y = f(x) This means that there must be a point, c, where the Figure 1
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1 Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile over the intervl (, ), then there is t lest one numer, c, in the intervl (, ), such tht f ( ) f '(. This mens tht there must e point, c, where the c grdient of the tngent to f is equl to the verge grdient. 1. Point to point speed cmers tht operte on the Hume Freewy re networked nd synchronised to mesure the verge speed of vehicle over long distnce. A truck is detected pssing cmer t Seymour t 8:5 m. A cmer t Bddginnie gin detects it t 9:16 m, 85 km from Seymour. The speed of the truck pst ech cmers ws 110 km/h. Figure 1 y y f() () Clculte the verge speed of the truck, in km/h, to the nerest integer. () t 41 min 41/60 h. d 85 v t 41/ 60 The verge speed ws 124 km/h, correct to the nerest integer. () Assuming tht the truck did not stop etween Seymour nd Bddginnie nd tht the cmers re ccurte, eplin, in terms of the Men Vlue Theorem, whether the truck driver cn e legitimtely fined for eceeding the 110 km/h speed limit. () By the MVT, there is t lest one point when the instntneous speed is equl to the verge speed. Therefore, t lest once, the truck s speed must hve een 124 km/h. Therefore the police cn llege tht the truck trvelled t 124 km/h. Hence the driver cn e fined for eceeding 110 km/h. ( The driver s lwyer chllenged the fine on the grounds tht the Men Vlue Theorem doesn t pply, ecuse the truck stopped for 5 minutes etween Seymour nd Bddginnie. Is this vlid defence? Eplin your nswer. ( This is not vlid defence, s there ws no discontinuity in time. Furthermore, if stopped for 5 minutes, the trip ws completed in only 6 minutes of trvel time. Hence the verge speed, while the truck ws in motion, ws even higher thn 124 km/h. 2. Consider the function f, shown in Figure 1, over the intervl. () Write the coordintes of the endpoints, in terms of nd. () Write n epression for the grdient of the line segment joining the endpoints, in terms of nd. ( Write n epression for the grdient of the tngent to f t the point (c, f(). A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 1
2 () Endpoints re (, f()) nd (, f()). f ( ) () m ( The grdient of the tngent t c is f (. If the grdient of the tngent t c is prllel to the line f ( ) segment pssing through the endpoints, then f (.. Consider the function g: [0, 4] R, where g() () Clculte the verge grdient of the function on the intervl [0, 4]. g(4) g(0) m m 1 4 TI-8 TI-89/ 92/ Voyge 200 () Find the coordintes of ll points in g where the grdient of the tngent is equl to the verge grdient. Approimte the vlues of the coordintes to two deciml plces. The grdient of the tngent equls 1 TI-8 TI-89/ 92/ Voyge g ( c 12c ± 2 c Coordintes re: 6 ± 2 6 ± 2, g 6 ± 2 6 ± 10, 9 Correct to two deciml plces: g(0.845) g(.1547) (0.85, 5.08) nd (.15, 2.08) solve in ctlog menu solve(function, vrile, guess) or use solver in the MATH menu. Keystrokes: yê(ctlog). Select solve :nderiv( ~ :Y-Vrs À ¹À ÀËÁ Í Ect coordintes ( Sketch nd lel grph of g, showing the line segment of verge grdient nd the tngents. Lel the coordinte(s) of the turning point(s) nd the coordinte(s) of the point(s) of intersections of g nd the tngents, to two deciml plces TI-8 TI-89/ 92/ Voyge 200 A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 2
3 Prt 2: Applying the Men Vlue Theorem for Differentition 4. An musement prk ride hs pltform tht moves up nd down, on tower, with incresing mplitude. For prt of the ride, the height of the pltform, h metres ove the ground, t time t seconds, is modelled y the function h: [0, 11] R, h ( t) pt cos(5 ( t + 1) ) + q, where the prmeters p nd q hve positive rel vlues. () The initil height of the pltform is 10 m, nd t t 11 seconds the height is m. Find the vlue of p nd q, correct to the nerest integer. Hence write the rule for h(t). h( t) pt cos(5 h(0) 10 p 0 cos(5 q 10 h(11) pcos(5 ( t + 1) + q (0 + 1) + q 10 (11+ 1) p cos(5 12) Correct to the nerest integer, h ( t) t cos(5 ( t + 1)) + 10 TI-8 solve from ctlog menu Synt is: solve(function, vrile, guess) Alterntively, use solver from mth menu. TI-89/ 92/ Voyge 200 () Sketch the grph of h, over the specified domin, lelling the coordintes of the endpoints, correct to two deciml plces. (There is no need to work out the coordintes of the turning points). TI-8 TI-89/ 92/ Voyge 200 ( Comment on the key fetures of the grph. The grph is continuous nd smooth on [0,11], nd therefore differentile on (0,11). It hs two locl mim nd two locl minim. (d) At wht times, in seconds, correct to two deciml plce, is the velocity of the pltform equl to zero? A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori
4 Velocity is zero h (t) 0 Grphiclly, find coordintes of the sttionry points. Numericlly, solve h (t) 0, using pproprite ounds. Correct to 2 deciml plces, h (t) 0 for t 0.87s, t 2.76 s, t 5.50 s, t 9.05 s Grphicl solution (All TIs) TI-89/ 92/ Voyge 200 Only one solution found lgericlly To find the other solutions, use numericl solve, shifting the ounds ech time. The grphicl method is identicl to tht of the TI-8. (e) Wht is the height of the pltform, in metres, correct to two deciml plce, t the times when the velocity of the pltform is zero? The heights cn e found grphiclly, or numericlly, y sustituting the vlues of t into h(t). TI-8 Numericl solution TI-89/ 92/ Voyge 200 h(0.87) m h(2.76) 7.4 m h(5.50) 15.41m h(9.05) 1.04 m (f) Wht re the mimum nd minimum heights reched in the intervl 0 t 11? The mimum height reched is m. The minimum height reched is 1.04 m. (g) Clculte the verge velocity of the pltform, correct to two deciml plces, on t [0.87, 9.05]. h(9.05) h(0.87) v v The verge velocity is 1.19 m/s. TI-8 TI-89/ 92Plus /Voyge 200 A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 4
5 (h) Find the eqution of the line joining the endpoints of the intervl [0.87, 9.05]. Give your nswers correct to two deciml plces. The eqution of the line segment: m v v 1.19 nd psses through (0.87, 10.74) TI-8 TI-89/ 92/ Voyge 200 y y m( 1) 1 y ( 0.87) y Eqution of the line is h(t) 1.19 t (i) Find the times, correct to two deciml plces, t which the instntneous velocity of the pltform is equl to the verge velocity over the intervl [0.87, 9.05]. The instntneous velocity of the pltform is equl to the verge velocity when h (t) 1.19 Grphiclly, or solving numericlly, t 1.15 s, 2.51 s, 5.72 s, 8.84 s. TI-8 Numericl Solution Grphicl solution (ll TIs) TI-89/ 92/ Voyge 200 (j) Sketch nd lel grph of h, showing the line segment joining the endpoints of the intervl [0.87, 9.05] nd one of the tngents to h tht represent the instntneous velocity of the pltform eing equl to the verge velocity over the intervl [0.87, 9.05]. TI-8 TI-89/ 92/ Voyge 200 Prt : Men Vlue Theorem s generlistion of Rolle s Theorem The Men Vlue Theorem is generlistion of nother theorem, clled Rolle s Theorem Rolle s Theorem: let f e defined nd continuous over the intervl [, ], nd differentile over the intervl (, ). If f() f(), then there is t lest one point, c, in the intervl (, ) for which f '( 0. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 5
6 5. Eplin the geometric menings of f() f() nd f '( 0, where c (, ). Hence illustrte Rolle s Theorem geometriclly. y f ( 0 f() f() mens tht, t nd t, f hs the sme y-vlue. f '( 0 represents sttionry point t c. (c, f() Hence Rolle s theorem sys tht if f is smooth function nd the y-vlues t nd t re equl, then there must e sttionry point t c, etween nd. (, f()) (, f()) c 6. To prove the men vlue theorem, consider the function f ( ) g( ) ( ) + f ( ) f ( ), [, ] () (i) If f() 2, find the eqution of g() for 1 nd. (ii) On the sme set of es, drw the grphs of f(), g() nd the line segment pssing through the points (, f()) nd (, f()). TI-8 TI-89/ 92/ Voyge 200 f ( ) g( ) ( ) + f ( ) f ( ) 2 1 g( ) ( 1) 1 g( ) (4( 1) + 1) g( ) + 4 The line pssing through (1, 1) nd (, 9) 9 1 m 4 1 y 1 4( 1) y 4 () From the sitution illustrted in prt () ove, give geometric interprettion to the function g, defined on [, ]. f ( ) Note tht y ( ) + f ( ) is just the eqution of the stright line pssing through (, f()) nd (, f()). Tht is, ( ) f ( ) y y1 m 1, where 1, y 1 f() nd m. Hence, for ny vlues nd, let h() e the eqution of the line pssing through (, f()) nd (, f()). In ll cses, g() h() f(). Hence, geometriclly, g() cn e otined y ddition of ordintes: h() + ( f()). g() is trnsformtion of f() such tht f() is reflected in the -is, nd trnslted such tht the -intercepts re (, 0) nd (, 0). ( If f() is continuous on [, ] nd differentile on (, ), eplin why there must e point c in (, ) such tht g ( 0. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 6
7 The -intercepts of g() re lwys (, 0) nd (, 0). Since f() is smooth function, y Rolle s Theorem, there must e vlue c on the intervl (, ), such tht g ( 0. f ( ) (d) If g( ) ( ) + f ( ) f ( ), [, ], find g (). Hence find g (. TI-8 TI-89/ 92/ Voyge 200 f ( ) g( ) ( ) + f ( ) f ( ) f ( ) g'( ) + 0 f '( ) g'( f ( ) f '( (e) Given tht g ( 0, rerrnge the epression in prt (d) ove, to mke f ( the suject of the eqution. g ( 0, from Rolle s Theorem f ( ) 0 f '( f ( ) f '( (f) Eplin the significnce of the result otined in prt (e) ove. The result is n epression of the Men Vlue Theorem (MVT). Thus, MVT cn e proved from Rolle s Theorem. Rolle s theorem is specil cse of the MVT, where the verge grdient of the function, on [, ], is zero. Prt 4: Men Vlue Theorem for Integrtion nd Averge Vlue of function y Let y f() e function which is continuous on the closed intervl [, ]. Then there eists t lest one vlue, c, in the f() intervl [, ] such tht f ( ) d f ( ( ) eqution 1 f( gives the verge vlue of f from to. Rerrnging eqution 1: 1 f ( ( f ( )) d eqution 2 f( f() c Figure Consider the reltionship etween Figure 2 nd eqution 2. () Eplin the geometric mening of the following, with reference to Figure 2: (i) ( f ( )) d (ii) 1 () Hence eplin why f ( )) d ( gives the verge vlue of f on the intervl [, ]. y f() A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 7
8 () (i) ( f ( )) d gives the re ounded y the curve, the -is, nd. Tht is, the re of the shded region Figure 2. (ii) ( ) is the width of the shded region in Figure 2. () The re of the shded region is given y the width multiplied y the verge vlue (verge height ). 1 Therefore, the re divided y the width, f ( )) d (, gives the verge vlue of g on [, ]. (This is nlogous to rerrnging the formul for re of trpezium to find the verge length of the prllel sides). 8. At certin ltitude, the length of time, hours, from sunrise to sunset, t dys fter Spring 2π equino, is modelled y the function ( t) 4sin t + 12, 0 t () Wht re the mimum nd minimum hours of dylight, ccording to this model? Mimum hours Minimum hours TI-8 TI-89/ 92/ Voyge 200 () Use the verge vlue of function formul to clculte the verge numer of dylight hours per dy during the dys following the Spring Equino (i.e. over 0 t ). Give the ect vlue, in hours, nd n pproimtion to two deciml plces. The integrl cn e evluted y hnd or with the use of technology. TI-8 TI-89/ 92/ Voyge πt 4sin dt πt cos t 56 π π 4(π + 2) π The verge length of dylight is hours (2 deciml plces). The shded res ove nd elow the men vlue re equl. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 8
9 9. Consider the function g() , 0. () Find g(, the verge vlue of g on [0, ] () Find ll vlues of c, correct to 2 deciml plces. ( On grph of g, for [0, ], shde the regions representing g ( )) d. On the sme es, drw the grph of y g(. Lel the coordintes of the endpoints nd the coordintes of ll vlues of c. ( 0 () The integrl cn e evluted y hnd or with the use of technology ( ( )) ( 8 + 6) f d 0 d g( g( 5 9 g( The verge vlue is or () To find ll vlues of c, solve g( ) 1.8. Using technology, or 2.62 The vlues of c re 1.07 or 2.62 TI-8 TI-89/ 92/ Voyge 200 A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 9
10 Clculus Investigtion: Student Booklet A Men Slope NAME: Prt 1: Understnding the Men Vlue Theorem for differentition The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile over the intervl (, ), then there is t lest one numer, c, in the intervl (, ), such tht f ( ) f '(. This mens tht there must e point, c, where the grdient of the tngent to f is equl to the verge grdient. y y f() c Figure 1 1. Point to point speed cmers tht operte on the Hume Freewy re networked nd synchronised to mesure the verge speed of vehicle over long distnce. A truck is detected pssing cmer t Seymour t 8:5 m. A cmer t Bddginnie gin detects it t 9:16 m, 85 km from Seymour. The speed of the truck pst ech cmers ws 110 km/h. () Clculte the verge speed of the truck, in km/h, to the nerest integer. () Assuming tht the truck did not stop etween Seymour nd Bddginnie, nd tht the cmers re ccurte, eplin, in terms of the Men Vlue Theorem, whether the truck driver cn e legitimtely fined for eceeding the 110 km/h speed limit. ( The driver s lwyer chllenged the fine on the grounds tht the Men Vlue Theorem doesn t pply, ecuse the truck stopped for 5 minutes etween Seymour nd Bddginnie, creting discontinuity. Is this vlid defence? Eplin your nswer. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 10
11 2. Consider the function f, shown in Figure 1, over the intervl. () Write the coordintes of the endpoints, in terms of nd. () Write n epression for the grdient of the line segment joining the endpoints, in terms of nd. ( Write n epression for the grdient of the tngent to f t the point (c, f(), in terms of nd.. Consider the function g: [0, 4] R, where g() () Clculte the verge grdient of the function on the intervl [0, 4]. () Find the coordintes of ll points in g where the grdient of the tngent is equl to the verge grdient. Approimte vlues of the coordintes to two deciml plces. ( Sketch nd lel grph of g, showing the line segment of verge grdient nd the tngents. Lel the coordinte(s) of the turning point(s) nd the coordinte(s) of the point(s) of intersections of g nd the tngents, to two deciml plces. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 11
12 g() Prt 2: Applying the Men Vlue Theorem for Differentition 4. An musement prk ride hs pltform tht moves up nd down, on tower, with incresing mplitude. For prt of the ride, the height of the pltform, h metres ove the ground, t time t seconds, is modelled y the function h: [0, 11] R, h ( t) pt cos(5 ( t + 1) ) + q, where the prmeters p nd q hve positive rel vlues. () The initil height of the pltform is 10 m, nd t t 11 seconds the height is m. Use these dt to find the vlue of p nd q, correct to the nerest integer. Hence write the rule for h(t). () Sketch the grph of h, over the specified domin, lelling the coordintes of the endpoints, correct to two deciml plces. (There is no need to work out the coordintes of the turning points). h t ( Comment on the key fetures of the grph. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori
13 (d) At wht times, in seconds, correct to two deciml plce, is the velocity of the pltform equl to zero? (e) Wht is the height of the pltform, in metres, correct to two deciml plce, t the times when the velocity of the pltform is zero? (f) Wht re the mimum nd minimum heights reched in the intervl 0 t 11? (g) Clculte the verge velocity of the pltform, correct to two deciml plces, on t [0.87, 9.05]. (h) Find the eqution of the line joining the endpoints of the intervl [0.87, 9.05]. Give your nswers correct to two deciml plces. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 1
14 (i) Find the times, correct to two deciml plces, t which the instntneous velocity of the pltform is equl to the verge velocity over the intervl [0.87, 9.05]. (j) Sketch nd lel grph of h, showing the line segment joining the endpoints of the intervl [0.87, 9.05] nd the tngents to h tht represent the instntneous velocity of the pltform is equl to the verge velocity over the intervl [0.87, 9.05]. Include the coordintes of the points where the touch the curve. h t Prt : Men Vlue Theorem s generlistion of Rolle s Theorem The Men Vlue Theorem is generlistion of nother theorem, clled Rolle s Theorem Rolle s Theorem: let f e defined nd continuous over the intervl [, ], nd differentile over the intervl (, ). If f() f(), then there is t lest one point, c, in the intervl (, ) for which f '( Eplin the geometric menings of f() f() nd f '( 0, where c (, ). Hence illustrte Rolle s Theorem geometriclly. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 14
15 6. To prove the men vlue theorem, consider the function f ( ) g( ) ( ) + f ( ) f ( ), [, ] () (i) If f() 2, find the eqution of g() for 1 nd. (ii) On the sme set of es, drw the grphs of f(), g() nd the line segment pssing through the points (, f()) nd (, f()). y () From the sitution illustrted in prt () ove, give geometric interprettion to the function g, defined on [, ]. ( If f() is continuous on [, ] nd differentile on (, ), eplin why there must e point c in (, ) such tht g ( 0. A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 15
16 f ( ) (d) If g( ) ( ) + f ( ) f ( ), [, ], find g (), find g (. (e) Given tht g ( 0, rerrnge the epression in prt (d) ove, to mke f ( the suject of the eqution. (f) Eplin the significnce of the result otined in prt (e) ove. Prt 4: Men Vlue Theorem for Integrtion nd Averge Vlue of function Let y f() e function which is continuous on the closed intervl [, ]. Then there eists t lest one vlue, c, in the intervl [, ] such tht f ( ) d f ( ( ) eqution 1 f( gives the verge vlue of f from to. Rerrnging eqution 1: f( f() 1 f ( ( f ( )) d eqution 2 7. Consider the reltionship etween Figure 2 nd eqution 2. c Figure 2. () Eplin the geometric mening of the following, with reference to Figure 2: (i) ( f ( )) d (ii) f() y y f() A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 16
17 1 () Hence eplin why f ( )) d ( gives the verge vlue of f on the intervl [, ]. 8. At certin ltitude, the length of time, hours, from sunrise to sunset, t dys fter Spring 2π equino, is modelled y the function ( t) 4sin t + 12, 0 t ( Wht re the mimum nd minimum hours of dylight, ccording to this model? (d) Use the verge vlue of function formul to clculte the verge numer of dylight hours per dy during the dys following the Spring Equino (i.e. over 0 t ). Give the ect vlue, in hours, nd n pproimtion to two deciml plces. 9. Consider the function g() , 0. () Find g(, the verge vlue of g on [0, ] A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 17
18 () Find ll vlues of c, correct to 2 deciml plces. ( On grph of g(), for [0, ], shde the regions representing g ( )) d. On the sme es, drw the grph of y g(. Lel the coordintes of the endpoints nd the coordintes of ll vlues of c. g() ( 0 A Men Slope Techer Solutions: pges 1 9 Student ooklet pges: Frnk Moy 2004, PD scholrship winner 2004 Frnkston High School, Victori 18
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