( ) 4 f () S0 f( ) f() f () S0 (8 4) 8 4( ) 4 4 8 4 4 8 4 8 4 8 4 Note tat in te eample above, we could ave found te derivative of te function f() 4 at a particular value of, sa 3, b evaluating te derivative formula at tat value: In addition to f () 8 so f (3) 8(3) 4 f (), te derivative at an point ma be denoted b d d,, d d f(), D, or D f() We can, of course, use variables oter tan and to represent functions and teir derivatives. For eample, we can represent te derivative of te function defined b p q 1 b dp dq. Ceckpoint 1. Find te average rate of cange of f() 30 over [1, 4].. For te function f() 1, find f( ) f() (a) f( ) f() (b) f( ) f() (c) f () (d) f () S0 In Section 1.6, pplications of Functions in Business and Economics, we defined te marginal revenue for a product as te rate of cange of te total revenue function for te product. If te total revenue function for a product is not linear, we define te marginal revenue for te product as te instantaneous rate of cange, or te derivative, of te revenue function.
614 Capter 9 Derivatives Marginal Revenue Suppose tat te total revenue function for a product is given b R R(), were is te number of units sold. Ten te marginal revenue at units is provided tat te limit eists. MR R () S0 R( ) R() Note tat te marginal revenue (derivative of te revenue function) can be found b using te steps in te Procedure/Eample table on te preceding page. Tese steps can also be combined, as te are in Eample 4. ] EXMPLE 4 Revenue (pplication Preview) Suppose tat an oil compan s revenue (in tousands of dollars) is given b te equation were is te number of tousands of barrels of oil sold eac da. (a) Find te function tat gives te marginal revenue at an value of. (b) Find te marginal revenue wen 0,000 barrels are sold (tat is, at 0). Solution (a) Te marginal revenue function is te derivative of R(). R () S0 R( ) R() R R() 100, 0 S0 3100( ) ( ) 4 (100 ) S0 100 100 ( ) 100 100 (100 ) 100 S0 S0 Tus, te marginal revenue function is MR R () 100. (b) Te function found in (a) gives te marginal revenue at an value of. To find te marginal revenue wen 0 units are sold, we evaluate R (0). R (0) 100 (0) 60 Hence te marginal revenue at 0 is $60,000 per tousand barrels of oil. Because te marginal revenue is used to approimate te revenue from te sale of one additional unit, we interpret R (0) 60 to mean tat te epected revenue from te sale of te net tousand barrels (after 0,000) will be approimatel $60,000. [Note: Te actual revenue from tis sale is R(1) R(0) 1659 1600 59 (tousand dollars).] Tangent to a Curve s mentioned earlier, te rate of cange of revenue (te marginal revenue) for a linear revenue function is given b te slope of te line. In fact, te slope of te revenue curve gives us te marginal revenue even if te revenue function is not linear. We will sow tat te slope of te grap of a function at an point is te same as te derivative at tat point. In order to sow tis, we must define te slope of a curve at a point on te curve. We will define te slope of a curve at a point as te slope of te line tangent to te curve at te point.
9.3 verage and Instantaneous Rates of Cange: Te Derivative 615 In geometr, a tangent to a circle is defined as a line tat as one point in common wit te circle. (See Figure 9.0(a).) Tis definition does not appl to all curves, as Figure 9.0(b) sows. Man lines can be drawn troug te point tat touc te curve onl at. One of te lines, line l, looks like it is tangent to te curve. l Figure 9.0 (a) (b) We can use secant lines (lines tat intersect te curve at two points) to determine te tangent to a curve at a point. In Figure 9.1, we ave a set of secant lines s 1, s, s 3, and s 4 tat pass troug a point on te curve and points Q 1, Q, Q 3, and Q 4 on te curve near. (For points and secant lines to te left of point, tere would be a similar figure and discussion.) Te line l represents te tangent line to te curve at point. We can get a secant line as close as we wis to te tangent line l b coosing a second point Q sufficientl close to point. s we coose points on te curve closer and closer to (from bot sides of ), te limiting position of te secant lines tat pass troug is te tangent line to te curve at point, and te slopes of tose secant lines approac te slope of te tangent line at. Tus we can find te slope of te tangent line b finding te slope of a secant line and taking te limit of tis slope as te second point Q approaces. To find te slope of te tangent to te grap of f() at ( 1, f( 1 )), we first draw a secant line from point to a second point Q( 1, f( 1 )) on te curve (see Figure 9.). Q 4 Q 3 l s 4 s3 Q s ( 1, f( 1 )) Q( 1 +, f( 1 + )) f( 1 + ) f( 1 ) Q 1 s 1 = f() Figure 9.1 Figure 9. Te slope of tis secant line is m Q f( 1 ) f( 1 ) s Q approaces, we see tat te difference between te -coordinates of tese two points decreases, so approaces 0. Tus te slope of te tangent is given b te following.
616 Capter 9 Derivatives Slope of te Tangent Te slope of te tangent to te grap of f() at point ( 1, f( 1 )) is m S0 f( 1 ) f( 1 ) if tis limit eists. Tat is, m f ( 1 ), te derivative at 1. EXMPLE 5 Slope of te Tangent Find te slope of f() at te point (, 4). Solution Te formula for te slope of te tangent to Tus for f(), we ave f() at (, 4) is m f () S0 f( ) f() m f () S0 ( ) Taking te limit immediatel would result in bot te numerator and te denominator approacing 0. To avoid tis, we simplif te fraction before taking te limit. 4 4 m 4 4 (4 ) 4 S0 S0 S0 Tus te slope of te tangent to at (, 4) is 4 (see Figure 9.3). 10 = 8 6 4 (, 4) Tangent line m = 4 Figure 9.3 6 4 4 6 Te statement te slope of te tangent to te curve at (, 4) is 4 is frequentl simplified to te statement te slope of te curve at (, 4) is 4. Knowledge tat te slope is a positive number on an interval tells us tat te function is increasing on tat interval, wic means tat a point moving along te grap of te function rises as it moves to te rigt on tat interval. If te derivative (and tus te slope) is negative on an interval, te curve is decreasing on te interval; tat is, a point moving along te grap falls as it moves to te rigt on tat interval.
9.3 verage and Instantaneous Rates of Cange: Te Derivative 617 EXMPLE 6 Tangent Line Given f() 3 11, find (a) te derivative of f() at an point (, f()). (b) te slope of te curve at (1, 16). (c) te equation of te line tangent to 3 11 at (1, 16). Solution (a) Te derivative of f() at an value is denoted b f () S0 f( ) f() f () and is S0 33( ) ( ) 114 (3 11) S0 3( ) 11 3 11 S0 6 3 S0 (6 3 ) 6 (b) Te derivative is f () 6, so te slope of te tangent to te curve at (1, 16) is f (1) 6(1) 8. (c) Te equation of te tangent line uses te given point (1, 16) and te slope m 8. Using 1 m( 1 ) gives 16 8( 1), or 8 8. Te discussion in tis section indicates tat te derivative of a function as several interpretations. Interpretations of te Derivative For a given function, eac of te following means find te derivative. 1. Find te velocit of an object moving in a straigt line.. Find te instantaneous rate of cange of a function. 3. Find te marginal revenue for a given revenue function. 4. Find te slope of te tangent to te grap of a function. Tat is, all te terms printed in boldface are matematicall te same, and te answers to questions about an one of tem give information about te oters. For eample, if we know te slope of te tangent to te grap of a revenue function at a point, ten we know te marginal revenue at tat point. Calculator Note Note in Figure 9.3 tat near te point of tangenc at (, 4), te tangent line and te function look coincident. In fact, if we graped bot wit a graping calculator and repeatedl zoomed in near te point (, 4), te two graps would eventuall appear as one. Tr tis for ourself. Tus te derivative of f() at te point were a can be approimated b finding te slope between (a, f(a)) and a second point tat is nearb.
618 Capter 9 Derivatives In addition, we know tat te slope of te tangent to f() at a is defined b f (a) S0 f(a ) f(a) Hence we could also estimate f (a) tat is, te slope of te tangent at a b evaluating f(a ) f(a) wen 0 and 0 EXMPLE 7 pproimating te Slope of te Tangent Line f(a ) f(a) (a) Let f() 3 6 5. Use and two values of to make estimates of te slope of te tangent to f() at 3 on opposite sides of 3. (b) Use te following table of values of and g() to estimate g (3). 1 1.9.7.9.999 3 3.00 3.1 4 5 g() 1.6 4.3 11.4 10.8 10.513 10.5 10.474 10.18 6 5 Solution Te table feature of a graping utilit can facilitate te following calculations. (a) We can use 0.0001 and 0.0001 as follows: Wit 0.0001: Wit 0.0001: f(3 0.0001) f(3) f (3) 0.0001 f(3.0001) f(3) 0.001 0 0.0001 f(3 ( 0.0001)) f(3) f (3) 0.0001 f(.9999) f(3) 19.9988 0 0.0001 (b) We use te given table and measure te slope between (3, 10.5) and anoter point tat is nearb (te closer, te better). Using (.999, 10.513), we obtain g (3) 1 10.5 10.513 0.013 1 3.999 0.001 13 Calculator Note Most graping calculators ave a feature called te numerical derivative (usuall denoted b nder or nderiv) tat can approimate te derivative of a function at a point. On most calculators tis feature uses a calculation similar to our metod in part (a) of Eample 7 and produces te same estimate. Te numerical derivative of f() 3 6 5 wit respect to at 3 can be found as follows on man graping calculators: nderiv( 3 6 5,, 3) 0 Differentiabilit and Continuit So far we ave talked about ow te derivative is defined, wat it represents, and ow to find it. However, tere are functions for wic derivatives do not eist at ever value of. Figure 9.4 sows some common cases were f (c) does not eist but were f () eists for all oter values of. Tese cases occur were tere is a discontinuit, a corner, or a vertical tangent line.
9.3 verage and Instantaneous Rates of Cange: Te Derivative 619 c Discontinuit c Corner c Vertical tangent c Vertical tangent (a) Not differentiable at = c (b) Not differentiable at = c (c) Not differentiable at = c (d) Not differentiable at = c Figure 9.4 From Figure 9.4 we see tat a function ma be continuous at c even toug f (c) does not eist. Tus continuit does not impl differentiabilit at a point. However, differentiabilit does impl continuit. Differentiabilit Implies Continuit If a function f is differentiable at c, ten f is continuous at c. EXMPLE 8 Water Usage Costs Te montl carge for water in a small town is given b (a) Is tis function continuous at 0? (b) Is tis function differentiable at 0? Solution (a) We must ceck te tree properties for continuit. Tus f() is continuous at 0. (b) Because te function is defined differentl on eiter side of 0, we need to test to see weter f (0) eists b evaluating bot f(0 ) f(0) f(0 ) f(0) (i) lim and (ii) lim S0 S0 and determining weter te are equal. (i) lim S0 18 if 0 0 f() b 0.1 16 if 0 1. f() 18 for 0 so f(0) 18. lim f() S0 S0 18 18 lim f() r 1 lim f() 18 S0 S0 (0.1 16) 18 S0 3. f() f(0) f(0 ) f(0) 18 18 lim S0 S0 S0 0 0
60 Capter 9 Derivatives (ii) f(0 ) f(0) 30.1(0 ) 164 18 lim S0 S0 0.1 S0 0.1 0.1 S0 Because tese limits are not equal, te derivative f (0) does not eist. Ceckpoint 3. Wic of te following are given b f (c)? (a) Te slope of te tangent wen c (b) Te -coordinate of te point were c (c) Te instantaneous rate of cange of f() at c (d) Te marginal revenue at c, if f() is te revenue function 4. Must a grap tat as no discontinuit, corner, or cusp at c be differentiable at c? Calculator Note We can use a graping calculator to eplore te relationsip between secant lines and tangent lines. For eample, if te point (a, b) lies on te grap of, ten te equation of te secant line to from (1, 1) to (a, b) as te equation 1 b 1 b 1 ( 1), or ( 1) 1 a 1 a 1 Figure 9.5 illustrates te secant lines for tree different coices for te point (a, b). 5 5 5-6 6-5 (a) Figure 9.5-6 6-5 (b) -6 6-5 (c) We see tat as te point (a, b) moves closer to (1, 1), te secant line looks more like te tangent line to at (1, 1). Furtermore, (a, b) approaces (1, 1) as a S 1, and te slope of te secant approaces te following limit. b 1 lim as1 a 1 a 1 as1 a 1 (a 1) as1 Tis limit,, is te slope of te tangent line at (1, 1). Tat is, te derivative of at (1, 1) is. [Note tat a graping utilit s calculation of te numerical derivative of f() wit respect to at 1 gives f (1). ]
9.3 verage and Instantaneous Rates of Cange: Te Derivative 61 Ceckpoint Solutions 1. f(4) f(1) 4 1. (a) (b) (c) 10 8 3 18 3 f( ) f() 3( ) ( ) 14 ( 1) 1 1 f( ) f() 1 6 f( ) f() f () ( 1) S0 S0 1 (d) f () 1, so f () 3. 3. Parts (a), (c), and (d) are given b f (c). Te -coordinate were c is given b f(c). 4. No. Figure 9.4(c) sows suc an eample. 9.3 Eercises In Problems 1 4, for eac given function find te average rate of cange over eac specified interval. 1. f() 1 over (a) 30, 54 and (b) 3 3, 104. f() 6 over (a) 3 1, 4 and (b) 31, 104 3. For f() given b te table, over (a) 3, 54 and (b) 33.8, 44 0.5 3 3.8 4 5 f() 14 0 19 17 16 30 4. For f() given in te table, over (a) 33, 3.54 and (b) 3, 64 1 3 3.5 3.7 6 f() 40 5 18 15 18 38 5. Given f(), find te average rate of cange of f() over eac of te following pairs of intervals. (a) 3.9, 34 and 3.99, 34 (b) 33, 3.14 and 33, 3.014 (c) Wat do te calculations in parts (a) and (b) suggest te instantaneous rate of cange of f() at 3 migt be? 6. Given f() 3 7, find te average rate of cange of f() over eac of te following pairs of intervals. (a) 31.9, 4 and 31.99, 4 (b) 3,.14 and 3,.014 (c) Wat do te calculations in parts (a) and (b) suggest te instantaneous rate of cange of f() at migt be? 7. In te Procedure/Eample table in tis section we were given f() 4 and found f () 8. Find (a) te instantaneous rate of cange of f() at 4. (b) te slope of te tangent to te grap of f() at 4. (c) te point on te grap of f() at 4. 8. In Eample 6 in tis section we were given f() 3 11 and found f () 6. Find (a) te instantaneous rate of cange of f() at 6. (b) te slope of te tangent to te grap of f() at 6. (c) te point on te grap of f() at 6. 9. Let f(). (a) Use te definition of derivative and te Procedure/ Eample table in tis section to verif tat f () 4 1. (b) Find te instantaneous rate of cange of f() at 1. (c) Find te slope of te tangent to te grap of f() at 1. (d) Find te point on te grap of f() at 1. 10. Let f() 9 1. (a) Use te definition of derivative and te Procedure/ Eample table in tis section to verif tat f (). (b) Find te instantaneous rate of cange of f () at.
6 Capter 9 Derivatives (c) Find te slope of te tangent to te grap of f() at. (d) Find te point on te grap of f() at. In Problems 11 14, te tangent line to te grap of f() at 1 is sown. On te tangent line, P is te point of tangenc and is anoter point on te line. (a) Find te coordinates of te points P and. (b) Use te coordinates of P and to find te slope of te tangent line. (c) Find f (1). (d) Find te instantaneous rate of cange of f() at P. 11. 1. In Problems 3 and 4, use te given tables to approimate f (a) as accuratel as ou can. 3. 1.0 1.99 13 13.1 a 13 f() 1.41 17.4 17.11.84 4. 7.4 7.50 7.51 7 f().1.351.38 4.1 a 7.5 In te figures given in Problems 5 and 6, at eac point and B draw an approimate tangent line and ten use it to complete parts (a) and (b). (a) Is f () greater at point or at point B? Eplain. (b) Estimate f () at point B. 5. P = f() P = f() 10 8 6 4 B = f () 13. 14. P = f() P = f() 6. 5 4 1 3 4 5 6 7 For eac function in Problems 15 18, find (a) te derivative, b using te definition. (b) te instantaneous rate of cange of te function at an value and at te given value. (c) te slope of te tangent at te given value. 15. 16. 17. 18. For eac function in Problems 19, approimate f (a) in te following was. (a) Use te numerical derivative feature of a graping utilit. f(a ) f(a) (b) Use wit 0.0001. (c) Grap te function on a graping utilit. Ten zoom in near te point until te grap appears straigt, pick two points, and find te slope of te line ou see. 19. 0. 1. f() 4 1; 3 f() 16 4 ; 1 p(q) q 4q 1; q 5 p(q) q 4q 5; q f () for f() 3 4 7 5 f ( 1) for f() 3 11 9 f (4) for f() ( 1) 3. f (3) for () 3 1 5 3 1 B = f () 10 0 30 40 50 60 70 In Problems 7 and 8, a point (a, b) on te grap of f() is given, and te equation of te line tangent to te grap of f () at (a, b) is given. In eac case, find f (a) and f(a). 7. ( 3, 9); 5 3 8. ( 1, 6); 10 59 9. If te instantaneous rate of cange of f() at (1, 1) is 3, write te equation of te line tangent to te grap of f() at 1. 30. If te instantaneous rate of cange of g() at ( 1, ) is 1, write te equation of te line tangent to te grap of g() at 1. Because te derivative of a function represents bot te slope of te tangent to te curve and te instantaneous rate of cange of te function, it is possible to use information about one to gain information about te oter. In Problems 31 and 3, use te grap of te function f() given in Figure 9.6.
9.3 verage and Instantaneous Rates of Cange: Te Derivative 63 a b Figure 9.6 c B C 31. (a) Over wat interval(s) (a) troug (d) is te rate of cange of f() positive? (b) Over wat interval(s) (a) troug (d) is te rate of cange of f() negative? (c) t wat point(s) troug E is te rate of cange of f() equal to zero? 3. (a) t wat point(s) troug E does te rate of cange of f() cange from positive to negative? (b) t wat point(s) troug E does te rate of cange of f() cange from negative to positive? 33. Given te grap of f() in Figure 9.7, determine for wic -values, B, C, D, or E te function is (a) continuous. (b) differentiable. 34. Given te grap of f() in Figure 9.7, determine for wic -values F, G, H, I, or J te function is (a) continuous. (b) differentiable. B C D E F G H I J Figure 9.7 = f() In Problems 35 38, (a) find te slope of te tangent to te grap of f() at an point, (b) find te slope of te tangent at te given -value, (c) write te equation of te line tangent to te grap of f() at te given point, and (d) grap bot f() and its tangent line (use a graping utilit if one is available). 35. (a) f() 36. (a) f() 3 (b) (b) 1 (c) (, 6) (c) ( 1, ) 37. (a) f() 3 3 38. (a) f() 5 3 (b) 1 (b) 1 (c) (1, 4) (c) ( 1, 3) D d E PPLICTIONS 39. Total cost Suppose total cost in dollars from te production of printers is given b C() 0.0001 3 0.005 8 3000 Find te average rate of cange of total cost wen production canges (a) from 100 to 300 printers. (b) from 300 to 600 printers. (c) Interpret te results from parts (a) and (b). 40. verage velocit If an object is trown upward at 64 ft/s from a eigt of 0 feet, its eigt S after seconds is given b S() 0 64 16 Wat is te average velocit in te (a) first seconds after it is trown? (b) net seconds? 41. Demand If te demand for a product is given b D(p) 1000 1p 1 wat is te average rate of cange of demand wen p increases from (a) 1 to 5? (b) 5 to 100? 4. Revenue If te total revenue function for a blender is R() 36 0.01 were is te number of units sold, wat is te average rate of cange in revenue R() as increases from 10 to 0 units? 43. Total cost Suppose te figure sows te total cost grap for a compan. rrange te average rates of cange of total cost from to B, B to C, and to C from smallest to greatest, and eplain our coice. Tousands of dollars 50 40 30 0 10 C() B C 0 40 60 80 100 Tousands of Units
64 Capter 9 Derivatives 44. Students per computer Te following figure sows te number of students per computer in U.S. public scools for te scool ears tat ended in 1984 troug 00. (a) Use te figure to find te average rate of cange in te number of students per computer from 1990 to 000. Interpret our result. (b) From te figure, determine for wat two consecutive scool ears te average rate of cange of te number of students per computer is closest to zero. 150 100 50 0 15.0 83 84 75.0 Students Per Computer in U.S. Public Scools 50.0 37.0 3.0 5.0.0 0.0 18.0 16.0 14.0 10.5 10.0 7.8 6.1 5.7 5.4 5.0 4.9 84 85 85 86 86 87 87 88 88 89 89 90 90 91 45. Marginal revenue Sa te revenue function for a stereo sstem is R() 300 dollars were denotes te number of units sold. (a) Wat is te function tat gives marginal revenue? (b) Wat is te marginal revenue if 50 units are sold, and wat does it mean? (c) Wat is te marginal revenue if 00 units are sold, and wat does it mean? (d) Wat is te marginal revenue if 150 units are sold, and wat does it mean? (e) s te number of units sold passes troug 150, wat appens to revenue? 46. Marginal revenue Suppose te total revenue function for a blender is R() 36 0.01 dollars were is te number of units sold. (a) Wat function gives te marginal revenue? (b) Wat is te marginal revenue wen 600 units are sold, and wat does it mean? (c) Wat is te marginal revenue wen 000 units are sold, and wat does it mean? (d) Wat is te marginal revenue wen 1800 units are sold, and wat does it mean? 47. Labor force and output Te montl output at te Olek Carpet Mill is 91 9 Source: Qualit Education Data, Inc., Denver, Co. Reprinted b permission. Q() 15,000 units, (40 60) 9 93 93 94 94 95 95 96 96 97 97 98 98 99 99 00 000 01 001 0 were is te number of workers emploed at te mill. If tere are currentl 50 workers, find te instantaneous rate of cange of montl output wit respect to te number of workers. Tat is, find Q (50). 48. Consumer ependiture Suppose tat te demand for units of a product is 10,000 100p were p dollars is te price per unit. Ten te consumer ependiture for te product is E(p) p p(10,000 100p) 10,000p 100p Wat is te instantaneous rate of cange of consumer ependiture wit respect to price at (a) an price p? (b) p 5? (c) p 0? In Problems 49 5, find derivatives wit te numerical derivative feature of a graping utilit. 49. Profit Suppose tat te profit function for te montl sales of a car b a dealersip is P() 500 100 were is te number of cars sold. Wat is te instantaneous rate of cange of profit wen (a) 00 cars are sold? Eplain its meaning. (b) 300 cars are sold? Eplain its meaning. 50. Profit If te total revenue function for a to is R() and te total cost function is C() 100 0. wat is te instantaneous rate of cange of profit if 10 units are produced and sold? Eplain its meaning. 51. Heat inde Te igest recorded temperature in te state of laska was 100 F and occurred on June 7, 1915, at Fort Yukon. Te eat inde is te apparent temperature of te air at a given temperature and umidit level. If denotes te relative umidit (in percent), ten te eat inde (in degrees Fareneit) for an air temperature of 100 F can be approimated b te function f() 0.009 0.139 91.875 (a) t wat rate is te eat inde canging wen te umidit is 50%? (b) Write a sentence tat eplains te meaning of our answer in part (a). 5. Receptivit In learning teor, receptivit is defined as te abilit of students to understand a comple concept. Receptivit is igest wen te topic is introduced and tends to decrease as time passes in a lecture. Suppose
9.4 Derivative Formulas 65 tat te receptivit of a group of students in a matematics class is given b g(t) 0.t 3.1t 3 were t is minutes after te lecture begins. (a) t wat rate is receptivit canging 10 minutes after te lecture begins? (b) Write a sentence tat eplains te meaning of our answer in part (a). 53. Marginal revenue Suppose te grap sows a manufacturer s total revenue, in tousands of dollars, from te sale of cellular telepones to dealers. (a) Is te marginal revenue greater at 300 cell pones or at 700? Eplain. (b) Use part (a) to decide weter te sale of te 301st cell pone or te 701st brings in more revenue. Eplain. 80 60 R() 54. Social Securit beneficiaries Te grap sows a model for te number of millions of Social Securit beneficiaries (actual to 000 and projected beond 000). Te model was developed wit data from te 000 Social Securit Trustees Report. (a) Was te instantaneous rate of cange of te number of beneficiaries wit respect to te ear greater in 1960 or in 1980? Justif our answer. (b) Is te instantaneous rate of cange of te number of beneficiaries projected to be greater in 000 or in 030? Justif our answer. Millions of Beneficiaries 80 40 1970 1990 010 030 Year 40 0 00 400 600 800 1000 OBJECTIVES 9.4 To find derivatives of powers of To find derivatives of constant functions To find derivatives of functions involving constant coefficients To find derivatives of sums and differences of functions Derivative of f() n Derivative Formulas ] pplication Preview For more tan 30 ears, U.S. total personal income as eperienced stead growt. Wit Bureau of Economic nalsis, U.S. Department of Commerce data for selected ears from 1975 to 00, U.S. total personal income I, in billions of current dollars, can be modeled b I I(t) 5.33t 81.5t 8 were t is te number of ears past 1970. We can find te rate of growt of total U.S. personal income in 007 b using te derivative I (t) of te total personal income function. (See Eample 8.) s we discussed in te previous section, te derivative of a function can be used to find te rate of cange of te function. In tis section we will develop formulas tat will make it easier to find certain derivatives. We can use te definition of derivative to sow te following: If f(), ten f (). If f() 3, ten f () 3. If f() 4, ten f () 4 3. If f() 5, ten f () 5 4.